The Lexicographic First Occurrence of a I-II-III pattern

TL;DR

This paper investigates the distribution of the first lexicographically ordered occurrence of a specific pattern (I-II-III) in random permutations, providing insights into pattern appearance timing in discrete probability.

Contribution

It introduces the study of the first occurrence of an I-II-III pattern in permutations under lexicographic order, extending understanding of pattern distribution in permutation theory.

Findings

Analysis of the expected position of the first pattern occurrence

Conditions under which the pattern never appears in permutations

Distributional properties of pattern occurrence times

Abstract

Consider a random permutation $\pi\in{\cal S}_n$. In this paper, perhaps best classified as a contribution to discrete probability distribution theory, we study the {\it first} occurrence $X=X_n$ of a I-II-III-pattern, where "first" is interpreted in the lexicographic order induced by the 3-subsets of $[n]=\{1,2,...,n\}$. Of course if the permutation is I-II-III-avoiding then the first I-II-III-pattern never occurs, and thus $\e(X)=\infty$ for each $n$; to avoid this case, we also study the first occurrence of a I-II-III-pattern given a bijection $f:{\bf Z}^+\to{\bf Z}^+$.