Encoding labelled $p$-Riordan graphs by words and pattern-avoiding permutations

TL;DR

This paper introduces $p$-Riordan words to encode $p$-Riordan graphs and explores their representations via pattern-avoiding permutations and balanced words, providing new insights and alternative proofs for known enumerative results.

Contribution

It defines $p$-Riordan words and demonstrates their use in encoding $p$-Riordan graphs, including special cases with pattern-avoiding permutations and balanced words, advancing combinatorial graph representations.

Findings

Encoding of $p$-Riordan graphs via $p$-Riordan words

Alternative encoding for Riordan and oriented Riordan graphs

New proof of enumeration of closed walks in the cube

Abstract

The notion of a $p$-Riordan graph generalizes that of a Riordan graph, which, in turn, generalizes the notions of a Pascal graph and a Toeplitz graph. In this paper we introduce the notion of a $p$-Riordan word, and show how to encode $p$-Riordan graphs by $p$-Riordan words. For special important cases of Riordan graphs (the case $p=2$) and oriented Riordan graphs (the case $p=3$) we provide alternative encodings in terms of pattern-avoiding permutations and certain balanced words, respectively. As a bi-product of our studies, we provide an alternative proof of a known enumerative result on closed walks in the cube.