Stirling permutations on multisets

TL;DR

This paper introduces Stirling permutations on multisets, explores their associated Stirling polynomials, and generalizes classical Stirling numbers with combinatorial interpretations, including new types like odd and central factorial numbers.

Contribution

It develops a comprehensive theory of Stirling polynomials for multisets, generalizes classical Stirling numbers, and introduces new variants with combinatorial significance.

Findings

Derived Stirling polynomials for multisets

Generalized classical Stirling numbers with combinatorial interpretations

Introduced Stirling numbers of odd type and central factorial number generalizations

Abstract

A permutation $\sigma$ of a multiset is called Stirling permutation if $\sigma(s)\ge \sigma(i)$ as soon as $\sigma(i)=\sigma(j)$ and $i<s<j.$ In our paper we study Stirling polynomials that arise in the generating function for descent statistics on Stirling permutations of any multiset. We develop generalizations of the classical Stirling numbers and present their combinatorial interpretations. Particularly, we apply the theory of $P$-partitions. Using certain specifications we also introduce the Stirling numbers of odd type and generalizations of the central factorial numbers.