Water Cells in Compositions of 1s and 2s

TL;DR

This paper investigates compositions of integers into parts of 1 and 2 with water cells, deriving generating functions, recurrences, and connections to Riordan arrays, thus expanding understanding of restricted compositions.

Contribution

It introduces new combinatorial results on compositions with water cells, including generating functions, recurrences, and links to Riordan arrays, for compositions with parts 1 and 2.

Findings

Derived generating functions for compositions with water cells.

Established recurrences within the array of counts.

Connected the compositions to Riordan arrays.

Abstract

Mansour and Shattuck introduced the notion of water cells for integer compositions in 2018. We focus on compositions with parts restricted to 1 and 2 and consider the array of counts for such compositions of $n$ with $k$ water cells, establishing generating functions for the columns and diagonal sums, recurrences within the array in the spirit of Pascal's lemma, and connections to other restricted compositions. Most of our proofs are combinatorial, but we also make connections to Riordan arrays.