The metric space of limit laws for $q$-hook formulas

In earlier work, Billey--Konvalinka--Swanson studied the asymptotic distribution of the coefficients of Stanley's $q$-hook length formula, or equivalently the major index on standard tableaux of straight shape and certain skew shapes. We extend those investigations to Stanley's $q$-hook-content formula related to semistandard tableaux and $q$-hook length formulas of Bj\"orner--Wachs related to linear extensions of labeled forests. We show that, while their coefficients are ``generically'' asymptotically normal, there are uncountably many non-normal limit laws. More precisely, we introduce and completely describe the compact closure of the metric space of distributions of these statistics in several regimes. The additional limit distributions involve generalized uniform sum distributions which are topologically parameterized by certain decreasing sequence spaces with bounded $2$-norm. The closure of these distributions in the L\'evy metric gives rise to the space of DUSTPAN distributions. As an application, we completely classify the limiting distributions of the size statistic on plane partitions fitting in a box.

The famed Frame-Robinson-Thrall hook length formula is a rational product formula for counting the number of standard Young tableaux of a given partition shape λ [FRT54], denoted SYT(λ). Stanley's q-analogue of the hook length formula [Sta99,Cor. 7.21.5] is a remarkably simple generalization for the polynomial generating function of the major index statistic on SYT(λ). His q-hook length formula replaces each integer n with the corresponding q-integer [n] q := 1 + q + · · · + q n−1 , times an overall shift of q r(λ) where r(λ) := i≥1 (i − 1)λ i : (1) Consequently, (1) encodes probabilistic information concerning the distribution of the major index statistic when sampling from SYT(λ) uniformly at random. [BKS20] considered the distribution of maj on SYT(λ). Given a sequence of partitions, [BKS20] completely determined when the corresponding sequence of standardized random variables converges in distribution. Equivalently, this determines the asymptotic distribution of the coefficients of Stanley's q-hook length formula. For these random variables, countably many continuous limit laws are possible: one gets the normal distribution "generically" and, in certain degenerate regimes, the Irwin-Hall distributions. A key technical tool in [BKS20] is an exact formula for the cumulants of the underlying random variables, which follows easily from work of Chen-Wang-Wang [CWW08] and Hwang-Zacharovas [HZ15] together with Stanley's q-hook length formula (1).
The present work generalizes the explorations of [BKS20] to the next most famous qanalogues of the hook length formula: Stanley's q-hook-content formula for semistandard tableaux, and formulae of Björner-Wachs for linear extensions of labeled forests. See Table 1 for a summary of the q-hook-type formulas we use. The limit laws in these cases turn out to be much more intricate than in [BKS20], with uncountably many rather than countably many possible limits.
Typical central limit theorems are based on an integer sequence so they "let n → ∞," even when the limit laws are complicated such as in the work of Chatterjee-Diaconis [CD14]. By contrast, the combinatorial statistics considered here and in [BKS20] have much more complex indexing sets involving objects like integer partitions and forests. We address this complication by considering sets of standardized distributions as metric spaces under the Lévy metric on all distributions, together with a corresponding space of parameters. Our overarching goal is to describe the closure of these metric spaces and to completely classify which sequences tend to which limit points in terms of the relevant parameter spaces.  A key step in our approach is the introduction of a new family of continuous univariate distributions which we call DUSTPAN distributions 1 . These distributions involve convolutions of the normal law with a countable family of uniform measures supported on some intervals. More precisely, we have the following abstract description. See Definition 3.24 for the concrete version.
Theorem 1.1. The family of DUSTPAN distributions with variance 1 is uniquely characterized as the smallest family F of standardized real-valued distributions such that: (i) U[0, 1] * ∈ F (ii) If X, Y ∈ F, then the standardized independent sum random variable αX+βY √ α 2 +β 2 belongs to F for any α, β ∈ R not both 0. (iii) F is closed under convergence in distribution.
The general strategy of our arguments is as follows. First, we convert formulas involving ratios of q-integers into explicit expressions for the cumulants. In most cases these expressions involve significant cancellation. Next comes the difficult step where we find an asymptotically cancellation-free approximation to the cumulants in a suitable regime; see for instance Lemma 5.8. Finally, in all cases considered in this paper, we use the approximate cumulants to identify the limiting standardized distributions pertaining to SSYT's and linear extensions of trees as some particular DUSTPAN distribution. While the first step is quite generic, the combinatorial arguments and inequalities underlying the second step are highly domainspecific and expand on the corresponding approach from [BKS20].
In Section 1.1, we summarize the results of [BKS20] and reframe them in terms of metric spaces as a prelude to our new, more technical results on semi-standard tableaux and forests. To keep this introduction to a manageable length and avoid frequent digressions, we assume familiarity with tableaux combinatorics and cumulants. Detailed background on these topics is provided in [BKS20,§2] or [Sta99,Ch.7]. The main new results in this paper are outlined in Section 1.2 and Section 1.3. See Section 2 for background necessary for the new material.
1.1. Standard tableaux. Let X λ [maj] denote the random variable associated with maj on SYT(λ), sampled uniformly at random. Then the probability P(X λ [maj] = k) = a λ k /f λ where SYT(λ) maj (q) = a λ k q k and f λ = SYT(λ) maj (1) is the number of standard Young tableaux of shape λ. Hence, studying the distribution of the random variable X λ [maj] and the sequence of coefficients {a λ k : k ≥ 0} for SYT(λ) maj (q) are essentially equivalent. Furthermore, any polynomial in q with nonnegative integer coefficients can be associated to a random variable in a similar way.
For the sake of understanding limiting distributions, we typically standardize the random variables involved so they have mean 0 and variance 1. In general, given any random variable X with mean µ and standard deviation σ > 0, let X * := (X − µ)/σ denote the corresponding standardized random variable with mean 0 and variance 1. To avoid overemphasizing trivialities, we implicitly ignore degenerate distributions with σ = 0 throughout the paper without further comment, so every distribution we consider does have a standardization. Write X n ⇒ X to mean that the sequence X n converges in distribution to X . Let N (µ, σ 2 ) denote a normal distribution, and let IH M denote the M th Irwin-Hall distribution, obtained by summing M independent continuous uniform [0, 1] random variables. These distributions are also referred to as uniform sum distributions in the literature. Note that the normal and Irwin-Hall distributions are continuous, while each of the random variables coming from q-hook formulas below determine discrete distributions.
(ii) X λ (N ) [maj] * ⇒ IH * M if and only if aft(λ (N ) ) → M < ∞. Theorem 1.2 shows that the set Z ≥1 ∪ {∞} parameterizes the set of all possible limit distributions associated to the q-hook length formulas and the standardized random variables X λ [maj] * . If we instead parameterize the limit distributions by 1 n : n ∈ Z ≥1 ∪ {0}, we get a parameter space and a distribution space which are homeomorphic as topological spaces. Hence, we introduce the notion of a metric space of standardized distributions. Endow M IH and M SYT with the topology inherited from the topology of distributions of real-valued random variables under the Lévy metric, which is characterized by convergence in distribution [Bil95, Ex. 14.5].
By the Central Limit Theorem, M IH = M IH ∪ {N (0, 1)}. In light of Theorem 1.2, we have the following very precise description of the minimal compactification of the metric space of SYT distributions.
Corollary 1.4. In the Lévy metric, which is compact. Moreover, the set of limit points of M SYT is exactly M IH .
When the set parametrizing our combinatorial statistics has a natural topology, one might hope that it is homeomorphic to the space of distributions. For example, let be the Irwin-Hall parameter space. We endow P IH ⊂ [0, 1] with the topology of pointwise convergence, so P IH = P IH ⊔ {0}. Since IH * M ⇒ N (0, 1) as M → ∞, the bijection P IH → M IH given by 1 M → IH * M and 0 → N (0, 1) is a homeomorphism. It is less clear how to impose a topology on standard Young tableaux, but a characterization of the multiset of hook lengths would be a key consideration. See [BKS20, Thm. 7.1].
Remark 1.5. Recent work of Kim-Lee identified certain normal [KL20] and bivariate normal [KL21] distributions as limits of normalizations of des and (des, maj) over conjugacy classes in the symmetric group. In their context, the space of limit distributions is parameterized by real numbers in [0, 1].
