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

TL;DR

This work analyzes the capacity statistic on $\mathcal{B}_n$, the set of compositions of $n$ into $1$'s and $2$'s, by deriving recurrences, explicit distribution formulas, and combinatorial proofs, and by expanding to a polynomial refinement with additional parameters. Through generating-function techniques, the authors obtain a closed form for the bivariate generating function $F(x,y)=\sum_{n\ge0}\sum_{k\ge0} w(n,k) x^n y^k$ and extract explicit formulas for $w(n,k)$ (for $k\ge1$), the total capacity $\sum_k k w(n,k)$, and the sign-balance of the capacity distribution on $\mathcal{B}_n$. They provide combinatorial proofs of prior recurrences, establish two recurrences for $b_n$, and give a detailed analysis of the joint distribution with two extra parameters. The paper further generalizes to a polynomial refinement ${\bf b}_n(y;p,q)$ with a rich generating function $F(x,y;p,q)$, yielding explicit counts $|\mathcal{B}_{n,k,j}|$ and a generalized total-capacity formula involving Fibonacci polynomials $F_n(p)$, thereby connecting capacity statistics to Fibonacci and tiling structures. Overall, the results offer a thorough combinatorial and algebraic treatment of the capacity distribution on constrained compositions and extend the framework to a multi-parameter setting with broader implications for related discrete structures.

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.

Figures (2)

• Figure 1: The bargraph of $\pi \in \mathcal{C}_{30}$ with cap($\pi$)=10, where the water cells are shaded.
• Figure 2: The bargraphs of $212 \in\mathcal{B}_{5,1}$ and of $211212,\,221112 \in \mathcal{B}_{9,3}$.

Theorems & Definitions (17)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Corollary 4
• proof
• Theorem 5
• proof