Permutations avoiding 312 and another pattern, Chebyshev polynomials and longest increasing subsequences

TL;DR

This paper investigates the longest increasing subsequence lengths in permutations avoiding the pattern 312 and another pattern, providing exact and asymptotic formulas, especially for monotone and length-four patterns, linking to Chebyshev polynomials.

Contribution

It offers new exact and asymptotic formulas for LIS in pattern-avoiding permutations, extending understanding for specific pattern classes.

Findings

Exact formulas for LIS lengths in certain pattern-avoiding permutations

Asymptotic behavior characterized for large permutations

Connections established with Chebyshev polynomials

Abstract

We study the longest increasing subsequence problem for random permutations avoiding the pattern $312$ and another pattern $\tau$ under the uniform probability distribution. We determine the exact and asymptotic formulas for the average length of the longest increasing subsequences for such permutation classes specifically when the pattern $\tau$ is monotone increasing or decreasing, or any pattern of length four.