1.2. Semistandard tableaux and plane partitions. Stanley's hook-content formula is a rational product formula for counting the set SSYT ≤m (λ) of semistandard tableaux of shape λ with entries at most m. He gave a natural q-analogue of this formula, which is in fact the polynomial generating function for the rank statistic on SSYT ≤m (λ). A second rational product formula for rank on SSYT ≤m (λ) with important representation-theoretic meaning is given by the type A case of the q-Weyl dimension formula. Explicitly, See Section 2.2 for more details. Let X λ;m [rank] denote the random variable associated with the rank statistic on SSYT ≤m (λ), sampled uniformly at random. In Section 2, we derive simple explicit cumulant formulas from these rational expressions which allow us to study the possible limiting distributions for X λ;m [rank] * . While the closures M SYT and M IH are completely characterized above, the closure of the metric space of SSYT distributions, is much more complicated. In particular, we show that the following generalization of the Irwin-Hall distributions are related to limit laws for X λ;m [rank] * . Definition 1.6. Given a finite multiset t of non-negative real numbers, let where we assume the summands are independent and U[a, b] denotes the continuous uniform distribution supported on [a, b]. If t consists of M copies of 1, then S t + M 2 = IH M . By convention, we consider the multiset t as a weakly decreasing sequence of real numbers t = {t 1 ≥ t 2 ≥ · · · ≥ t m } where t m ≥ 0. We call the distribution associated to S t a finite generalized uniform sum distribution.
Certain sequences of random variables X λ;m [rank] * which converge to a finite generalized uniform sum distribution are completely characterized by an auxiliary multiset called the distance multiset. This auxiliary set also comes up in the Turnpike Reconstruction Problem, which is essentially the problem of identifying all possible sequences t from the following multiset ∆t, which has applications in DNA sequencing and X-ray crystallography [Wei95, Sect. 10.5.1]. The Turnpike Reconstruction Problem is a potential candidate for being in NP-Intermediate. See [LSS03] for further computational complexity considerations.
Theorem 1.8. Let λ be an infinite sequence of partitions with ℓ(λ) < m where λ 1 /m 3 → ∞. Let t(λ) = (t 1 , . . . , t m ) ∈ [0, 1] m be the finite multiset with t k := λ k λ 1 for 1 ≤ k ≤ m. Then X λ;m [rank] * converges in distribution if and only if the multisets ∆t(λ) converge pointwise. In that case, the limit distribution is Theorem 1.8 suggests we consider the metric space of distance distributions and its associated parameter space P DIST defined in Section 3.4. By padding with 0's, we consider P DIST ⊂ R N as a sequence space with the topology of pointwise convergence. The metric space of distance distributions is significantly more complex than the metric space of Irwin-Hall distributions. Nonetheless, a careful analysis involving the topology of the parameter space of distance multisets done in Section 3.4 yields the following results. We will show that both P DIST and M DIST have natural one point compactifications, where 0 is the infinite sequence of 0's. Furthermore, in analogy with Corollary 1.4, we will show that the map P DIST → M DIST given by d → S * d and 0 → N (0, 1) is a homeomorphism between sequentially compact spaces. See Theorem 3.32. Therefore, Theorem 1.8 and Theorem 3.32 combine to give the following complete characterization of the possible limit laws for a particular family of semistandard tableaux in analogy with Corollary 1.4. Corollary 1.9. For any fixed ϵ > 0, let which is compact. Moreover, the set of limit points of M ϵ SSYT is M DIST .
Corollary 1.9 already indicates that the limiting distributions associated to semistandard tableaux are much more varied than the case of standard Young tableaux. See Summary 4.20 for a synopsis of all of the asymptotic limits we have identified for X λ,m [rank] * . This includes several "generic" asymptotic normality criteria and a partial analogue of aft, called weft, which controls asymptotic normality in many cases of interest. A complete description of the closure of M SSYT akin to Theorem 1.2 and Corollary 1.4 remains open.
Open Problem 1.10. Describe M SSYT in the Lévy metric. What are all possible limit points?
By studying one more special family of semistandard tableaux, we will show that the Irwin-Hall distributions are also among the limit points. Thus, the strongest statement we have shown for the metric space of limit laws for Stanley's q-hook-content formula is Using a well-known bijection, the two product formulas in (3) imply product formulas for the generating function of the size statistic on the set PP(a × b × c) of plane partitions fitting in a box. See the second and third rows of Table 1. Let X a×b×c [size] similarly denote the random variable associated with the size statistic on PP(a × b × c). In the theorem below, we give a complete characterization of the limit laws for plane partitions and {X a×b×c [size] * }. This leads to an analog of Corollary 1.4 for the metric space of plane partition distributions, Theorem 1.11. Let a, b, c each be a sequence of positive integers. ( Corollary 1.12. In the Lévy metric, which is compact. Moreover, the set of limit points of M PP is exactly M IH . 1.3. Linear extensions of forests. Knuth [Knu73,p. 70] gave a rational product formula for counting the set L(P ) of linear extensions of a forest P , analogous to the Frame-Robinson-Thrall hook length formula. Using a fixed bijection w : P → [n], one may interpret L(P ) as a set of permutations L(P, w) ⊂ S n and consider the distribution of the major index or inversion number statistics on these permutations. Stanley [Sta72] and Björner-Wachs [BW89] gave q-analogues of Knuth's formula for major index and number of inversions using certain labelings w. All of these statistics agree up to an overall shift. See the fourth row of Table 1 and Section 2.3 for details. Let X P denote the random variable associated with the maj or inv statistic on L(P, w) where w is order-preserving. The distribution of X * P is independent of the choice of statistic and the choice of w. Let M Forest := {X * P : P is a forest} be the metric space of forest distributions. We show that the behavior of the possible limiting distributions for X * P breaks into two distinct regimes. The first "generic" regime exhibits classic asymptotic normality, while the second "degenerate" regime allows even more continuous limit laws than have appeared in the theory for standard or semistandard tableaux.
Let rank(P ) denote the length of a maximal chain in P . Let |P | denote the number of vertices. For example, the rank of a complete binary tree with 2 n − 1 vertices is n, so rank(P ) ≈ log 2 |P |. Typically, rank(P ) is much smaller than |P |, so the following theorem covers the "generic" regime.
Theorem 1.13. Given a sequence of forests P , the corresponding sequence of random variables X * P is asymptotically normal if |P | → ∞ and lim sup rank(P ) |P | < 1.
In the "degenerate" regime, rank(P ) ∼ |P |, so the number of vertices not in a chosen maximal chain is relatively small. We completely describe the possible limit distributions when |P | − rank(P ) = o(|P | 1/2 ). To do so, we generalize both the distance distributions and the Irwin-Hall distributions to the distributions associated to countable sums of independent, continuous, uniform random variables with finite mean and variance. We call these generalized uniform sum distributions. Again we can reduce to sums of independent centralized random variables S t exactly as in (4), except now we consider countably infinite multisets t = {t 1 ≥ t 2 ≥ . . . } of nonnegative real numbers. See Section 3.1 for details such as cumulants, the density function, and the relation to pointwise convergence in R N .
The variance of a uniform sum random variable S t is closely related to the 2-norm of t, In this notation, Var[S t ] = B 2 2 |t| 2 2 , where B 2 = 1 6 is a Bernoulli number. Thus, in order for S t to be well defined, it must have finite variance, so |t| 2 < ∞ is required. Let ℓ 2 := {t = (t 1 , t 2 , . . .) : t 1 ≥ t 2 ≥ · · · ≥ 0, |t| 2 < ∞}. The standardized general uniform sum distributions are indexed by the decreasing sequences t ∈ ℓ 2 such that 1 = Var[S t ] = B 2 2 |t| 2 2 , so |t| 2 2 = 2 B 2 = 12. Thus, we will see the number 12 coming up in several places. In particular, define the hat-operation on t ∈ ℓ 2 with positive 2-norm by so that Var[S t ] = 1 and S t = S * t . Now, we can return to the limiting distributions of forests in the "degenerate" regime. We show in Remark 2.22 that it suffices to consider only standardized trees in order to characterize all of M Forest . In Definition 5.7, we associate to each tree P an elevation multiset e depending on a maximal chain in P . These multisets determine a new type of limiting distribution related to the generalized uniform sum distributions, but with another normal summand.
