The Eulerian distribution on involutions is indeed unimodal

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:113

Article

Guo, Victor J. W. ; Zeng, Jiang

关键词: involutions;    descent number;    unimodality;    Eulerian polynomial;    Zeilberger's algorithm;   

DOI  :  10.1016/j.jcta.2005.10.002

来源: Elsevier

【 摘 要 】

Let I-n,I-k (respectively J(n,k)) be the number of involutions (respectively fixed-point free involutions) of {1,...,n) with k descents. Motivated by Brenti's conjecture which states that the sequence I-n,(0),I-n,(1),...,I-n,(n-1) is log-concave, we prove that the two sequences I-n,I-k and J(2n,k) are unimodal in k, for all n. Furthermore, we conjecture that there are nonnegative integers a(n,k) such that (k=0)Sigma(n-1) I(n,k)t(k =) (k=0)Sigma([(n-1)/2]) a(n,k)t(k)(1+t)(n-2k-1). This statement is stronger than the unimodality of I-n,I-k but is also interesting in its own right. (c) 2005 Elsevier Inc. All rights reserved.

【 授权许可】

Free