In 2013, Zhai proved that most numerical semigroups of a given genus have depth at most \(3\) and that the number \(n_g\) of numerical semigroups of a genus \(g\) is asymptotic to \(S\varphi^g\), where \(S\) is some positive constant and \(\varphi \approx 1.61803\) is the golden ratio. In this paper, we prove exponential upper and lower bounds on the factors that cause \(n_g\) to deviate from a perfect exponential, including the number of semigroups with depth at least \(4\). Among other applications, these results imply the sharpest known asymptotic bounds on \(n_g\) and shed light on a conjecture by Bras-Amorós (2008) that \(n_g \geq n_{g-1} + n_{g-2}\). Our main tools are the use of Kunz coordinates, introduced by Kunz (1987), and a result by Zhao (2011) bounding weighted graph homomorphisms.

Mathematics Subject Classifications: 20M14, 05A15, 05A16

Keywords: Numerical semigroup, genus, Kunz coordinate, graph homomorphism

The classical hook length formula counts the number of standard tableaux of straight shapes. In 1996, Okounkov and Olshanski found a positive formula for the number of standard Young tableaux of a skew shape. We prove various properties of this formula, including three determinantal formulas for the number of nonzero terms, an equivalence between the Okounkov-Olshanski formula and another skew tableaux formula involving Knutson-Tao puzzles, and two \(q\)-analogues for reverse plane partitions, which complement work by Stanley and Chen for semistandard tableaux. We also give several reformulations of the formula, including two in terms of the excited diagrams appearing in a more recent skew tableaux formula by Naruse. Lastly, for thick zigzag shapes we show that the number of nonzero terms is given by a determinant of the Genocchi numbers and improve on known upper bounds by Morales-Pak-Panova on the number of standard tableaux of these shapes.

Mathematics Subject Classifications: 05A15, 05A15, 05A19, 05A05