Theorem 1.14. Let P be an infinite sequence of standardized trees with |P | − rank(P ) = o(|P | 1/2 ). Then X * P converges in distribution if and only if the multisets e converge pointwise to some element t ∈ ℓ 2 . In that case, the limit distribution is S t +N (0, σ 2 ) where |t| 2 2 /12+σ 2 = 1. Inspired by Theorem 1.14, we begin the study of DUSTPAN distributions associated to random variables of the form S t + N (0, σ 2 ), assuming the two random variables are independent, t ∈ ℓ 2 , and σ ∈ R ≥0 . The nomenclature DUSTPAN refers to a distribution associated to a uniform sum for t plus an independent normal distribution. The generalized uniform sum distributions with variance 1 are the special case when σ = 0. Let (9) P DUST := t ∈ ℓ 2 : |t| 2 2 ≤ 12 be the standardized DUSTPAN parameter space, considered as a sequence space with the topology of pointwise convergence. Define the metric space of standardized DUSTPAN distributions to be (10) M DUST := {S t + N (0, σ 2 ) : |t| 2 2 /12 + σ 2 = 1}. The standardized DUSTPAN parameter space P DUST is a closed subset of the sequence space ℓ 2 ⊂ R N considered as a Fréchet space (rather than a Banach space). See e.g. [MV97, Ex. 5.18(1)] for more details on this structure. In fact, M DUST is closed as well, and we will show we have the following homeomorphism of compact spaces.  Remark 1.18. The foundational idea of information geometry is to endow spaces of distributions with the structure of Riemannian manifolds. Consequently, one may be tempted to recast Theorem 1.15 in the context of manifold theory. However, the infinite-dimensional case is generally "not mathematically easy" [Ama16, §2.5, p.39]. Here, ℓ 2 is a Hilbert manifold and a Banach manifold under the ℓ 2 -norm, as well as a Fréchet manifold under pointwise convergence. There does not appear to be a generally agreed-upon Hilbert, Banach, or Fréchet manifold structure which the closed subset P DUST inherits from ℓ 2 , though it could perhaps be thought of as a manifold with corners. In any case, the inherited Hilbert and Banach topology on P DUST disagrees with the Fréchet topology, so for our purposes, Theorem 1.15 requires us to use the Fréchet structure of pointwise convergence. It is consequently unclear if a useful differentiable structure exists for P DUST More generally, it is natural to ask which limit laws are possible for the coefficients of arbitrary q-hook-type formulas, namely polynomials with nonnegative integer coefficients of In [BS20], we call such q-integer quotients cyclotomic generating functions (CGF's) and study their properties from a variety of algebraic and probabilistic perspectives. Let M CGF denote the corresponding metric space of standardized distributions. By Prohkorov's Theorem, M CGF is compact.
Open Problem 1.19. Describe M CGF in the Lévy metric. What are all possible limit points? Is M CGF ∪ M DUST the metric space of limit laws for q-hook formulas, referring back to the title of this article?
1.4. Paper organization. The rest of the paper is organized as follows. In Section 2, we provide background for the hook and cumulant formulas summarized in Table 1. In Section 3, we analyze the metric space of generalized uniform sum distributions and its variations in order to prove Theorem 1.15 and its analog for the distance distributions. The analysis of M SSYT and M PP is in Section 4. The analysis of M Forest is in Section 5. Some additional open questions and avenues for future work are listed in Section 6.

Background
In this section, we briefly recall statements from the literature we will need related to asymptotic distributions, semistandard tableaux and forests. All of our arguments for determining asymptotic distributions use the Method of Moments/Cumulants. Using work of Hwang-Zacharovas, we explain a key insight for this paper, namely that rational product formulas such as appear in Table 1 give rise to explicit formulas for cumulants of the corresponding distributions. See [BKS20, §2-3] for a more extensive exposition aimed at an audience familiar with enumerative combinatorics. See [Bil95] for background in probability.
2.1. Asymptotic distributions. Let X be a real-valued random variable. For d ∈ Z ≥0 , the dth moment X is The moment-generating function of X is which for us will always have a positive radius of convergence. The characteristic function of which exists for all t ∈ R and which is the Fourier transform of the density or mass function associated to X . We will need the following technical details for the proofs in future sections.
Example 2.3. Let U = U[0, 1] be the continuous uniform random variable whose density takes the value 1 on the interval [0, 1] and 0 otherwise. Then the moment generating function is M U (t) = 1 0 e tx dx = (e t − 1)/t, so the cumulant generating function log M U (t) coincides with the exponential generating function for the divided Bernoulli numbers B d d for d ≥ 1. Their exponential generating function E D (t) satisfies Recall from Section 1, IH m is the Irwin-Hall distribution obtained by adding m independent U[0, 1] random variables. By additivity, the dth cumulant of IH m is mB d /d. More generally, let S := m k=1 U[α k , β k ] be the sum of m independent uniform continuous random variables. Then the dth cumulant of S for d ≥ 2 is by the homogeneity and additivity properties of cumulants.
The Method of Moments/Cumulants is based on the following theorem. All random variables we encounter will have moments of all orders.
Theorem 2.4 (Frechét-Shohat Theorem, [Bil95, Theorem 30.2]). Let X 1 , X 2 , . . . be a sequence of real-valued random variables, and let X be a real-valued random variable. Suppose the moments of X n and X all exist and the moment generating functions all have positive radius of convergence. If Corollary 2.5. A sequence X 1 , X 2 , . . . of real-valued random variables on finite sets is asymptotically normal if for all d ≥ 3 we have For a positive integer n, define the associated q-integer to be the polynomial [n] q = 1 + q + q 2 + · · · + q n−1 = (1 − q n )/(1 − q). The q-integers factor into cyclotomic polynomials over the integers. Therefore, the hook length formulas considered in this paper are all products of cyclotomic polynomials. Because these rational product formulas are polynomial, all cancellation can be done efficiently by taking the multiset difference between the numerator and denominator of the cyclotomic factors.
In a forthcoming paper [BS20], we investigate general properties of generating functions which are products of cyclotomic polynomials with nonnegative coefficients. For this paper, we just need two facts. The first theorem first appeared explicitly in the work of Hwang Let X be a discrete random variable with P[X = k] = c k /P (1). Then the dth cumulant of X is The following corollary is proved in [BS20]. It also follows from the tail decay bound in [HZ15, Lemma 2.8]. We need this for our current investigations for hook length formulas.
Lemma 2.7 (Converse of Frechét-Shohat for CGF's). Suppose X 1 , X 2 , . . . is a sequence of random variables corresponding to polynomials of the same form as (18). If X * n ⇒ X for some random variable X , then X is determined by its cumulants and, for all d ∈ Z ≥1 ,

Semistandard Young tableaux and plane partitions.
We briefly recall the definition and notation for Schur functions, semistandard tableaux and plane partitions. For more information on symmetric functions and their connection with the enumeration of plane partitions and tableaux, see [Sta99,Ch. 7]. A partition λ = (λ 1 ≥ λ 2 ≥ · · · ≥ λ k ) is a finite decreasing sequence of positive integers. Let ℓ(λ) = k denote the length of λ. We think of λ in terms of its Young diagram, which is a left justified array of ℓ(λ) rows with λ i cells on row i and index the cells in matrix notation.
A semistandard Young tableau, or just semistandard tableau for short, of shape λ is a filling of the cells of λ with positive integer labels, possibly repeated, such that the labels weakly increase to the right in rows and strictly increase down columns. The set of semistandard Young tableaux of shape λ is denoted SSYT(λ). The subset of SSYT(λ) filled with integers no greater than m is denoted SSYT ≤m (λ), which is a finite set. The type of a semistandard tableau T is the composition α(T ) = (α 1 , α 2 , . . .) where α i is the number of times i appears in T . The Schur function is the type generating function for all semistandard tableaux of shape λ, where x α := x α 1 1 x α 2 2 · · · . The rank of a semistandard tableau T is a nonnegative integer statistic depending only on the type. It is defined by For example, for a fixed partition λ, the smallest possible rank of any T ∈ SSYT(λ) occurs for the tableau with all 1's in the first row, all 2's in the second row, etc. in the diagram of λ. Therefore, the minimal rank is (i − 1)λ i , which we denote as rank(λ). The rank generating function for SSYT(λ) is given by the principal specialization of the Schur function, The motivation for considering this particular specialization comes from the q-analog of the Weyl dimension formula in representation theory. Stembridge [Ste94, §2.2-2.3, Prop. 2.4] put a ranked poset structure on the weights of a semisimple Lie algebra, which in type A reduces to rank(α). The following rational product formula for s λ (1, q, q 2 , . . . , q m−1 ) follows easily from the classical ratio of determinants definition of Schur polynomials.
