Block patterns in Stirling permutations

TL;DR

This paper introduces block patterns in Stirling permutations, providing a framework to compute generating functions for pattern occurrences, with applications to Wilf equivalence, Bessel polynomials, and labeled trees.

Contribution

It develops a general method for analyzing block patterns in Stirling permutations and connects these patterns to various combinatorial objects and polynomials.

Findings

Derived generating functions for block pattern occurrences

Established Wilf equivalence results for block patterns

Provided new interpretations of Bessel polynomials

Abstract

We introduce and study a new notion of patterns in Stirling and $k$-Stirling permutations, which we call block patterns. We prove a general result which allows us to compute generating functions for the occurrences of various block patterns in terms of generating functions for the occurrences of patterns in permutations. This result yields a number of applications involving, among other things, Wilf equivalence of block patterns and a new interpretation of Bessel polynomials. We also show how to interpret our results for a certain class of labeled trees, which are in bijection with Stirling permutations.