SHELLING THE m = 1 AMPLITUHEDRON

. The amplituhedron A n,k,m was introduced by Arkani-Hamed and Trnka [AHT14] in order to give a geometric basis for calculating scattering amplitudes in planar N = 4 supersymmetric Yang–Mills theory. It is a projection inside the Grassmannian Gr k,k + m of the totally nonnegative part of Gr k,n . Karp and Williams [KW19] studied the m = 1 amplituhedron A n,k, 1 , giving a regular CW decomposition of it. Its face poset R n,l (with l := n − k − 1) consists of all projective sign vectors of length n with exactly l sign changes. We show that R n,l is EL-shellable, resolving a problem posed in [KW19]. This gives a new proof that A n,k, 1 is homeomorphic to a closed ball, which was originally proved in [KW19]. We also give explicit formulas for the f -vector and h -vector of R n,l , and show that it is rank-log-concave and strongly Sperner. Finally, we consider a related poset P n,l introduced by Machacek [Mac], consisting of all projective sign vectors of length n with at most l sign changes. We show that it is rank-log-concave, and conjecture that it is Sperner.


Let Gr ≥0
k,n denote the totally nonnegative Grassmannian [Pos07,Lus94], comprised of all kdimensional subspaces of R n whose Plücker coordinates are nonnegative.Motivated by the physics of scattering amplitudes, Arkani-Hamed and Trnka [AHT14] introduced a generalization of Gr ≥0 k,n , called the (tree) amplituhedron and denoted A n,k,m (Z).It is defined as the image of Gr ≥0 k,n under (the map induced by) a linear surjection Z : R n → R k+m whose (k + m) × (k + m) minors are all positive.While the definition of A n,k,m (Z) depends on the choice of Z, it is expected that its geometric and combinatorial properties only depend on n, k, and m.The amplituhedron may be regarded as a generalization of both a cyclic polytope (which we obtain when k = 1) and the totally nonnegative Grassmannian Gr ≥0 k,n (which we obtain when k + m = n).
Karp and Williams posed the problem [KW19, Problem 6.19] of showing that the poset R n,l is shellable.We resolve this problem: Theorem 1.1.The poset R n,l with a minimum and a maximum adjoined is EL-shellable.
The motivation behind [KW19, Problem 6.19] was the following.Karp and Williams showed that the m = 1 amplituhedron A n,k,1 (Z) is a regular CW complex which can be identified with the bounded complex of a certain generic arrangement of n hyperplanes in R k (namely, a cyclic arrangement).It then follows from a general result of Dong [Don08] that A n,k,1 (Z) is homeomorphic to a k-dimensional closed ball.Karp and Williams observed that rather than appealing to [Don08], one could reach the same conclusion by showing that the face poset R n,l is shellable, using a result of Björner [Bjö84,Proposition 4.3(c)].(This relies on the regular CW decomposition, along with the fact that every cell of codimension one is contained in the closure of at most two maximal cells.)Therefore, as a consequence of Theorem 1.1, we obtain a new proof that A n,k,1 (Z) is homeomorphic to a closed ball: Corollary 1.2 ([KW19, Corollary 6.18]).The m = 1 amplituhedron A n,k,1 (Z) is homeomorphic to a k-dimensional closed ball.
We expect that for any m ≥ 1, the amplituhedron A n,k,m (Z) has a shellable regular CW decomposition and is homeomorphic to a closed ball, thereby generalizing the situation which holds when n = k + m.Indeed, in this case A n,k,n−k (Z) is the totally nonnegative Grassmannian Gr ≥0 k,n ; Williams [Wil07] showed that the face poset of Gr ≥0 k,n is EL-shellable, and Galashin, Karp, and Lam [GKL22b,GKL22a] showed that Gr ≥0 k,n is a regular CW complex homeomorphic to a closed ball.See Remark 2.10 for further discussion of related work.In the case we consider here, m = 1, we make use of the explicit description of the face poset R n,l of a cell decomposition of A n,k,m (Z).No such description is known as yet for general m.For work in this direction, see [KWZ20,EZLT] for the case m = 4, and [ Luk, BH, LPW] for the case m = 2.