Stanley gave an alternate rational product formula for s λ (1, q, . . . , q m−1 ), which is called the q-hook-content formula. Here the content of a cell u in row i, column j in λ is defined as c u := j − i. Also, the hook length of cell u, denoted h u , is the number of cells directly east of u, plus the number of cells directly south of u in the diagram of λ.
Combining Theorem 2.6 and Theorem 2.9, we get an exact formula for the cumulants of the random variable associated to the rank function on semi-standard Young tableaux on the alphabet [m] chosen uniformly. This cumulant formula is the key to analyzing the asymptotic distributions.
Corollary 2.10. Fix a partition λ. If κ λ;m d is the dth cumulant of the random variable associated to rank on SSYT ≤m (λ), then, for d > 1, Observe, the summands in (23) can be negative, but the summands in (22)  Definition 2.11. A plane partition is a finite collection of unit cubes in the positive orthant of R 3 stacked towards the origin. More formally, it is a finite lower order ideal in Z 3 ≥1 under the component-wise partial order. We may imagine a plane partition ρ as a matrix with entry ρ ij recording the number of cells with x-coordinate i and y-coordinate j. The size of a plane partition ρ is the number of cubes, denoted |ρ| = ρ ij . We write PP(a × b × c) for the set of all plane partitions fitting inside an a by b by c rectangular prism.
There is a straightforward bijection between plane partitions and semistandard Young tableaux of rectangular shape, All T and ρ in the bijection are rectangular arrays with a rows and b columns with entries labeled using matrix indexing conventions. Letting |T | := i,j T ij , note that |T | = rank(T ) + ab and |T | + |ρ| = abc + b a+1 2 is constant. Hence, the unique element of minimal size in PP(a × b × c), namely ∅, maps to the unique maximal rank tableau in SSYT ≤a+c ((b a )) with values c − i in row i for each 1 ≤ i ≤ a.
By Theorem 2.8 and Theorem 2.9, we know SSYT(λ) rank (q) is symmetric up to an overall q-shift. Similarly, PP(a × b × c) is closed under box complementation, so it follows from the bijection and (20) that The later two product formulas are originally due to MacMahon. See the proof of [Sta99, Thm. 7.21.7] for more details and [Sta99, pp. 402-403] for historical references. In particular, the cumulants of size on PP(a × b × c) are given by (22) or (23) where λ = (b a ) and m = a + c.
2.3. Linear extensions of forests. Next, we summarize the relevant terminology and results from [BW89]. Briefly recall, a tree is a finite, connected simple graph with no cycles. A forest is a finite disjoint union of trees. A tree is rooted if it has a distinguished vertex, called the root. A forest is rooted if each of its trees is rooted. The Hasse diagram of a partially ordered set (poset) P is the graph with vertex set P where there is an edge between x and y if y covers x, i.e. x < P y and there does not exist u ∈ P such that x < P u < P y. We refer to a poset as a forest if its Hasse diagram is a forest with roots as maximal elements, or equivalently if every element of P is covered by at most one element.
Definition 2.12. Let P be a finite partially ordered set. The rank of P is the maximum number of elements in any chain u 1 < u 2 < · · · < u k in P . For instance, if P is a singleton, its rank is 1. Note that this definition is one larger than the standard definition in [Sta12, Ch.3], but it is more convenient for our purposes.
Definition 2.13. Let P be a poset. A labeling of P is a bijection w : P → [n], and a labeled poset is a pair (P, w) where w is a labeling of P . A labeling w of P for which w(p) ≤ w(q) whenever p ≤ P q is called a natural labeling. A labeling w of P is regular if for all x < P z and y ∈ P , if w(x) < w(y) < w(z) or w(x) > w(y) > w(z) then x < P y or y < P z. Regular labelings of forests include the postorder, preorder, and inorder labelings, which are commonly used in computer science.
Definition 2.14. A linear extension of P is an ordered list p 1 , . . . , p n of the elements of P such that i ≤ j whenever p i ≤ P p j . If (P, w) is a labeled poset, a linear extension can be thought of as the permutation i → w(p i ) of [n]. The set L(P, w) is the set of all permutations obtained in this fashion from linear extensions of the labeled poset (P, w). It is often convenient to use a natural labeling w of P so that id ∈ L(P, w). Choosing labelings which are not natural forces inversions to appear in any σ ∈ L(P, w). Finding the minimum number of inversions in any linear extension of an arbitrarily labeled poset motivates the following analogues related to inversions and descents in permutations.
Definition 2.15. Let (P, w) be a labeled poset. Set where the hook length of an element x ∈ P is (29) h x := #{t ∈ P : t ≤ P x}. Remark 2.17. One can consider a partition λ as a poset on its cells where (u, v) ≤ (x, y) if and only if u ≤ x and v ≤ y. However, the hook lengths of λ do not agree with (29) except when λ is a single row or column. For example, the hook lengths for the partition (2, 2) are 1, 2, 2, 3, while the hook lengths for the diamond poset are 1, 2, 2, 4.
Mallows and Riordan first studied the inversion enumeration on labeled rooted trees [MR68], and connected it to cumulants of the lognormal distribution. Knuth gave a hook length formula for |L(P, w)| [Knu73, p. 70] for posets which are forests. Björner-Wachs [BW89] and Stanley [Sta72] generalized Knuth's result to q-hook length formulas using the inv and maj statistics on L(P, w). Stanley considered only the case when w is natural, i.e. when inv(P, w) = maj(P, w) = 0, for the maj generating function. Recently Zaguia has studied linear extensions of forests and proved the "1/3-2/3 Conjecture" holds on such posets [Zag19]. if and only if P is a forest. Similarly, if and only if (P, w) is a regularly labeled forest. Moreover, if P is a forest, [n]q! u∈P [hu]q has symmetric and unimodal coefficients.
The hook lengths of the underlying poset are 1, 1, 3, 4. One can verify the formulas in Theorem 2.18 hold in each of these cases, but they don't hold for the diamond poset.
Given a forest P , define the polynomial and let X P the associated random variable. Note, the distribution of X P does not depend on the choice of labeling of the vertices of P since L P (q) depends only on the unlabeled poset structure. We also get simple formulas for the associated cumulants in the next two statements.
Remark 2.20. By the unimodality result in Theorem 2.18, we know L P (q) := [n] q !/ u∈P [h u ] q has nonzero coefficients in an interval, so it has no internal zeros. The degree of L P (q) is and the mean of X P is half the degree.
Corollary 2.21. Let P be a forest with n elements. Suppose d ∈ Z ≥2 . Let κ P d denote the dth cumulant of the random variable X P . Then, Remark 2.22. In order to characterize all possible limit laws for the standardized random variables associated with maj and inv on labeled forests, we only need to consider the set of all distributions associated with standardized trees as follows. Given any forest P , we may turn P into a tree by adding a new vertex covering the roots of all the trees of P . It is easy to see that the quotient in (30) is unchanged, so the cumulants and the corresponding distributions are the same. Similarly, if P is a tree and the root has exactly one child, we may delete the root while preserving the fact that P is a tree, and the quotient in (30) is again unchanged. Consequently, we say a forest is standardized if it is a tree and the root has at least two children. Therefore, M Forest := {X * P : P is a forest} = {X * P : P is a standardized tree}.
2.4. Riemann integral estimates. Many of our theorems depend on approximations using a mixture of combinatorics and analysis. In particular, we return to certain basic sums over and over again. Let h d (a, b) = d j=0 a j b d−j denote the complete homogeneous symmetric function on two inputs.
Lemma 2.23. For positive integers a, b, and d > 1, we have Proof. Use a Riemann integral estimate. □ 2.5. Standard notation for approximations. We use the following standard Bachmann-Landau asymptotic notation without further comment. We write f (n) = Θ(g(n)) to mean there exist constants a, b > 0 such that for n large enough, we have ag(n) ≤ f (n) ≤ bg(n).

Metric spaces related to uniform sum distributions
Motivated by applications to M SSYT and M Forest in the next two sections, we first analyze the distributions of finite and infinite sums of uniform continuous random variables. We parameterize these distributions using certain sequence spaces and precisely relate weak convergence of the underlying distributions to pointwise convergence of the parametrizing sequences. The closure of the space of all possible distributions associated to standardized sums of independent uniform random variables leads us to define the metric space of DUSTPAN distributions. We also describe a closed subset of the DUSTPAN distributions related to distance multisets, which appear in the study of M SSYT .
3.1. Generalized uniform sum distributions and decreasing sequence space. The Irwin-Hall distributions, also known as uniform sum distributions, are the distributions associated to finite sums of independent, identically distributed, uniform random variables supported on [0, 1]. First, we relax the requirement that they be identically distributed, and then we relax the requirement that they are finite sums.
