Recurrence relations for patterns of type $(2,1)$ in flattened permutations

TL;DR

This paper develops recurrence relations and explicit formulas for counting specific permutation patterns in flattened permutations, revealing equalities in average pattern occurrences and pattern avoidance counts.

Contribution

It introduces a unified recurrence-based approach to enumerate and analyze pattern distributions in flattened permutations, including explicit formulas and pattern avoidance counts.

Findings

Derived recurrence relations for pattern distributions in flattened permutations.

Obtained explicit formulas for pattern avoidance and average occurrence counts.

Discovered equalities in average pattern occurrences and avoidance counts among different patterns.

Abstract

We consider the problem of counting the occurrences of patterns of the form $xy-z$ within flattened permutations of a given length. Using symmetric functions, we find recurrence relations satisfied by the distributions on $\mathcal{S}_n$ for the patterns 12-3, 21-3, 23-1 and 32-1, and develop a unified approach to obtain explicit formulas. By these recurrences, we are able to determine simple closed form expressions for the number of permutations that, when flattened, avoid one of these patterns as well as expressions for the average number of occurrences. In particular, we find that the average number of 23-1 patterns and the average number of 32-1 patterns in $\text{Flatten}(\pi)$, taken over all permutations $\pi$ of the same length, are equal, as are the number of permutations avoiding either of these patterns. We also find that the average number of 21-3 patterns in $\text{Flatten}(\pi)$ over all $\pi$ is the same as it is for 31-2 patterns.