Another consequence of Theorem 1.1 is that the poset R n,l has a nonnegative h-vector.In particular, by a result of Björner [Bjö80] and Stanley [Sta72], h i equals the number of maximal chains of R n,l with exactly i descents with respect to the EL-labeling of Theorem 1.1 (see Theorem 3.5).We give an alternative description of the h-vector using generating functions (see Theorem 3.14), which is explicit but non-positive.
We observe that when l = 0, the poset R n,l is the Boolean algebra B n (consisting of all subsets of {1, . . ., n} ordered by containment) with the minimum removed.Maximal chains of R n,0 correspond to permutations of {1, . . ., n} with the usual notion of descent, and h i is the Eulerian number n i (see Proposition 3.9).Therefore the h-vector of R n,l provides a generalization of the Eulerian numbers.
Two further well-known properties of the Boolean algebra B n are that its rank sizes form a log-concave sequence and that it is strongly Sperner (see e.g.[Eng97]).We show that R n,l also has these properties: Figure 1.The Hasse diagram of the poset R 3,1 , with elements labeled as sign vectors (left) and as tuples of sets (right).
Theorem 1.3.The poset R n,l is rank-log-concave.It also admits a normalized flow, and hence is strongly Sperner.
Finally, we consider a poset closely related to R n,l , denoted P n,l , introduced by Machacek [Mac].It consists of projective sign vectors of length n with at most (rather than exactly) l sign changes, under the relation (1.1).For example, P 3,1 is depicted in Figure 7.The poset P n,l can be regarded as a quotient of the face poset of a certain simplicial complex B(l, n) studied by Klee and Novik [KN12].Notice that P n,l also specializes to B n when l = 0. Machacek [Mac] showed that the order complex of P n,l is a manifold with boundary which is homotopy equivalent to RP l , and homeomorphic to RP n−1 when l = n − 1.Although Pn,l is not shellable in general, Bergeron, Dermenjian, and Machacek [BDM20] showed that when l is even or l = n − 1, the order complex of P n,l is partitionable.This is a weaker property which still implies that the h-vector is nonnegative, and they showed that the h-vector counts certain type-D permutations with respect to type-B descents.
We prove that P n,l is rank-log-concave (see Theorem 5.2), and we conjecture that it is Sperner (see Conjecture 5.3).We prove this conjecture when l equals 0, 1, or n − 1 by constructing a normalized flow (see Proposition 5.4).
The remainder of this paper is organized as follows.In Section 2 we give some background on poset topology and prove Theorem 1.1 (see Theorem 2.8).In Section 3 we consider the f -vector and h-vector of R n,l .In Section 4 we give background on unimodality, log-concavity, and the Sperner property, and prove Theorem 1.3.In Section 5 we consider the poset P n,l .
Acknowledgments.We thank Isabella Novik and Bruce Sagan for helpful comments, and anonymous reviewers for their valuable feedback.
We assume the reader has some familiarity with posets; we refer to [Sta12,Wac07] for further background.We use to denote cover relations in a poset, i.e., x y if and only if x < y and there does not exist z such that x < z < y.
Definition 2.1.Let P be a finite poset.We say that P is graded (or pure) if every maximal chain has the same length d, which we call the rank of P .Definition 2.2.Let P be a poset.We define the bounded extension as the poset P obtained from P by adjoining a new minimum 0 and a new maximum 1.
We now recall the definition of an EL-labeling, due to Björner [Bjö80, Definition 2.1].We slightly modify the original definition, following Wachs [Wac07]; see [Wac07, Remark 3.2.5]for further discussion.
Definition 2.3 ([Wac07, Definition 3.2.1]).Let P be a finite graded poset.An edge labeling of P is a function λ from the set of edges of the Hasse diagram of P (i.e. the cover relations of P ) to a poset (Λ, ).An increasing chain is a saturated chain x 0 x 1 • • • x r in P whose edge labels strictly increase in Λ: We call λ an EL-labeling of P if the following properties hold for every closed interval [x, y] in P : (EL1) there exists a unique increasing maximal chain C 0 in [x, y]; and (EL2) if x z ≤ y such that z = x 1 , where x x 1 is the first edge of C 0 , then λ(x x 1 ) ≺ λ(x z). 2jörner showed that a finite graded poset with an EL-labeling is shellable [Bjö80, Theorem 2.3].