Consider a random variable defined as the sum of m independent uniform continuous random variables of the form S := m k=1 U[α k , β k ] with α k ≤ β k for each k. We call the distribution of S a generalized uniform sum distribution. See Figure 3 for example density functions. We note that each of the generalized uniform sum distributions is non-normal, though the histograms may look quite similar. By Example 2.3, the dth cumulant of S for d ≥ 2 is which only depends on the differences t k := β k − α k . It is useful to compare (31) to the cumulants in (19). The random variable S can be expressed as a constant overall shift c = 1 2 m k=1 (α k + β k ) plus a uniform sum random variable associated to t (32)   Figure 3. Plots of density functions for the distributions S t with t = (6, 5, 1) and t = (6, 5, 5, 5, 1).
We will show below that a similar classification holds for the distributions associated to countable sums of independent continuous uniform random variables, which are defined provided the expectation and variance are finite. Again we have a nice formula for the cumulants of infinite sums of uniform random variables simply by letting m → ∞. Observe that (31) is very similar to the definition of the p-norm for a real vector space. (1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9).
The sequence space with finite p-norm ℓ p := {t = (t 1 , t 2 , . . .) ∈ R N ≥0 : |t| p < ∞} is commonly used in functional analysis and statistics. Here we define a related concept for analyzing sums of central continuous uniform random variables.
The elements of ℓ p may equivalently be thought of as the set of countable multisets of nonnegative real numbers with finite p-norm. Any finite multiset of non-negative real numbers can be considered as an element of ℓ p with finite support by sorting the multiset and appending 0's. The multisets in ℓ p are uniquely determined by their p-norms. In fact, any subsequence of p-norm values injectively determines the multiset provided the sequence goes to infinity.
Lemma 3.4. Let t, u ∈ ℓ p for some 1 ≤ p ≤ ∞. Suppose |t| p j = |u| p j for some sequence p j → ∞. Then t = u.
Proof. We have We may remove the first elements from both t and u to obtain multisets (t 2 , t 3 , . . .) and (u 2 , u 3 , . . .) which are both in ℓ p and have equal p j -norms again. While removing these largest elements alters the p j -norms, it does so by the same amount for both t and u. Repeating the argument, t i = u i for all i, so t = u. □ Theorem 3.5. Finite generalized uniform sum distributions are bijectively parameterized by R × {t ∈ ℓ 2 : t has finite support}.
Proof. As noted above, every such distribution is defined by a random variable of the form c + S t for some c ∈ R and t = (t 1 , . . . , t m , 0, 0, . . . ) ∈ ℓ 2 . To show uniqueness, suppose S t = S u . By (33), we know Therefore, since the even Bernoulli numbers are non-zero, we have |t| d = |u| d for each d even, which is a sequence approaching infinity. Hence, by Lemma 3.4, t = u. □ The probability density functions (PDF) for any finite generalized uniform sum distributions can be determined as a convolution. We will not need this formula in the rest of this paper, but we note it here for completeness. It was used to generate Figure 4.
Proof. For the case m = 1, Let * denote convolution. One can check that for all u > 0, we have the convolution identity The probability density function of the sum of independent random variables is the convolution of their density functions. Therefore, the general case of the lemma now follows by applying the m = 1 case and the convolution identity inductively. □ Remark 3.7. When t 1 = · · · = t m = 1, the formula in Lemma 3.6 collapses to Hence we recover the density formula for the Irwin-Hall distributions [JKB94, p. 296] Remark 3.8. A similar formula for the cumulative distribution function of S t as a sum over the vertices of the hypercube is given in [BS79]. See also [JKB94, p. 298-300] for relevant discussion.
We now turn to infinite sums of independent uniform continuous random variables. Our next goal is to generalize Theorem 3.5 to this setting. To do so, we must first extend the uniformsum distributions S t to countably infinite multisets t, and discuss the basic properties of these random variables including existence, characteristic functions, and cumulants. Existence depends on the following well-known result, which often appears in treatments of the law of large numbers. See, for example, [Dur10, Thm. 2.5.3].
Theorem 3.9 (Kolmogorov's Two-Series Theorem). Let X 1 , X 2 , . . . be a sequence of independent real-valued random variables. Suppose E[X k ] = 0 and ∞ k=1 Var[X k ] < ∞. Then ∞ k=1 X k converges almost surely. Almost sure convergence implies convergence in distribution. Therefore, by Kolmogorov's Two-Series Theorem, we are lead to the following definition.
Definition 3.10. A generalized uniform sum distribution is any distribution associated to a random variable with finite mean and variance given as a countable sum of independent continuous uniform random variables. As in the finite case, such random variables are given by a constant overall shift plus a uniform sum random variable ] diverges with positive probability, so the assumption |t| 2 2 < ∞ is essential. In this way we also see that uncountably many non-zero summands of independent continuous uniform random variables must diverge. Thus, we cannot extend Definition 3.10 beyond countable sums.
We claim that each uniform sum random variable S t for t ∈ ℓ 2 gives rise to a distinct distribution. In order to prove the claim, we need to verify the relationship between the p-norms and the cumulants of the infinite sums is as expected. To do so, we describe the characteristic and moment-generating functions of S t .
Lemma 3.11. Let t = (t 1 , t 2 , . . .) ∈ ℓ 2 . Then S t exists, has moments of all orders, and is determined by its moments. The characteristic function is the entire function Proof. As mentioned above, the assumption t ∈ ℓ 2 and Theorem 3.9 together imply S t exists.
where sinc(0) := 1. Consequently, the nth partial sum S n = n k=1 U − t k 2 , t k 2 has characteristic function ϕ Sn (s) = n k=1 sinc(st k /2). Almost sure convergence implies convergence in distribution, so S n ⇒ S t . Thus, by Lévy's Continuity Theorem, we have for each s ∈ R that ϕ S t (s) = ∞ k=1 sinc(st k /2). By Lemma 3.12 below, the product form for ϕ S t is entire and hence complex analytic on an open ball, so (35) follows from Remark 2.1. Likewise, S t has moments of all orders and S t is determined by its moments.
Since the entire functions ϕ Sn (s) converge uniformly on compact subsets of C to ϕ S t (s), it follows that the d th moment can be determined by the constant term of the d th derivative of the characteristic function for all d ≥ 1. The moments of any random variable determine its cumulants and vice versa. Therefore, the cumulant formula now follows from (33), including the first two moments. □ Lemma 3.12. Let t = (t 1 , t 2 , . . .) ∈ ℓ 2 . As a function of s, the infinite product converges to an entire function in the complex plane. Moreover, for |s| < 1/|t| 2 , Proof. For each D > 0, the entire function 1−sinc(z) z 2 is bounded on |z| < D by some constant Thus, the sum converges uniformly on compact subsets of {|s| < 2D/ sup{t i }}. Taking D → ∞, the sum converges uniformly on compact subsets of all of C. The result now follows by standard criteria for infinite product convergence such as [Rud87, Thm. 15.6].
In particular, if t, u ∈ ℓ 2 with t ̸ = u, then S t ̸ = S u . Furthermore, S * t = S * u if and only if t, u differ by a scalar multiple.
Proof. The first and second claims follow exactly as in Theorem 3.5 using the cumulant formula in Lemma 3.11. For the third claim, we can assume |t| ∞ = |u| ∞ by rescaling if necessary and S * t = S * u . From Lemma 3.11 and the general properties of cumulants, it follows that for all d even, Since |t| ∞ = |u| ∞ , we have |t| 2 = |u| 2 , which hence gives |t| d = |u| d for all d even. Again by Lemma 3.4, we have t = u. □ Example 3.14. Infinite sums of independent continuous uniform random variables have appeared elsewhere in the literature, though rarely. For instance, when t = (1, 1/2, 1/4, 1/8, . . .), the cumulative distribution function of S t = ∞ k=1 U[−1/2 k , 1/2 k ] is the so-called Fabius function, [Fab66], which is a known example of a C ∞ -function on an interval which is nowhere analytic. The characteristic function is nonetheless entire by Lemma 3.12.
3.2. Pointwise convergence and convergence in even norms. The decreasing sequence space ℓ 2 has a natural notion of pointwise convergence. In this subsection, we relate pointwise convergence to convergence of p-norms for all positive even p ≥ 4, assuming the 2-norms are bounded.

