Enumeration rises according to parity in compositions

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 \( \pi_i \equiv s \pmod{2} \), \( \pi_{i+1} \equiv t \pmod{2} \), 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 \( s, t \).