2.2.Edge labeling.We now study the bounded extension Rn,l of R n,l .Recall that R n,l consists of all projective sign vectors of length n with exactly l sign changes, under the relation (1.1).We begin by giving an alternative definition of R n,l .
We observe that the bounded extension Rn,l of R n,l is graded.Explicitly, the minimum 0 has rank 0, the maximum 1 has rank n−l +1, and (A 1 , . . ., A l+1 ) has rank We now divide the cover relations of R n,l into two types; see Remark 2.7 for motivation.
Definition 2.5.Let 0 ≤ l < n, and let x = (A 1 , . . ., A l+1 ) ∈ R n,l .Note that the elements of R n,l which cover x are precisely those that can be obtained from it by adding some element a ∈ (We take the inequality above to be a < min(A 2 ) when i = 1, and max(A l ) < a when i = l + 1.) We say that such a cover relation is of type α if a < max(A i ), and of type β if a > max(A i ).
Remark 2.7.We were led to the construction in Definition 2.6 in part so that the following property holds (though we will not end up using it).Let x ∈ Rn,l such that x is not covered by 1, and let y 1 , . . ., y r be the elements of Rn,l which cover x, ordered so that the labels of x y 1 , . . ., x y r are increasing in (Λ n,l , ).Then y 1 , . . ., y r are in increasing order in the lexicographic order on (l + 1)-tuples.For example, see Figure 3.In fact, one can show that ordering the atoms of [x, 1] lexicographically for all x ∈ Rn,l defines a recursive atom ordering of Rn,l (see e.g.[Wac07, Section 4.2]), where the order of the atoms of [x, 1] does not depend on a choice of maximal chain of [ 0, x].Li [Li21, Lemma 1.1] showed that any finite, bounded, and graded poset admitting such a recursive atom ordering is EL-shellable, so this provides an alternative way to prove Theorem 1.1.We omit the proof of this fact, and instead find it simplest to work only with the edge labeling in Definition 2.6.
Theorem 2.8.The edge labeling of Rn,l in Definition 2.6 is an EL-labeling.
Proof.We must verify that (EL1) and (EL2) hold for every closed interval [x, y] in Rn,l .We consider four cases, depending on whether x = 0 and y = 1.When x = 0 we write x = (A 1 , . . ., A l+1 ), and when y = 1 we write y = (B 1 , . . ., B l+1 ).In each case, we explicitly describe the unique maximal chain of [x, y], thereby proving (EL1).It will then be apparent from the form of this maximal chain that (EL2) holds.
Case 1: x = 0, y = 1.The maximal chains of [x, y] are obtained by adding, in some order, all the elements of B i \ A i to the ith block (for i ∈ [l + 1]).The unique increasing chain is given by adding these elements in the following order: • for i = 1, . . ., l + 1, we add the elements of B i \ A i which are less than max(A i ) to the ith block, in increasing order (in cover relations of type α); • for i = l + 1, . . ., 1, we add the elements of B i \ A i which are greater than max(A i ) to the ith block, in increasing order (in cover relations of type β).We see that (EL2) holds.
Case 2: x = 0, y = 1.The first edge of any maximal chain of [ 0, y] is labeled by an element of [n]  l+1 , and so if it is increasing, after the first edge it must pass through edges only of type β.Therefore the unique increasing maximal chain of [ 0, y] begins with the edge 0 ({b 1 }, . . ., {b l+1 }), where b i := min(B i ) for i ∈ [l + 1] (whence (EL2) is satisfied), and after that follows the unique increasing chain from ({b 1 }, . . ., {b l+1 }) to y, as in Case 1.
Case 3: x = 0, y = 1.The last edge of any maximal chain of [x, 1] is labeled by (β, l + 1, n + 1), and so if it is increasing, before the final edge it must pass through edges only of type α or with a label (β, l + 1, * ).Therefore the unique increasing maximal chain of [x, 1] ends with the edge (C 1 , . . ., C l+1 ) 1, where and before that follows the unique increasing chain from x to (C 1 , . . ., C l+1 ), as in Case 1.We see that (EL2) holds.
