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

For $τ\in S_3$, let $S_n(τ)$ denote the set of permutations in $S_n$ which avoid the pattern $τ$, and let $E_n^τ$ denote the expectation with respect to the uniformly random probability measure on $S_n(τ)$. Let $\mathcal{I}_n(σ)$ denote the number of inversions in $σ\in S_n$. We study $E_n^τ\mathcal{I}_n$ for $τ\in\{231,132,213,312\}\subset S_3$. 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 $E_n^{231}\mathcal{I}_n=E_n^{312}\mathcal{I}_n\sim\frac{\sqrtπ}2n^\frac32.$ We also show that the variance $\text{Var}_{P_n^τ}(\mathcal{I}_n)$ of $\mathcal{I}_n$ under $P_n^τ$ satisfies $\text{Var}_{P_n^τ}(\mathcal{I}_n)\sim (\frac56-\frac\pi4)n^3\approx 0.048n^3,\ \text{for}\ τ\in\{231,132,213,312\}.$

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