Further Results for the Capacity Statistic Distribution on Compositions of 1's and 2's

TL;DR

This paper explores the capacity distribution on compositions of n with 1's and 2's, providing new recurrences, formulas, and combinatorial proofs, along with analysis of joint distributions with additional parameters.

Contribution

It introduces new recurrences and formulas for the capacity distribution on compositions of 1's and 2's, with algebraic and combinatorial proofs, and examines joint distributions with other parameters.

Findings

New recurrences for capacity distribution

Formulas for total capacity and sign balance

Analysis of joint distribution with additional parameters

Abstract

In this paper, we study additional aspects of the capacity distribution on the set $\mathcal{B}_n$ of compositions of $n$ consisting of $1$'s and $2$'s. Among our results are further recurrences for this distribution as well as formulas for the total capacity and sign balance on $\mathcal{B}_n$. We provide algebraic and combinatorial proofs of our results. We also give combinatorial explanations of some prior results where such a proof was requested. Finally, the joint distribution of the capacity statistic with two further parameters on $\mathcal{B}_n$ is briefly considered.