The result follows by the Method of Moments/Cumulants. □
In light of Lemma 3.23, pointwise convergence in ℓ 2 leads to us to study an additional family of sums of random variables. Note, the sum of two generalized uniform sum random variables is another generalized uniform sum of random variables. Also, the sum of two normal distributions is normal, so we have reached a natural limit to the generalizations.
Definition 3.24. A DUSTPAN distribution is a distribution associated to a uniform sum for t plus an independent normal distribution S t + N (0, σ 2 ), assuming the two random variables are independent, t ∈ ℓ 2 , and σ ∈ R ≥0 .
From Definition 3.24, it follows that the characteristic functions of DUSTPAN distributions have nice properties. Recall that a normal family of holomorphic functions in some open set U ⊂ C is one where every infinite sequence has a subsequence which converges uniformly on compact subsets of U .  N (0, σ (N ) ) ∈ M DUST converges in distribution to some X . Then E[X d ] < ∞ exists for all d ∈ Z ≥1 , X is determined by its moments, and Proof. By Lévy's Continuity Theorem, ϕ X N (s) → ϕ X (s) for all s ∈ R. By Lemma 3.26, we may replace X N if necessary with a subsequence for which ϕ X N (s) converges uniformly on compact subsets so that we can assume ϕ X (s) is entire. Therefore, the moment generating function of X has positive radius of convergence, moments of all order exist, X is determined by its moments, and the limit of the moments is the moment of the limit. □ We may now restate and prove Theorem 1.15 from the introduction.
Theorem 1.15. The map Φ : P DUST → M DUST given by t → S t + N (0, σ) where σ := 1 − |t| 2 2 /12 is a homeomorphism between sequentially compact spaces. Proof. The parameter space P DUST is closed under pointwise convergence by Lemma 3.18. Moreover, it is sequentially compact under pointwise convergence, either by Tychonoff's Theorem applied to [0, √ 12] N or by a simple diagonalization argument. Since P DUST and M DUST are metrizable and Φ is a bijection by Theorem 3.13, we need only show that Proof. Since {S t : t ∈ ℓ 2 , |t| 2 2 = 12, t is finite} ⊂ M DUST by definition and M DUST is closed by Corollary 3.28, we know {S t : t ∈ ℓ 2 , |t| 2 2 = 12, t is finite} ⊂ M DUST = M DUST . For the other inclusion, we just need to show each t ∈ ℓ 2 with |t| 2 2 ≤ 12 is the pointwise limit of a sequence t (N ) ∈ ℓ 2 with |t (N ) | 2 2 = 12 and t finite by Theorem 1.15. As noted above, this follows from Lemma 3.21. □ 3.4. The metric space of distance distributions. For convenience, we recall some of the definitions and notation from the introduction. For each t ∈ ℓ 2 with |t| 2 > 0, let be the rescaled sequence such that | t| 2 2 = 12 and S t = S * t . By definition of the hatoperation, t ∈ P DUST and Φ( t) = S t + N (0, 0) = S t = S * t . The distance multiset of t = {t 1 ≥ t 2 ≥ · · · ≥ t m } is the multiset ∆t := {t i − t j : 1 ≤ i < j ≤ m}, and the metric space of distance distributions is Thus, the parameter space of distance multisets, mentioned in Section 1, is defined as By padding with 0's, consider P DIST ⊂ P DUST ⊂ ℓ 2 as a sequence space with the topology of pointwise convergence.
Lemma 3.30. The closure of P DIST is P DIST ⊔ {0}.
Proof. Let d (N ) ∈ P DIST be a sequence converging pointwise to d. By Theorem 1.15, we can assume d ∈ P DUST . By definition, each d (N ) = ∆t (N ) for some finite sequence of real numbers   N (0, 1) is a homeomorphism between (sequentially) compact spaces.
Proof. First note that P DIST ⊂ P DUST and M DIST ⊂ M DUST by construction, so Φ DIST is the restriction of Φ : P DUST −→ M DUST . Therefore, by Theorem 1.15, Φ DIST is a homeomorphism. Since closed subsets of compact spaces are compact, we know P DIST is compact by Lemma 3.30. Furthermore, Φ(P DIST ) = M DIST ⊔ {N (0, 1)} is closed and compact. □

Metric spaces related to SSYT ≤m (λ) distributions
We next consider the family of generating functions for semistandard tableaux given by the principal specialization of Schur polynomials, or equivalently the rank statistic on SSYT ≤m (λ), as described in Section 1.2 and Section 2.2. An interesting special case is given by MacMahon's formula for the size statistic on the set PP(a × b × c) of plane partitions inside an (a × b × c) box, given in (25). In particular, we will prove Theorem 1.8 and Theorem 1.11. We provide a wide variety of limit law classification results for these statistics in various regimes. The subsections are divided into four natural special cases: n/m → 0, n/m → ∞, cases based on the number of distinct parts of λ, and plane partitions. See Summary 4.20 for a summary.
We begin with a simple analogue of Theorem 1.2. This will be our only use of the hookcontent-based cumulant formula; all of our other results rely on the q-Weyl dimension-based cumulant formula.
Theorem 4.1. Let λ denote an infinite sequence of partitions with |λ| = n. If n m → 0, then for each fixed d ∈ Z ≥2 , the corresponding sequence of cumulants is Furthermore, we can characterize convergence in distribution in the case n m → 0 depending on the limiting value of n.
(i) If n converges to a finite value N , then X λ [rank] * converges in distribution to IH * N . (ii) If n → ∞, then X λ [rank] is asymptotically normal.
Proof. Given a sequence in M ϵ , if m is bounded, then so is n = |λ|, so there are only finitely many distinct (λ, m) in the sequence and convergence occurs if and only if the sequence is eventually constant. On the other hand, if m → ∞, then n < m 1−ϵ yields n m < m −ϵ → 0, so (43) follows immediately from Theorem 4.1.
By these observations, every infinite sequence of distinct points in M ϵ has a limit point in M IH , so M ϵ consists entirely of isolated points and M IH consists entirely of limit points. Sequential compactness is similarly clear. □

4.2.
Limit laws with |λ|/m → ∞ and distance distributions. At the other extreme, we may consider the case when |λ|/m → ∞. As we will see, the possible behavior is vastly more varied in this limit. Among the sequences of partitions λ with |λ|/m → ∞, the easiest case to consider is when λ 1 /m 3 → ∞. This includes the case where m converges to a fixed finite value and |λ| → ∞. For a partition λ = (λ 1 , . . . , λ m ), recall from Theorem 1.8 that t(λ) = (t 1 , . . . , t m ) ∈ [0, 1] m is the finite multiset with t j := λ j λ 1 for 1 ≤ j ≤ m. By Definition 1.7, the corresponding distance multiset is which is the d th cumulant of the rescaled uniform sum S ∆t(λ) /λ 1 .
Proof. Note that Divide through by λ d 1 and consider the upper bound. Setting t j := λ j λ 1 for 1 ≤ j ≤ m, we find Summing over all 1 ≤ i < j ≤ m and considering both bounds gives It follows that The result now follows from (22) and (33). □ We use the results on generalized uniform sum distributions from Section 3 to characterize convergence in distribution in the next theorem. It is a more explicit statement of Theorem 1.8.
Theorem 4.5. Let λ denote an infinite sequence of partitions, with ℓ(λ) < m and |λ| = n. If n m → ∞ in such a way that λ 1 /m 3 → ∞, then for each fixed d ∈ Z ≥2 , the standardized cumulants are approximately Furthermore, we can characterize convergence in distribution when it occurs.
by the definition of the hat-operation (8). By (33), B d d | ∆t(λ)| d d is the d th cumulant for the uniform sum random variable S ∆t(λ) .
By the Method of Moments/Cumulants (Theorem 2.4) together with its converse in this context (Lemma 2.7), the sequence X λ;m [rank] * converges in distribution to some X if and only if the limit of the standardized cumulants (κ λ;m d ) * → κ X d < ∞ for each d ≥ 1, which happens if and only if κ and m (N ) = N → ∞. Thus X (2 N −1 ,2 N −2 ,...,1);N [rank] is asymptotically normal by Theorem 4.5(ii).

4.3.
Limit laws based on distinct values in λ and the weft statistic. We now describe a very general test for asymptotic normality of X λ;m [rank] based on a new statistic we call weft in analogy with aft for standard Young tableaux. This test depends on the number of distinct values in a partition, so we switch to exponential notation. Note, throughout the rest of this section k will denote the number of distinct values in λ.
