A proof of Lin's conjecture on inversion sequences avoiding patterns of relation triples

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:179

Article

Andrews, George E.¹  Chern, Shane¹ 

[1] Penn State Univ, Dept Math, University Pk, PA 16802 USA

关键词: Inversion sequence;    Pattern avoidance;    Relation triple;    Generating function;    Kernel method;   

DOI  :  10.1016/j.jcta.2020.105388

来源: Elsevier

【 摘 要 】

A sequence e = e(1)e(2) ... e(n) of natural numbers is called an inversion sequence if 0 <= e(i) <= i - 1 for all i is an element of{1, 2, ..., n}. Recently, Martinez and Savage initiated an investigation of inversion sequences that avoid patterns of relation triples. Let rho(1), rho(2) and rho(3) be among the binary relations {<, >, <=, >=, =, not equal, -}. Martinez and Savage defined I-n(rho(1), rho(2), rho(3)) as the set of inversion sequences of length nsuch that there are no indices 1 <= i < j < k <= n with e(i) rho(1) e(j), e(j) rho(2) e(k) and e(i) rho(3) e(k). In this paper, we will prove a curious identity concerning the ascent statistic over the sets I-n(>, not equal, >=) and I-n(=, not equal, >). This confirms a recent conjecture of Zhicong Lin. (C) 2020 Elsevier Inc. All rights reserved.

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