Enumeration of flattened $k$-Stirling permutations with respect to descents

TL;DR

This paper introduces a new class of flattened $k$-Stirling permutations, establishes a bijection with colored type $B$ set partitions, and derives generating functions and enumeration formulas for their descents.

Contribution

It provides the first enumeration and generating functions for flattened $k$-Stirling permutations, linking them to colored type $B$ partitions and weighted structures.

Findings

Derived exponential generating functions for flattened $k$-Stirling permutations.

Established bijection with colored type $B$ set partitions.

Provided formulas for permutations with small and maximum descents.

Abstract

A $k$-Stirling permutation of order $n$ is said to be "flattened" if the leading terms of its increasing runs are in ascending order. We show that flattened $k$-Stirling permutations of order $n+1$ are in bijection correspondence with a colored variant of type $B$ set partitions of $[-n,n]$, introduced by D.G.L. Wang. Using the theory of weighted labelled structures, we give the exponential generating functions of their cardinality and their descent enumerating polynomials. We also provide enumerative formulae for the number of flattened $k$-Stirling permutations of order $n$ with small number of descents and the number of flattened Stirling permutations with maximum number of descents.