Lemma 4.9. Take a partition λ = (λ 1 , . . . , λ m ) = ℓ e 1 1 · · · ℓ e k k . Then, uniformly for all d ≥ 2, Proof. Observe that we may restrict the sum in (45) to just the indices with λ i ̸ = λ j . Hence, we group the terms according to the distinct values λ i = ℓ a and λ j = ℓ b for 1 ≤ a < b ≤ k.
We use the cumulant formula in (48) to prove the asymptotic normality result. Write x ab := (ℓ a − ℓ b )e a e b and y ab := ℓ a − ℓ b − 1 + e a + · · · + e b for 1 ≤ a < b ≤ k. By (48), we have for all d ≥ 2 even Thus, as N → ∞ this statistic goes to infinity, so X δ N ;N [rank] is asymptotically normal by Theorem 4.10.
The characterization in Theorem 4.10 is powerful enough to prove asymptotic normality in many cases of interest. We will use the criteria in the next corollary to further simplify the arguments in the examples below and the applications to plane partitions. As mentioned in the introduction to this section, we can assume ℓ(λ) < m without loss of generality. We may include the case ℓ(λ) = m if desired by replacing ℓ 1 with ℓ 1 − ℓ k in the following result.
This latter bound is independent of the actual order of the f i . Consequently, 1≤i≤k e 2 i ≥ e [2] m 2 2k .
Remark 4.15. The limit shape of a randomly chosen partition of n as n → ∞ is well-known to be the curve e − π √ 6 x + e Thus, one heuristically expects X λ;m [rank] to be asymptotically normal for randomly chosen partitions. We do not attempt to make this precise.
Question 4.16. Suppose λ is a sequence of partitions with ℓ(λ) < m and k is the number of distinct parts of λ. Does k → ∞ ensure X λ;m [rank] is asymptotically normal?
4.4. Limit laws for plane partitions. We may use Corollary 4.12(ii) to deduce the complete characterization of the asymptotic limits for plane partitions in a box from the introduction.
On the other hand, if median{a, b, c} is bounded, we may suppose a ≤ b ≤ c, so that n = ab is bounded. If c → ∞, then the standardized limit distribution is IH * ab provided ab converges by Theorem 4.1(i). The result follows. □ We conclude this section by giving some sample applications of the preceding results to three natural scaling limits of partitions obtained by stretching rows and/or columns by scale factors tending to ∞. , · · · , λ m , . . . , λ m c (N ) ),

Metric spaces related to forest distributions
In this section, we consider the two q-analogs of the number of linear extensions of posets which come from trees and forests using variations on the inv and maj statistics for permutations as given by Björner-Wachs in [BW89]. Recall the background for these q-analogs from Section 2.3. As summarized in Section 1.3, we will show that the coefficients in the corresponding polynomials "generically" are asymptotically normal, but that the metric space of DUSTPAN distributions M DUST characterizes all possible limit laws in a certain degenerate regime. In particular, we prove Theorem 1.13, Theorem 1.14, and Corollary 1.17. 5.1. Generic asymptotic normality for trees and forests. Recall from Section 2.3 that for any forest P , there is an associated q-hook length polynomial and random variable X P . Here we show that the sequences of random variables X P for forests P are asymptotically normal if certain numerical conditions hold; see Theorem 1.13. This covers the "generic cases". We begin by describing a family of trees which maximize the sum of the hook lengths over all trees of rank r with n elements. We use this family of trees to identify good approximations for the cumulants corresponding with all trees.
Definition 5.1. Suppose n ∈ Z ≥1 and 1 < r ≤ n. Let H n,r be the tree obtained by starting with a rooted chain C with r elements and adding n − r elements each as children of the second-smallest node in the chain. See Figure 6. Lemma 5.2. Among all trees P with n elements and rank 1 < r ≤ n, H n,r is the unique maximizer of u∈P h u . Consequently, the degree of L P (q) is Proof. Let C be a maximal chain of P with r > 1 elements and second-smallest element y.
If P ̸ = H n,r , let x ∈ P − C be a leaf of P which is not a child of y. Let P ′ be the result of moving x to be a descendant x ′ of y, which preserves the rank and number of vertices. Since C is maximal, we can easily determine the change in the sum of the hook lengths: it increases by #{v ′ ∈ P ′ : v ′ ∈ C, v ′ ≥ y} = |C| − 1 = r − 1 and decreases by #{v ∈ P : v > x} ≤ r − 1.
This procedure always weakly increases the sum of the hook lengths and arrives at H n,r after a finite number of iterations, so the maximality claim follows. Observe the procedure strictly increases the sum of the hook lengths unless #{v ∈ P : v > x} = r − 1. In this case, let z be the unique cover of x in P . By construction, z ̸ ∈ C. After applying the procedure to x to get P ′ , applying the procedure again to all of z's children and then to z will strictly increase the sum of hook lengths. Thus, P has strictly smaller sum of hook lengths than H n,r , and the uniqueness claim follows.
For the equality in (51), we find v∈Hn,r □ Lemma 5.3. Suppose 0 ≤ α < 1 and fix d ∈ Z ≥2 even. Uniformly for all trees P with n elements and rank 1 < r ≤ αn, we have Explicitly, for a fixed d ∈ Z ≥1 , where x := 2 1−α 2 − 1 > 1, a := 1/x, and b := 1/(x + 1), so 0 < a, b < 1. For the lower bound, construct a labeling w of P by iteratively building up P as follows. Begin by labeling the root of P with 1 in w. At each step, increment all existing labels in w, pick an element of P which has not been labeled whose parent has been labeled, and label it with 1. Observe that the resulting labeling w : P → [n] is natural. Consider the quantity w(u) − h u during this procedure. When u has initially been labeled, we have w(u) − h u = 1 − 1 = 0. After u has been labeled, when adding a new vertex v, if v ≤ u then both w(u) and h u are incremented, while if v ̸ ≤ u then only w(u) in incremented. Consequently, the final value of w(u) − h u counts the number of elements v added after u such that v ̸ ≤ u. In particular, w(u) − h u ≥ 0.
Using the real numbers a, b, x defined in the statement of the lemma, let M := {u ∈ P : w(u) − h u ≥ bn}. We claim #M ≥ an. To prove the claim, suppose to the contrary that #M < an. By definition, 0 < a, b < 1. Consequently, Thus asymptotic normality follows when n−r n 11/12 → ∞, or equivalently when n − r = ω(n 11/12 ). By contrast, Theorem 1.14 classifies limit laws when n − r = o(n 1/2 ). Analyzing the possible asymptotic behavior between these extremes is still an open problem.

Degenerate forests and DUSTPAN distributions.
We now consider sequences of random variables associated to the "degenerate" trees with n − r = o(n 1/2 ). Note, n − r = o(n 1/2 ) implies r/n → 1, so these sequences are not covered by Theorem 1.13. For such trees, we give a simple numerical estimate for the cumulants in terms of multisets of elevations, and use them to characterize asymptotic normality as well as the other limiting distributions in terms of the metric space of DUSTPAN distributions M DUST .
Remark 5.6. To avoid certain redundancies, we restrict to standardized trees in the sense of Remark 2.22. As an example of behavior which is prohibited by this assumption, consider the trees H n,n−k for fixed k, which are not standardized. This sequence of trees has rank r = n − k, so lim r/n = 1 as n → ∞, and Theorem 1.13 does not apply. Indeed, it is easy to see that Therefore, X * H n,n−k has the same discrete distribution for all n > k, so the limit distribution is discrete.
On the other hand, if n − r → ∞, the length of the support of X P tends to ∞ by Remark 2.20 and Lemma 5.2. Hence each distribution appears only finitely many times in such a sequence. Moreover, since the coefficients are unimodal, any sequence X * P with n − r → ∞ cannot converge to a discrete distribution.
We begin with a series of estimates relating the cumulants κ P d to the following auxiliary combinatorial quantity on P .
Definition 5.7. Let C be a fixed maximal chain in a forest P with |C| = r. For each u ∈ P − C, define the elevation of u to be e u := #{v ∈ C : u ̸ ≤ v}. Figure 7. Let s k (P, C) be the number of elements in P − C with elevation at least k − n + r, s k (P, C) := #{u ∈ P − C : e u ≥ k − n + r}.

See
For example, if u is attached to the root of the tree which is the maximal element of C, then the elevation is e u = r − 1. If u is attached to the second-smallest element of C, then e u = 1. We see that e u = r if and only if u is not connected by a path to C. Thus, if P is a tree, then 1 ≤ e u < r, so s n−r (P, C) = n − r, and s n (P, C) = 0.