Remark 2.9.There are results in the literature which imply that various special families of posets are shellable.However, as far as we know, Rn,l is not contained in such a family.For example, Provan and Billera [PB80, Section 3.4.2]showed that all distributive lattices (cf.[Sta12, Section 3.4]) are shellable.While Rn,l is a lattice, it is not distributive unless l = 0 (in which case R n,l is the Boolean algebra B n with the minimum removed) or l = n − 1 (in which case R n,l has a single element).For example, one can see from Figure 2 that R3,1 is not distributive.Also, Björner [Bjö80, Theorem 3.1] showed that all semimodular lattices (cf.[Sta12, Section 3.3]) are shellable.However, Rn,l is not upper-semimodular unless l = 0 or l = n − 1, and Rn,l is not lower-semimodular unless l = 0, l = n − 1, or (n, l) = (3, 1).For example, one can see from Figure 2 that R3,1 is not upper-semimodular.We omit the proofs of these claims.
Remark 2.10.Recall that R n,l is the face poset of the amplituhedron A n,k,m (Z) when m = 1.Another interesting special case of A n,k,m (Z) is n = k + m, whence it becomes the totally nonnegative Grassmannian Gr ≥0 k,n .Williams [Wil07] and Bao and He [BH21, Theorem 4.1] showed that the face poset of Gr ≥0 k,n with a minimum 0 adjoined is EL-shellable, and Knutson, Lam, and Speyer [KLS13, Section 3.5] showed that the face poset (without 0 adjoined) is dual EL-shellable.We point out that R n,l with 0 adjoined (but not 1) is an induced subposet of the face poset of Gr ≥0 k,n with 0 adjoined [KW19, Theorem 5.17], and so it is EL-shellable by [Wil07,BH21].Therefore the main difficulty in proving Theorem 1.1 is in dealing with the adjoined maximum 1.Our EL-labeling of Rn,l does not use the labelings of [Wil07, KLS13, BH21], and it is not clear to us how our labeling is related to these.We plan to study this further in future work.

f -vector and h-vector
In this section we examine the f -vector and h-vector of R n,l , as well as their refinements by ranks, namely the flag f -vector and flag h-vector.We give a combinatorial interpretation for the h-vector in terms of the EL-labeling of Section 2.2, and also prove explicit formulas for the f -vector and h-vector.
3.1.Background.We refer to [Sta96,Sta12] for background on the f -vector and h-vector.Definition 3.1 ([Sta12, Section 3.13]).Let P be a finite graded poset of rank d − 1, with ranks labeled from 1 to d.For S ⊆ [d], we let α S be the number of chains of P supported exactly at the ranks in S; we call α the flag f -vector of P .We also define the flag h-vector β of P by Alternatively, let P S denote the induced subposet of P consisting of all elements whose rank lies in S. Then α S is the number of maximal chains of P S , and (−1) |S|+1 β S is the Möbius invariant µ( PS ) of the bounded extension of P S .
In particular, P and P have the same (flag) h-vector, and the (flag) f -vector of P is easily determined from P .Therefore enumerative results for R n,l apply as well to Rn,l , and viceversa.Keeping this connection in mind, we will label the ranks of R n,l from 1 to n − l (rather than from 0 to n − l − 1).

Combinatorial interpretations. Björner [Bjö80, Theorem 2.7], based on work of
Stanley [Sta72, Theorem 1.2], gave a combinatorial interpretation for the flag h-vector of any poset with an edge labeling satisfying (EL1).We state it here in the special case of Rn,l , with the edge labeling defined in Definition 2.6.Definition 3.4.Given a maximal chain 0 = x 0 x 1 • • • x n−l x n−l+1 = 1 of Rn,l , we say that i ∈ [n − l] is a descent of C when λ(x i−1 x i ) λ(x i x i+1 ). 4heorem 3.5 (Björner and Stanley; cf.[Sta12, Theorem 3.14.2] 5 ).Recall the edge labeling of Rn,l in Definition 2.6.For all S ⊆ [n − l], we have that β S equals the number of maximal chains of Rn,l with descent set S. Thus for all 0 ≤ d ≤ n − l, we have that h d equals the number of maximal chains of Rn,l with exactly d descents.
We also have the following explicit description of all the maximal chains of R n,l (and hence also Rn,l ): difference for our edge labeling of Rn,l , since the label set Λ n,l is totally ordered and no label is repeated in any maximal chain.
