Let S_n be the set of permutations of length n, let SC_231:S_n→ S_n be the stack-sorting map whose stack avoids consecutive occurrences of 231, and let operatornameAv_n(132,231) denote the set of permutations in S_n avoiding 132 and 231 consecutively. Defant and Zheng's conjecture. For any permutation pi of length n≥ 3, SC_231^2n-4(pi)∈ operatornameAv_n(132,231). Also, for every n≥ 3, there exists tau∈ S_n for which SC_231^2n-5(tau)notin operatornameAv_n(132,231). The paper states this as a conjecture of Defant and Zheng and presents a counterexample, so the asserted universal claim is refuted; the first displayed bound is instead proved in the paper's main theorem.

Defant and Zheng's maximal-time conjecture for the consecutive-pattern-avoiding stack-sorting map

Let $S_n$ be the set of permutations of length $n$ , let $SC_{231}:S_n\to S_n$ be the stack-sorting map whose stack avoids consecutive occurrences of $231$ , and let $\operatorname{Av}_n(132,231)$ denote the set of permutations in $S_n$ avoiding $132$ and $231$ consecutively. Defant and Zheng's conjecture. For any permutation $\pi$ of length $n\geq 3$ ,

SC_{231}^{2n-4}(\pi)\in \operatorname{Av}_n(132,231).

Also, for every $n\geq 3$ , there exists $\tau\in S_n$ for which

SC_{231}^{2n-5}(\tau)\notin \operatorname{Av}_n(132,231).

The paper states this as a conjecture of Defant and Zheng and presents a counterexample, so the asserted universal claim is refuted; the first displayed bound is instead proved in the paper's main theorem.

Defant and Zheng conjectured that every permutation reaches the consecutive-pattern-avoiding set after $2n-4$ iterations, with this bound sharp. Seidel and Sun found a length- $11$ counterexample, so the universal assertion is false.

Known results

The conjecture was verified for $n\leq 9$ by Defant and Zheng.
Seidel and Sun exhibited $(4,6,8,5,11,7,2,9,10,3,1)$ , which reaches a periodic point only after $19$ iterations, whereas $2n-4=18$ .
Seidel and Sun proved the general upper bound $f(n)\leq(n-1)(n-2)$ , where $f(n)$ is the maximum sort-number.
Periodic points are exactly the permutations avoiding $231$ and its reverse consecutively, and all have period $2$ .

April 2026 computational study

A paper by Seidel and Sun computes sort-numbers through length $14$ and estimates averages through length $1000$ . It proves an $(n-1)$ lower bound and the upper bound $f(n)\leq\frac{(n+1)(n-2)}{2}$ , while reported data suggest faster-than-linear growth; the asymptotic behavior remains unclear.

Current status (as of September 2026): The $2n-4$ universal conjecture is refuted by a published counterexample, but the exact maximum sort-number and its growth rate remain open.

Defant and Zheng's maximal-time conjecture for the consecutive-pattern-avoiding stack-sorting map

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