If P is a tree and C is a chain in P , then P − C is a forest so both have associated cumulants. We may relate κ P d and κ P −C d using the numbers s k (P, C) as follows.
Lemma 5.8. Let C be a maximal chain in a tree P with n elements and |C| = r. Then for each d ∈ Z ≥1 ,  Figure 7. A tree P with maximal chain C = {c 0 < c 1 < c 2 < c 3 < c 4 } and elevations of P − C labeled.
Proof. Let C = v n > v n−1 > · · · > v n−r+1 . Note that for u ∈ P − C, we have u < v k if and only if e u < k − n + r. Consequently, for all n − r < k ≤ n we have h v k = k − n + r + #{u ∈ P − C : u < v k } = k − n + r + #{u ∈ P − C : e u < k − n + r} = k − n + r + (n − r − #{u ∈ P − C : e u ≥ k − n + r}) = k − s k . Therefore, (53) follows from the cumulant formula in Corollary 2.21. □ If P is a standardized tree with maximal chain C of size |C| = r > 1, it has an element u ∈ P − C with e u = r − 1, so e u /r ∼ 1 for r large. As we saw in Section 3.1, renormalizing a multiset by the maximum value is a useful technique while not changing the corresponding standardized general uniform sum distribution. Consequently, we consider the re-scaled multiset of elevations e/r = {e u /r : u ∈ P − C}, which are then related to the rescaled cumulants κ P d /r d .
Lemma 5.9. Suppose we have a sequence of standardized trees P such that the number of elements n → ∞ and the rank r satisfies n − r = o(n 1/2 ), i.e. (n − r)/n 1/2 → 0. Let C be a maximal length chain in P . Then, for each d ∈ Z ≥1 , κ P −C is S t + N (0, σ) ∈ M DUST where σ := 1 − |t| 2 2 /12. In particular, the sequence of random variables X P are asymptotically normal if and only if (57) |e/r| 2 2 := u∈P −C e u r 2 → ∞.
Proof. Fix d ≥ 2 even. By hypothesis, n − r = o(n 1/2 ), so Lemma 5.9 shows that Since e/r is finite, |e/r| 2 exists, so the hat-operation is defined on e/r and e/r = e after cancellation. Hence, (56) follows from the definition of the hat-operation in (8).
By the Method of Moments/Cumulants (Theorem 2.4) together with Lemma 2.7, the sequence X * P converges in distribution to some X if and only if B d d | e| d d converges to κ X d for each d ∈ Z ≥1 . By Corollary 3.19 and the fact that | e| 2 2 = 12 by definition, this occurs if and only if e converges pointwise to some t. Therefore, by Theorem 3.32, we have t ∈ P DUST and X has the associated DUSTPAN distribution Φ(t) = S t + N (0, σ).
In particular, the limiting distribution of X * P is N (0, 1) if and only if e → 0. Now e → 0 if and only if |e/r| ∞ /|e/r| 2 = 1/|e/r| 2 → 0 since standardized trees have an element of elevation r − 1. In particular, the limit is N (0, 1) if and only if |e/r| 2 → ∞. □ Remark 5.13. We note that considering only standardized trees in Theorem 5.12 is essential for the "if and only if" conditions to hold. For example, consider a sequence of trees H n,r with maximal chain C of size r such that n → ∞ and n − r = o(n 1/2 ). Since L Hn,r (q) = [n − r + 1] q ! and n − r → ∞, X Hn,r is asymptotically normal by [Fel45]. However, we have elevation e u = 1 for all u ∈ H n,r − C. Therefore, u∈Hn,r−C (e u /r) 2 = (n − r)/r 2 → 0 rather than ∞.
Remark 5.14. One can construct sequences of standardized trees with n − r = o(n 1/2 ) where e/r converges to any prescribed finite multiset t = (t 1 ≥ t 2 ≥ · · · ≥ t m ) ∈ ℓ 2 with |t| ∞ = 1. For each N = m + 3, m + 4, . . ., let r N = N − m. To construct the tree P N , start with a chain C N = (v 0 < v 1 < · · · < v r N −1 ), and for each nonzero value 1 = t 1 ≥ t 2 ≥ · · · ≥ t m , add a child to v ⌈(rn−1)t i ⌉ . Finally, for each t i = 0, add one additional child to v 1 . As constructed n = |P N | = N , r = r N − 1 and n − r = m is constant. Since t 1 = 1 by assumption, the root of P N has at least one child so it is a standard tree. Furthermore, m = |P N − C N | so the elevation multiset of P N has exactly m elements. By construction, the multisets e/r = {e N /r N : u ∈ P N − C N } approaches t as N → ∞. Therefore, S t is the limiting distribution of X * P N . By Corollary 3.29, we know that the closure of {S t : t ∈ ℓ 2 , t is finite} is M DUST . Thus, M Forest ∪ M DUST ⊂ M Forest as claimed in Section 1.  which is (sequentially) compact. Moreover, the set of limit points of M ϵ TREE is M DUST .
Proof. By the construction in Remark 5.14, we know M ϵ TREE ⊃ M DUST , and M DUST is closed by Corollary 3.28. Furthermore, we have (|P | − rank(P ))/|P | 1/2 < |P | −ϵ → 0, so Theorem 5.12 applies. Thus, for every sequence of trees P ∈ ϵ TREE with |P | → ∞ such that the corresponding random variables X * P ∈ M ϵ TREE converge in distribution, we know the distribution must be a DUSTPAN distribution. On the other hand, for every sequence of trees P ∈ ϵ TREE such that the corresponding random variables X * P ∈ M ϵ TREE converge in distribution but |P | is bounded, we must have a subsequence where n = |P | is eventually constant. There are only a finite number of standardized trees of size n in ϵ TREE, so we can further restrict to a sequence where each P is a particular tree, in which case the limiting of X * P is itself X * P ∈ M ϵ TREE . □

Future work
In addition to the open problems mentioned in Section 1 and Question 4.16, we pose the following questions for future study.
Question 6.1. Suppose we have a sequence of standardized trees such that n → ∞ where n − r grows at least as fast as n 1/2 but no faster than n 11/12 in the sense that n − r ̸ ∈ o(n 1/2 ) and n − r ̸ ∈ ω(n 11/12 ). When is the corresponding sequence of distributions asymptotically normal? What non-normal limit laws are possible?
Question 6.2. Does weft(λ) → ∞ if and only if X λ;m [rank] is asymptotically normal? See (49). Question 6.3. Consider the set of rooted, unlabeled forests with n vertices, sampled uniformly at random. What is the expected value of the rank r, i.e. the maximum length of a path starting at a root of a tree in the forest? How does r compare to n asymptotically as n → ∞?
See [Pit94] for growth rates of the form r ≈ log n for certain random tree generation techniques. For the number of rooted, unlabeled forests with n vertices, t trees, and rank r, see [OEI20, A291336]. Thm. 3] showed that E[r] ∼ C √ n for an explicit constant C > 0 when considering rooted, unlabeled binary trees. The corresponding problem when order is imposed either by labeling the vertices (resulting in labeled trees) or by ordering the children (resulting in planar trees) is older, though the E[r] ∼ D √ n behavior is common throughout; see [BF12, p.1] for a summary and further references.
In [Swa20], the following q, t-analogue of the hook length formula (1) is given. Let (r, c) ∈ λ denote a cell in row r and column c. Then (60) [n] q ! (r,c)∈λ q r−1 + tq c−1 [h(r, c)] q is the generating function for a pair of statistics (maj, neg) on standard supertableaux of shape λ. The t = 0 case of (60) yields (1). While (60) is not literally a quotient of q-integers, it is evidently "nearly" such a quotient. Computational evidence suggests the distributions are "typically" bivariate normal with non-trivial covariance, which is strikingly similar to the distributions encountered by Kim-Lee [KL21] for (des, maj) on permutations in fixed conjugacy classes. See Figure 8 for sample data.
Question 6.4. What are the possible limiting distributions of the coefficients of the q, t-hook length formula (60)? What is the support of (60)? One referee asked the following natural question, saying "A result of this form could give a conceptual explanation for some of the results." The authors regard this as an important question, but we do not expect a simple answer.
Question 6.5. Can one give a formula for the statistic rank on SSYT λ,m as a natural sum of m natural statistics on the tableaux, and then to show that they are (asymptotically) independent, converging to the uniform law?