Overview
This research introduces a generalization of descent polynomials, focusing on permutations incorporating a specific subset of inversions referred to as $\mathbf{h}$-inversions. The study demonstrates that this generalized count, denoted as $\mathcal{I}_\mathbf{h}(S;n)$, maintains a polynomial structure. Three distinct explicit expansions for $\mathcal{I}_\mathbf{h}(S;n)$ are presented, and the coefficients for two of these expansions are shown to be log-concave. Additionally, the work defines a graded generalization of this concept.
Research Context
The foundation of this work lies in the established concept of descent polynomials. The descent polynomial $\mathcal{D}(I;n)$ is defined as the enumerator for the number of permutations within the symmetric group $S_n$ that possess a specific descent set $I$, where $I$ is a finite subset of positive integers.
Approach
The researchers generalized the concept of descent polynomials by introducing a condition based on a specific subset of common inversions, termed $\mathbf{h}$-inversions. This involves a sequence $\mathbf{h} = (\mathbf{h}(1), \mathbf{h}(2), \ldots )$ which is weakly increasing and consists of positive integers, with the additional constraint that $\mathbf{h}(i) \ge i$. The mathematical object under investigation is $\mathcal{I}_\mathbf{h}(S;n)$, which counts permutations in $S_n$ that include a specified subset $S$ of these $\mathbf{h}$-inversions.
Findings
- The count $\mathcal{I}_\mathbf{h}(S;n)$, which represents the number of permutations in $S_n$ with a specific subset $S$ of common $\mathbf{h}$-inversions, is a polynomial.
- Three explicit expansions for $\mathcal{I}_\mathbf{h}(S;n)$ were derived.
- The coefficients for two of these explicit expansions were proven to be log-concave.
- A graded generalization of this concept was defined.