Bijections and congruences involving lattice paths and integer compositions

Abstract

We prove new bijections between different variants of Dyck paths and integer compositions, which give combinatorial explanations of their simple counting formula $4^{n-1}$. These give relations between different statistics, such as the number of crossings of the $x$-axis in classes of Dyck bridges or the distribution of peaks in classes of Dyck paths, and furthermore relate them with $k$- and $g$-compositions. These allow us to find and prove congruence results for Dyck paths and parity results for compositions. Our investigation uncovers unexpected connections to mock theta functions, Hardinian arrays, little Schr\"oder paths, Fibonacci numbers, and irreducible pairs of compositions, offering new insights into the structures of paths, partitions and compositions.