The alternating run polynomials of permutations

2019/04/25 by Ma, Shi-Mei, Ma, Jun, Yeh, Yeong-Nan
#05A05 #05A15 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1904.11437

Abstract

In this paper, we first consider a generalization of the David-Barton identity which relate the alternating run polynomials to Eulerian polynomials. By using context-free grammars, we then present a combinatorial interpretation of a family of q-alternating run polynomials. Furthermore, we introduce the definition of semi-gamma-positive polynomial and we show the semi-gamma-positivity of the alternating run polynomials of dual Stirling permutations. A connection between the up-down run polynomials of permutations and the alternating run polynomials of dual Stirling permutations is established.

Related

• Determinantal representations of alternating run polynomials
• Alternating Eulerian polynomials and left peak polynomials
• A q-analog of the Stirling-Eulerian Polynomials
• The 1/k-Eulerian polynomials and k-Stirling permutations
• Eulerian polynomials, perfect matchings and Stirling permutations of the second kind