Binary compositions and semi-Pell compositions

TL;DR

This paper introduces semi-Pell and semi-m-Pell compositions, explores their bijections with weakly unimodal m-ary compositions, and presents generating functions, bijective proofs, and surprising congruences.

Contribution

It defines new composition classes related to semi-Fibonacci partitions and establishes their combinatorial properties and connections.

Findings

Bijection between semi-Pell compositions and weakly unimodal m-ary compositions

Derived generating functions for these compositions

Discovered unexpected congruences involving these objects

Abstract

In analogy with the semi-Fibonacci partitions studied recently by Andrews, we define semi-Pell compositions and semi-$m$-Pell compositions. We find that these are in bijection with certain weakly unimodal $m$-ary compositions. We give generating functions, bijective proofs, and a number of unexpected congruences for these objects.