Exact formula and asymptotic behavior for the expected number of inversions in a random permutation avoiding a pattern of length three

TL;DR

This paper derives exact formulas and asymptotic behaviors for the expected number of inversions in permutations avoiding certain patterns of length three, revealing growth rates and variance estimates.

Contribution

It provides explicit formulas and asymptotic analysis for the expected inversions in pattern-avoiding permutations of length three, a novel contribution in permutation pattern research.

Findings

Expected inversions grow as approximately 0.8862 * n^{3/2} for certain patterns.

Variance of inversions scales as about 0.048 * n^3.

Explicit formulas relate expectations for different pattern avoidances.

Abstract

For $\tau\in S_3$, let $S_n(\tau)$ denote the set of permutations in $S_n$ which avoid the pattern $\tau$, and let $E_n^\tau$ denote the expectation with respect to the uniformly random probability measure on $S_n(\tau)$. Let $\mathcal{I}_n(\sigma)$ denote the number of inversions in $\sigma\in S_n$. We study $E_n^\tau\mathcal{I}_n$ for $\tau\in\{231,132,213,312\}\subset S_3$. We prove that $$ E_n^{231}\mathcal{I}_n=E_n^{312}\mathcal{I}_n=\frac12\frac{n!(n+1)!4^n}{(2n)!}-\frac12(3n+1), $$ and that $$ E_n^{132}\mathcal{I}_n=E_n^{213}\mathcal{I}_n=\frac12(n-1)n-E_n^{231}\mathcal{I}_n. $$ From the first equation it follows that $$ E_n^{231}\mathcal{I}_n=E_n^{312}\mathcal{I}_n\sim\frac{\sqrt\pi}2n^\frac32. $$ We also show that the variance $\text{Var}_{P_n^{\tau}}(\mathcal{I}_n)$ of $\mathcal{I}_n$ under $P_n^\tau$ satisfies $$ \text{Var}_{P_n^{\tau}}(\mathcal{I}_n)\sim (\frac56-\frac\pi4)n^3\approx 0.048n^3,\ \text{for}\ \tau\in\{231,132,213,312\}. $$