Abstract: Publication date: March 2019Source: Advances in Applied Mathematics, Volume 104Author(s): Veronika Pillwein In this note we provide further evidence for a conjecture of Gillis, Reznick, and Zeilberger on the positivity of the diagonal coefficients of a multivariate rational function. Kauers had proven this conjecture for up to 6 variables using computer algebra. We present a variation of his approach that allows us to prove positivity of the coefficients up to 17 variables using symbolic computation.

Abstract: Publication date: March 2019Source: Advances in Applied Mathematics, Volume 104Author(s): Carolyn Chun, James Oxley Let M be a 3-connected binary matroid; M is internally 4-connected if one side of every 3-separation is a triangle or a triad, and M is (4,4,S)-connected if one side of every 3-separation is a triangle, a triad, or a 4-element fan. Assume M is internally 4-connected and that neither M nor its dual is a cubic Möbius or planar ladder or a certain coextension thereof. Let N be an internally 4-connected proper minor of M. Our aim is to show that M has a proper internally 4-connected minor with an N-minor that can be obtained from M either by removing at most four elements, or by removing elements in an easily described way from a special substructure of M. When this aim cannot be met, the earlier papers in this series showed that, up to duality, M has a good bowtie, that is, a pair, {x1,x2,x3} and {x4,x5,x6}, of disjoint triangles and a cocircuit, {x2,x3,x4,x5}, where M\x3 has an N-minor and is (4,4,S)-connected. We also showed that, when M has a good bowtie, either M\x3,x6 has an N-minor and M\x6 is (4,4,S)-connected; or M\x3/x2 has an N-minor and is (4,4,S)-connected. In this paper, we show that, when M\x3,x6 has no N-minor, M has an internally 4-connected proper minor with an N-minor that can be obtained from M by removing at most three elements, or by removing elements in a well-described way from a special substructure of M. This is a final step towards obtaining a splitter theorem for the class of internally 4-connected binary matroids.

Abstract: Publication date: March 2019Source: Advances in Applied Mathematics, Volume 104Author(s): Sandra Di Rocco, Christian Haase, Benjamin Nill Generalizing the famous Bernstein–Kushnirenko theorem, Khovanskĭi proved in 1978 a combinatorial formula for the arithmetic genus of the compactification of a generic complete intersection associated to a family of lattice polytopes. Recently, an analogous combinatorial formula, called the discrete mixed volume, was introduced by Bihan and shown to be nonnegative. By making a footnote of Khovanskĭi in his paper explicit, we interpret this invariant as the (motivic) arithmetic genus of the non-compact generic complete intersection associated to the family of lattice polytopes.

Abstract: Publication date: February 2019Source: Advances in Applied Mathematics, Volume 103Author(s): Mitchell Lee, Ashwin Sah Let π∈Sm and σ∈Sn be permutations. An occurrence of π in σ as a consecutive pattern is a subsequence σiσi+1⋯σi+m−1 of σ with the same order relations as π. We say that patterns π,τ∈Sm are strongly c-Wilf equivalent if for all n and k, the number of permutations in Sn with exactly k occurrences of π as a consecutive pattern is the same as for τ. In 2018, Dwyer and Elizalde [6] conjectured (generalizing a conjecture of Elizalde [8] from 2012) that if π,τ∈Sm are strongly c-Wilf equivalent, then (τ1,τm) is equal to one of (π1,πm), (πm,π1), (m+1−π1,m+1−πm), or (m+1−πm,m+1−π1). We prove this conjecture using the cluster method introduced by Goulden and Jackson in 1979 [12], which Dwyer and Elizalde used to prove that π1−πm = τ1−τm . A consequence of our result is the full classification of c-Wilf equivalence for a special class of permutations, the non-overlapping permutations. Our approach uses analytic methods to approximate the number of linear extensions of the “cluster posets” defined by Elizalde and Noy in 2012 [11].

Abstract: Publication date: February 2019Source: Advances in Applied Mathematics, Volume 103Author(s): Trevor K. Karn, Max D. Wakefield Restricted Whitney numbers of the first kind appear in the combinatorial recursion for the matroid Kazhdan–Lusztig polynomials. In the special case of braid matroids (the matroid associated to the partition lattice, the complete graph, the type A Coxeter arrangement and the symmetric group) these restricted Whitney numbers are Stirling numbers of the first kind. We use this observation to obtain a formula for the coefficients of the Kazhdan–Lusztig polynomials for braid matroids in terms of sums of products of Stirling numbers of the first kind. This results in new identities between Stirling numbers of the first kind and Stirling numbers of the second kind, as well as a non-recursive formula for the braid matroid Kazhdan–Lusztig polynomials.