Interlacing polynomials and the veronese construction for rational formal power series

Abstract

Fixing a positive integer r and 0 ≤ k ≤ r − 1, define f r,k for every formal power series f as f (x) = f r,0 (x r ) + xf r,1 (x r ) + · · · + x r−1 f r,r−1 (x r ). Jochemko recently showed that the polynomial U n r,k h(x) := ((1 + x + · · · + x r−1 ) n h(x)) r,k has only nonpositive zeros for any r ≥ deg h(x) − k and any positive integer n. As a consequence, Jochemko confirmed a conjecture of Beck and Stapledon on the Ehrhart polynomial h(x) of a lattice polytope of dimension n, which states that U n r,0 h(x) has only negative, real zeros whenever r ≥ n. In this paper, we provide an alternative approach to Beck and Stapledon's conjecture by proving the following general result: if the polynomial sequence. Our result has many other interesting applications. In particular, this enables us to give a new proof of Savage and Visontai's result on the interlacing property of some refinements of the descent generating functions for colored permutations. Besides, we derive a Carlitz identity for refined colored permutations.

Refined Eulerian numbers and ballot permutations

“…The close form of the generating function for {A(n, d, j)} in (2.2) doesn't seem to be seen elsewhere. The conclusion could also be obtained by generating function calculation through Zhang's lemma in [36]:…”

Face numbers of uniform triangulations of simplicial complexes

“…Proof. This is because, by [15,Theorem 3.1], barycentric subdivision transforms any h-polynomial with nonnegative coefficients to one with only real nonpositive roots and r-fold edgewise subdivision (for instance, by [14,Theorem 4.5.6] or [30,Corollary 3.4]) transforms any h-polynomial with only real nonpositive roots to one with the same property.…”

“…x(g(x)) r,r−i , where g(x) := c n−1,r (x) A n (x) = (1 + x + • • • + x r−1 ) n−1 A n (x). Since A n (x) is real-rooted, an application of [30,Corollary 3.4] shows that the sequence (g(x)) r,r−i 1≤i≤r is interlacing. Therefore, h F (σ n , x) − h F (∂σ n , x) is real-rooted and the proof follows.…”

On the Real-Rootedness of the Local $h$-Polynomials of Edgewise Subdivisions

“…The local h-polynomial ℓ V (2 V ) r , x can also be interpreted combinatorially as the ascent generating function of certain Smirnov words [14], see [3,Theorem 4.6] for more details. We would like to point out that the realrootedness of E r ( (1 + x + x 2 + · · · + x r−1 ) n ), which is slightly different from the right hand side of (1), has been studied in [11,19] and references therein.…”