bibkey: hoffmanrizzoloslivken2015pattern authors: Christopher Hoffman, Douglas Rizzolo, and Erik Slivken year: 2015 title: “Pattern-avoiding permutations and Brownian excursion, Part II: Fixed points” doi: 10.48550/arXiv.1506.04174 url: https://arxiv.org/abs/1506.04174v1 claim: “Theorem 1.1 gives fixed-point scaling limits for uniform 231- and 123-avoiding permutations, implying the 231 derangement proportion tends to 0 and the 123 proportion tends to 9/16.” strata_touched: [] license: citation-only triage: anchor

Fixed points in 123- and 231-avoiding permutations

Hoffman, Rizzolo, and Slivken’s Theorem 1.1(a) states that the fixed-point counting measure of a uniform 231-avoiding permutation, after multiplication by n^(-1/4), converges weakly to (2^(7/4) * sqrt(pi))^(-1) * e(t)^(-3/2) dt. Its total mass is positive almost surely. If N_n is the number of fixed points, then n^(-1/4) * N_n converges in distribution to a random variable X with P(X = 0) = 0. The closed-set Portmanteau bound therefore gives limsup P(N_n = 0) <= P(X = 0) = 0, so the 231-avoidance derangement proportion tends to zero.

Theorem 1.1(b) states that the centered fixed-point counting measure of a uniform 123-avoiding permutation converges weakly to A * delta_{-e(1/2)/2} + B * delta_{e(1/2)/2}, where A and B are independent Bernoulli random variables with parameter 1/4. Taking total mass gives N_n converging in distribution to A + B. Since these variables are integer-valued, the continuity interval (-1/2, 1/2) yields P(N_n = 0) -> P(A + B = 0) = (3/4)^2 = 9/16.

This is prior literature that settles Vatter’s 2026 Conjecture 4.1 and Question 4.2. It is not a new repository theorem and has no formal D5 declaration attached to it.

Verified locator

• arXiv: https://arxiv.org/abs/1506.04174v1
• HTML: https://arxiv.org/html/1506.04174v1, Theorem 1.1(a)-(b)
• DOI: https://doi.org/10.48550/arXiv.1506.04174
• Submitted June 12, 2015 (version v1)