5 Our conventions differ slightly from those in [Sta12], since in (EL1) we require edge labels to strictly (rather than weakly) increase.Nevertheless, the result [Sta12, Theorem 3.14.2]and its proof transfer easily to our setting.
Proof.We can verify that the map (A, π) → C(A, π) gives a bijection from [n+l]  2l+1 × S n−l−1 to the set of maximal chains of R n,l .
While Proposition 3.7 gives a simple description of the maximal chains of Rn,l , we are not able in general to translate Definition 2.6 into a simple description of the descents of C(A, π) in terms of A and π.However, in the special case l = 0, we do have such a simple description: maximal chains correspond to permutations of [n] with the usual notion of descent, as we now explain.Definition 3.8.Given π ∈ S n , we say that r ∈ [n − 1] is a descent of π if π(r) > π(r + 1).For 0 ≤ d ≤ n, we define the Eulerian number n d as the number of permutations in S n with exactly d descents.
For example, 3 1 = 4, corresponding to the permutations (in one-line notation) 132, 213, 231, and 312.We refer to [Pet15] for further details about Eulerian numbers.Proposition 3.9.There is a bijection between maximal chains of Rn,0 and permutations in S n which preserves descent sets.In particular, by Theorem 3.5, we have Proof.The bijection sends the permutation π ∈ S n to the maximal chain We can verify that the notions of descent in Definition 2.6 and Definition 3.8 agree.

Explicit formulas.
We now turn to giving explicit formulas for the f -vector and hvector of R n,l .
Proposition 3.10.The flag f -vector of R n,l is given by Proof.We enumerate the chains x 1 < • • • < x d of R n,l supported at ranks r 1 , . . ., r d as follows.Write x 1 = (A 1 , . . ., A l+1 ) and After relabeling the set [n], we may assume that Let s i denote the size of A i (for 1 ≤ i ≤ l + 1), so that s i ≥ 1 and For 1 and set b i := max(B i ).Then the a i,j 's and b i 's are arbitrary elements of [l + r d ] subject to The number of ways to choose the a i,j 's and b i 's is at which point x 1 and x d are fixed.Finally, the elements x 2 , . . ., x d−1 are determined by a set composition of (B .
We now use Proposition 3.10 to give a formula for the f -vector of R n,l .The following formula allows us to simplify the resulting sum, at the cost of introducing minus signs.
Proof.Corollary 3.12.Let 0 ≤ l < n and 0 ≤ d ≤ n − l − 1.The number of chains of R n,l of length d which begin at rank r and end at rank r + s equals Then f d is given by summing the quantity above over all r ≥ 1 and s ≥ 0 (or alternatively s ≥ d).
Proof.This follows from Proposition 3.10, using Lemma 3.11.
Example 3.13.Taking d = 0 in Corollary 3.12, we obtain the number of elements of R n,l : Finally, we use Corollary 3.12 to obtain the generating functions for the f -and h-vectors: Theorem 3.14.The generating functions for the f -and h-vectors of R n,l are given by We then obtain an explicit formula (albeit with negative signs) for h i by taking the coefficient of t i in H(t).
Proof.By Corollary 3.12 (replacing r − 1 by r), and then writing d = i + j and applying the negative binomial theorem, we obtain This proves the first equation.The second equation follows by applying (3.1).
Example 3.15.Let us set l = 0 in Theorem 3.14 to obtain the generating function for the h-vector of R n,0 : where we applied the binomial theorem twice.This yields a well-known generating function for the Eulerian numbers [Pet15, (1.10)], in agreement with Proposition 3.9.
Example 3.16.Let us use Theorem 3.14 to find h 1 for R n,l , by taking the coefficient of t in H(t): We can compute the latter sum using the identity the first equality above follows from the binomial theorem, and the second equality follows from the product rule for the derivative.Setting u = 1 gives For example, when l = 0 we obtain n 1 = h 1 = 2 n − n − 1.

Normalized flow
In this section we prove Theorem 1.3, which states that R n,l is rank-log-concave and strongly Sperner.We prove the former in Proposition 4.3, and the latter in Theorem 4.7 using Harper's notion of a normalized flow [Har74].
