Crucial and bicrucial permutations with respect to arithmetic monotone patterns

TL;DR

This paper investigates permutations avoiding certain arithmetic monotone patterns, establishing the existence of arbitrarily long crucial and bicrucial permutations, and determining their minimal lengths.

Contribution

It proves the existence of long crucial and bicrucial permutations for any k, l ≥ 3, and determines their minimal lengths.

Findings

Existence of arbitrarily long (k,l)-crucial and bicrucial permutations.

Minimal length of (k,l)-crucial permutations is max(k,l)*(min(k,l)-1).

Minimal length of (k,l)-bicrucial permutations is at most twice that of crucial permutations.

Abstract

A pattern $\tau$ is a permutation, and an arithmetic occurrence of $\tau$ in (another) permutation $\pi=\pi_1\pi_2...\pi_n$ is a subsequence $\pi_{i_1}\pi_{i_2}...\pi_{i_m}$ of $\pi$ that is order isomorphic to $\tau$ where the numbers $i_1<i_2<...<i_m$ form an arithmetic progression. A permutation is $(k,\ell)$-crucial if it avoids arithmetically the patterns $12... k$ and $\ell(\ell-1)... 1$ but its extension to the right by any element does not avoid arithmetically these patterns. A $(k,\ell)$-crucial permutation that cannot be extended to the left without creating an arithmetic occurrence of $12... k$ or $\ell(\ell-1)... 1$ is called $(k,\ell)$-bicrucial. In this paper we prove that arbitrary long $(k,\ell)$-crucial and $(k,\ell)$-bicrucial permutations exist for any $k,\ell\geq 3$. Moreover, we show that the minimal length of a $(k,\ell)$-crucial permutation is $\max(k,\ell)(\min(k,\ell)-1)$, while the minimal length of a $(k,\ell)$-bicrucial permutation is at most $2\max(k,\ell)(\min(k,\ell)-1)$, again for $k,\ell\geq3$.