Colored q-Stirling and q-Lah numbers: A new view continued

Abstract

Cai 與 Readdy 提出了一個新的架構以研究組合結構 S 的 q-analogue f(q)。研究目的是在組合結構 S 及其真子集 S′ 上找出兩個統計量，使得 f(q) 能夠透過 S′ 的元素表示成 q-(1 + q) 的形式，並進一步探討此形式在偏序集合與拓撲方面的詮釋。依照這個架構，Cai 與 Readdy 對兩類 Stirling 數有詳細的研究成果。在本研究中，我們將 Cai 與 Readdy 的結果推廣至有顏色的兩類 q-Stirling 數以及有顏色的 q-Lah 數。另外，我們也在這個研究架構中討論 type D 的 q-Stirling 數與 q-Lah 數。Cai and Readdy proposed a new framework for studying the q-analogue f(q) of a combinatorial structure S. Specifically, the aim is to identify two statistics over S and a proper subset S ′ of S such that f(q) represents the q-(1 + q)-expansion over S′, and to explore the poset and topological interpretations of this expansion. Cai and Readdy provided comprehensive profiles for classical Stirling numbers of both kinds within this framework. In this work, we extend Cai and Readdy’s results to colored q-Stirling numbers of both kinds, as well as colored q-Lah numbers. We also briefly discuss q-Stirling and q-Lah numbers of type D.