Definition 4.1.Let s = (s 1 , . . ., s d ) be a sequence of nonnegative real numbers.We say that s is unimodal if for some 1 ≤ j ≤ d, we have We say that s is log-concave if One can verify that if s is a log-concave sequence of nonnegative real numbers and has no internal zeros, then s is unimodal.We also observe that the entry-wise product of two log-concave sequences is log-concave.Definition 4.2.Let P be a finite graded poset of rank d − 1, with ranks labeled from 1 to d.For 1 ≤ r ≤ d, the rth Whitney number of the second kind W r is defined to be the number of elements of P of rank r.In terms of the flag f -vector, we have W r = α {r} .We say that P is rank-unimodal (respectively, rank-log-concave) if the sequence (W 1 , . . ., W r ) is unimodal (respectively, log-concave).We observe that if P is rank-log-concave, then it is rank-unimodal.
For example, from Figure 1 we see that for R 3,1 , we have (W 1 , W 2 ) = (3, 2).For general R n,l , we can read off W r from Proposition 3.10: Proposition 4.3.The Whitney numbers of the second kind of R n,l (with ranks labeled from 1 to n − l) are In particular, R n,l is rank-log-concave.
Proof.The formula for W r follows by taking S = {r} in Proposition 3.10.The sequence (W 1 , . . ., W n−l ) is log-concave because it is the entry-wise product of the log-concave sequences We now introduce the (strongly) Sperner property and normalized flows.Recall that an antichain in a poset is a subset of pairwise incomparable elements.Definition 4.4.Let P be a finite graded poset of rank d − 1, with ranks labeled from 1 to d.Given j ≥ 1, we say that P is j-Sperner if the maximum size of a union of j antichains is realized by taking the j largest ranks, i.e., for all antichains A 1 , . . ., A j ⊆ P.
We say that P is Sperner6 if P is 1-Sperner, and we say that P is strongly Sperner if P is j-Sperner for all j ≥ 1. Harper [Har74,Theorem p. 55] showed that if P admits a normalized flow, then P is strongly Sperner.In fact, it follows from work of Kleitman [Kle74] (see [Eng97, Theorem 4.5.1]) that such a P satisfies the stronger LYM inequality: ≤ 1 for all antichains A.
4.2.Construction of the normalized flow.We define a normalized flow on R n,l .Our definition will manifestly satisfy (NF1), and we will then check carefully that (NF2) holds.
Definition 4.6.Let 0 ≤ l < n.We define an edge labeling f on R n,l , with label set R ≥0 , as follows.Let x = (A 1 , . . ., A l+1 ) ∈ R n,l , and let a ∈ ).Consider all elements y x obtained from x by adding a to some block; there are exactly 1 or 2 such y.
There is a unique such y if and only if a < max(A 1 ) or a > min(A l+1 ), in which case, we set f (x y) := 1.
Otherwise, we have max(A i ) < a < min(A i+1 ) for some 1 ≤ i ≤ l.We can add a either to the ith block or to the (i + 1)th block, forming, say, y 1 and y 2 , respectively.We then set Note that in either case, given x and a, the sum of f (x y) over all y obtained from x by adding a to some block equals 1.For example, see Figure 5 and Figure 6.
Theorem 4.7.The edge labeling of R n,l in Definition 4.6 is a normalized flow.In particular, R n,l is strongly Sperner.
Then by construction, we have which is positive and depends only on r.Therefore (NF1) holds.Now we prove (NF2).Let y ∈ R n,l have rank r + 1, and write y = (B 1 , . . ., B l+1 ).Let  removing some element b of B i .The value f (x y) is determined according to the following three cases: l+r .The sum of the values f (x y), over all b in all three cases above (with y and i fixed), equals Note that this formula also gives the desired sum (i.e.0) when s i = 1.Therefore we obtain x, x y which is positive and depends only on r.This completes the proof.

The poset P n,l
In this section we consider the poset P n,l .Recall that P n,l is the poset of projective sign vectors of length n with at most l sign changes, under the relation (1.1) (see Figure 7).
It is natural to ask which properties of R n,l carry over to P n,l .First we consider shellability.Since P n,0 = R n,0 , by e.g.Theorem 1.1, we have that Pn,0 is EL-shellable.We can also verify directly that P2,1 is EL-shellable.We claim that in the remaining cases, Pn,l is not shellable.Indeed, if it were shellable, then the order complex of P n,l would be homeomorphic to a sphere or a closed ball of dimension n − 1 [Bjö84, Proposition 4.3].On the other hand, Machacek [Mac] showed that the order complex of P n,l is homotopy equivalent to RP l , which is homeomorphic to the sphere S 1 when l = 1, and is not homotopy equivalent to a sphere or a closed ball when l ≥ 2.
