Paired patterns in lattice paths

TL;DR

This paper investigates the occurrence of paired patterns within lattice paths from (0,0) to (n,n), analyzing their generating functions and providing insights into pattern matching in combinatorial path structures.

Contribution

It introduces a novel framework for studying paired pattern occurrences in lattice paths and derives their generating functions, advancing combinatorial pattern analysis.

Findings

Derived explicit generating functions for paired pattern matches.

Characterized the distribution of paired patterns in lattice paths.

Provided combinatorial formulas for pattern occurrence counts.

Abstract

Let $\mathcal{L}_n$ denote the set of all paths from $[0,0]$ to $[n, n]$ which consist of either unit north steps $N$ or unit east steps $E$ or, equivalently, the set of all words $L \in \{E,N\}^*$ with $n$ $E$'s and $n$ $N$'s. Given $L \in \mathcal{L}_n$ and a subset $A$ of $[n] = \{1, \ldots, n\}$, we let $ps_{L}(A)$ denote the word that results from $L$ by removing the $i^{th}$ occurrence of $E$ and the $i^{th}$ occurrence of $N$ in $L$ for all $i \in [n]-A$, reading from left to right. Then we say that a paired pattern $P \in \mathcal{L}_k$ occurs in $L$ if there is some $A \subseteq [n]$ of size $k$ such that $ps_L(A) = P$. In this paper, we study the generating functions of paired pattern matching in $\mathcal L_n$.