Pascal Triangle and Restricted Words

TL;DR

This paper explores combinatorial properties of functions related to Pascal's triangle and restricted words, providing explicit formulas and interpretations for various binomial coefficient cases.

Contribution

It introduces explicit formulas for functions based on Pascal's triangle and interprets them in terms of restricted words, extending previous work.

Findings

Derived explicit formulas for $c_1(n,k)$ in different Pascal triangle cases

Provided combinatorial interpretations in terms of restricted words

Extended analysis to functions $f_m$ and $c_m$ for $m>0$ in some cases

Abstract

We continue to investigate combinatorial properties of functions $f_m$ and $c_m$ considered in our previous papers. They depend on an initial arithmetic function $f_0$. In this paper, the values of $f_0$ are the binomial coefficients. We first consider the case when the values of $f_0$ are the binomial coefficients from a row of the Pascal triangle. The values of $f_0$ consider next are the binomial coefficients from a diagonal of the Pascal triangle. In two final cases, the values of $f_0$ are the central binomial coefficients and its adjacent neighbors. In each case, we derive an explicit formula for $c_1(n,k)$ and give its interpretation in terms of restricted words. In the first two cases, we also consider the functions $f_m$ and $c_m$, for $(m>0)$.