Cells of fixed height in Catalan words and restricted growth functions

Abstract

A word $w=w_1{w}_2\cdots w_n$ of length n over the set of positive integers is called a Catalan word whenever $w_1=1$ and $1\le w_k\le w_{k-1}+1$ for $k=2,3,\dots ,n$. A restricted growth function is defined as a word $w=w_1{w}_2\cdots w_n$ of length n over the set of positive integers where $w_1=1$ and for $k\ge 2$ we have $1\le w_k\le \max \{w_1,w_2,\cdots ,w_{k-1}\}+1$. We also define cells and heights of cells and we represent such words as bargraphs (otherwise known as polyominoes) where the ith column contains $w_i$ cells for $1\le i\le n$ and where all columns have their bottom cell on the x-axis. In the case of Catalan words, we prove a relationship between the number of cells at different heights and first terms of the expanded polynomial ${(1+x)}^{2n}$. In the case of restricted growth functions we find polynomials $F_n\left(x\right)$ where the coefficient of $x^j$ counts the number of cells of height j across all rgfs with n parts. In this case we also find bivariate generating functions for rgfs with k blocks, where the generating functions tracks the number of cells at a given height as well as the number of parts.