Combinatorics of Multicompositions

TL;DR

This paper explores the combinatorial structure of multicompositions, establishing bijections, counting formulas, and connections to Fibonacci-like sequences and classical integer sequences.

Contribution

It introduces a bijection between two types of compositions and derives new counting formulas and combinatorial proofs related to Fibonacci and classical sequences.

Findings

Bijection between colored and zero-including compositions

Counting formulas for multicompositions by parts and zeros

Combinatorial proofs of identities involving Jacobsthal and Pell sequences

Abstract

Integer compositions with certain colored parts were introduced by Andrews in 2007 to address a number-theoretic problem. Integer compositions allowing zero as some parts were introduced by Ouvry and Polychronakos in 2019. We give a bijection between these two varieties of compositions and determine various combinatorial properties of these multicompositions. In particular, we determine the count of multicompositions by number of all parts, number of positive parts, and number of zeros. Then, working from three types of compositions with restricted parts that are counted by the Fibonacci sequence, we find the sequences counting multicompositions with analogous restrictions. With these tools, we give combinatorial proofs of summation formulas for generalizations of the Jacobsthal and Pell sequences.