Laurent polynomials, Eulerian numbers, and Bernstein's theorem

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:124

Article

Liu, Ricky Ini

关键词: Eulerian number;    Bernstein's theorem;    Hypersimplex;   

DOI  :  10.1016/j.jcta.2014.02.003

来源: Elsevier

【 摘 要 】

Erman, Smith, and Varilly-Alvarado (2011) showed that the expected number of doubly monic Laurent polynomials f (z) = z(-m) + a(-m+1)z(-m+1) + ... + a(n-1)z(n-1) + z(n) whose first m + n - 1 powers have vanishing constant term is the Eulerian number < m+n-1 m-1 >, as well as a more refined result about sparse Laurent polynomials. We give an alternate proof of these results using Bernstein's theorem that clarifies the connection between these objects. In the process, we show that a refinement of Eulerian numbers gives a combinatorial interpretation for volumes of certain rational hyperplane sections of the hypercube. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free