Further results for the capacity statistic distribution on compositions of 1s and 2s

<p>In this paper, we study additional aspects of the capacity distribution on the set <span class="math inline">ℬ<sub><em>n</em></sub></span> of compositions of <span class="math inline"><em>n</em></span> consisting of <span class="math inline">1</span>’s and <span class="math inline">2</span>’s extending recent results of Hopkins and Tangboonduangjit. Among our results are further recurrences for this distribution as well as formulas for the total capacity and sign balance on <span class="math inline">ℬ<sub><em>n</em></sub></span>. 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 <span class="math inline">ℬ<sub><em>n</em></sub></span> is briefly considered.</p>