Enumeration rises according to parity in compositions

Abstract

Let $s,t$ be any numbers in $\{0,1\}$ and let $\pi ={\pi }_1{\pi }_2\cdots {\pi }_m$ be any word. We say that $i\in [m-1]$ is an $(s,t)$-parity-rise if ${\pi }_i\equiv s(\mathrm{mod}2)$, ${\pi }_{i+1}\equiv t(\mathrm{mod}2)$, and ${\pi }_i<{\pi }_{i+1}$. We denote the number of occurrences of $(s,t)$-parity-rises in $\pi$ by ${\text{rise}}_{s,t}(\pi )$. Also, we denote the total sizes of the $(s,t)$-parity-rises in $\pi$ by ${\text{size}}_{s,t}(\pi )$, that is, ${\text{size}}_{s,t}(\pi )=\sum _{{\pi }_i<{\pi }_{i+1}}({\pi }_{i+1}–{\pi }_i).$ A composition $\pi ={\pi }_1{\pi }_2\cdots {\pi }_m$ of a positive integer $n$ is an ordered collection of one or more positive integers whose sum is $n$. The number of summands, namely $m$, is called the number of parts of $\pi$. In this paper, by using tools of linear algebra, we found the generating function that counts the number of all compositions of $n$ with $m$ parts according to the statistics ${\text{rise}}_{s,t}$ and ${\text{size}}_{s,t}$, for all $s,t$.