We now show that P n,l , like R n,l , is rank-log-concave.We will use the following lemma, which appeared in talk slides of Mani [Man09].We give a proof following an argument of Semple and Welsh [SW08, Example 2.2], who showed that a similar sequence is log-concave.The sequence (W 1 , . . ., W n ) is log-concave, i.e., P n,l is rank-log-concave.
Proof.The set of elements of P n,l is the disjoint union of R n,i for 0 ≤ i ≤ l, where rank s of R n,i appears in P n,l in rank s + i. .
We can verify that the first sequence is log-concave, and the second sequence is log-concave by Lemma 5.1.Therefore (W 1 , . . ., W n ) is log-concave.
We conjecture that P n,l , like R n,l , is Sperner: Conjecture 5.3.For 0 ≤ l < n, the poset P n,l is Sperner.
We have verified that Conjecture 5.3 holds for all 0 ≤ l < n ≤ 8.We also show that it holds when l equals 0, 1, or n − 1: Proposition 5.4.The posets P n,0 , P n,1 , and P n,n−1 admit a normalized flow, and hence are strongly Sperner.
Proof.For P n,0 = R n,0 , this follows from Theorem 4.7.For P n,n−1 , the constant function 1 is a normalized flow.This is because P n,n−1 is biregular, i.e., any two elements of P n,n−1 of the same rank have the same up-degree and the same down-degree in the Hasse diagram.
Finally, we construct a normalized flow f on P n,1 , similar to the one defined on R n,1 in Definition 4.6.Let x ∈ P n,1 , and let a ∈ [n] such that x a = 0. Consider the elements covering x obtained by changing entry a to either + or −; there are exactly one or two of them.If there is one such element, say y, we set f (x y) := 1.If there are two such elements, say y 1 and y 2 , we set f (x y i ) := 1 2 for i = 1, 2. Then if x has rank r (with 1 ≤ r ≤ n − 1), there are exactly n − r possible values of a, so y, x y f (x y) = n − r.This is positive and depends only on r, which proves (NF1).Now we verify that (NF2) holds.Let 1 ≤ r ≤ n − 1, and let y ∈ P n,l have rank r + 1.Given a ∈ [n] such that y a = 0, let x y be obtained from y by changing entry a to 0, and let z be the sign vector obtained from y by flipping entry a (from + to − or vice versa).If z has at most one sign change, then f (x y) = 1 2 , while if z has at least two sign changes, then f (x y) = 1.We observe that the first case occurs for exactly 2 values of a, while the second case occurs for the remaining r − 1 values of a. Therefore

Lemma 5. 1 .= n r l i=0 r − 1 i for 1
Let l ∈ N. Then the sequence (s 1 , s 2 , . . . ) is log-concave, wheres r := l i=0 r − 1 i for r ≥ 1.Proof.We must show that s r+1 s r+3 ≤ s 2 r+2 for r ≥ 0. Using Pascal's identity n i = Therefore we can rewrite the inequality s r+1 s r+3 ≤ s 2 i ≤ l.Theorem 5.2.Let 0 ≤ l < n.The Whitney numbers of the second kind of P n,l (with ranks labeled from 1 to n) are W r ≤ r ≤ n.
Therefore by Proposition 4.3, we haveW r (P n,l ) = min(l,r−1) i=0 W r−i (R n,i ) = n r l i=0 r − 1 i for 1 ≤ r ≤ n.This proves the formula for W r .Now note that (W 1 , . . ., W n ) is the product of the two sequences
The edge labeling of R3,1 in Definition 2.6.
This follows from [Sta12, (1.94a)], since both sides equal d!S(s, d), where S(s, d) is a Stirling number of the second kind.Alternatively, we can prove this directly from the inclusion-exclusion principle [Sta12, Theorem 2.1.1].
Note that the elements x y are precisely those obtained from y by selecting some block i (1 ≤ i ≤ l + 1) with s i ≥ 2, and