A q-analog of the Stirling-Eulerian Polynomials

2024/04/15 by Yao Dong, Dong, Yao, Zhicong Lin +3 · 2 citations
Agricultural and Biological Sciences · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Botanical Research and Chemistry #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2404.09777

openalex publication_date 2024/04/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In 1974, Carlitz and Scoville introduced the Stirling-Eulerian polynomial A_n(x,y|α,β) as the enumerator of permutations by descents, ascents, left-to-right maxima and right-to-left maxima. Recently, Ji considered a refinement of A_n(x,y|α,β), denoted P_n(u₁,u₂,u₃,u₄|α,β), which is the enumerator of permutations by valleys, peaks, double ascents, double descents, left-to-right maxima and right-to-left maxima. Using Chen's context-free grammar calculus, Ji proved a formula for the generating function of P_n(u₁,u₂,u₃,u₄|α,β), generalizing the work of Carlitz and Scoville. Ji's formula has many nice consequences, one of which is an intriguing γ-positivity expansion for A_n(x,y|α,β). In this paper, we prove a q-analog of Ji's formula by using Gessel's q-compositional formula and provide a combinatorial approach to her γ-positivity expansion of A_n(x,y|α,β).

Cited by

• A bi-Stirling-Euler-Mahonian polynomial
• Symmetric decompositions and Euler-Stirling statistics on Stirling permutations

Related

• The (α,β)-Eulerian Polynomials and Descent-Stirling Statistics on Permutations
• The Gessel Correspondence and the Partial γ-Positivity of the Eulerian Polynomials on Multiset Stirling Permutations
• The Binomial-Stirling-Eulerian Polynomials
• Eulerian polynomials, Stirling permutations and increasing trees
• Enumeration of permutations by the parity of descent positions