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Asymptotic Behaviour of the Containment of Certain Mesh Patterns

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asymptotic-behaviour-of-the-containment-of-certain-mesh-patterns-3Algebraic Topologymath.ATmath.COposed by Dejan Govc, Jason P. Smithrecorded: open · 1 machine check, unexamined

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Statement

Can occurrences of mesh patterns be used to compute the Betti numbers of permutation complexes? Or can we define a set of mesh patterns P such that if π\pi avoids P its permutation complex is contractible?

Context

Candidate 3 of the open problems stated in "Asymptotic Behaviour of the Containment of Certain Mesh Patterns", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    Problem: Literal formalization of Govc–Smith Question 5.2: for each permutation π\pi, let Y(π)Y(\pi) be its permutation complex, i.e. the simplicial complex of increasing subsequences of π\pi. Can the Betti numbers βi(Y(π))\beta_i(Y(\pi)) be computed from mesh-pattern occurrence numbers? Also, does there exist a set P\mathcal P of mesh patterns such that π\pi avoiding P\mathcal P implies Y(π)Y(\pi) is contractible?

    This is ambiguous: the intended question likely asks for a finite, natural, or non-tautological family. Under the literal wording, no such restriction is stated.

    Result: Yes, literally.

    For σ∈Sm\sigma\in S_m, define the fully shaded mesh pattern

    Eσ=(σ,{0,…,m}×{0,…,m}).E_\sigma=(\sigma,\{0,\dots,m\}\times\{0,\dots,m\}).

    Then for any permutation π\pi,

    occ⁡Eσ(π)={1,π=σ,0,otherwise.\operatorname{occ}_{E_\sigma}(\pi)= \begin{cases} 1,&\pi=\sigma,\\ 0,&\text{otherwise}. \end{cases}

    Indeed, if ∣π∣=m|\pi|=m, the only possible occurrence uses all entries, so it occurs exactly when π=σ\pi=\sigma. If ∣π∣>m|\pi|>m, any chosen mm-subsequence leaves an unchosen point, which lies in some shaded cell, so no occurrence is allowed. If ∣π∣<m|\pi|<m, occurrence is impossible.

    Thus, for every ii,

    βi(Y(π))=∑σβi(Y(σ)) occ⁡Eσ(π),\beta_i(Y(\pi))=\sum_{\sigma} \beta_i(Y(\sigma))\,\operatorname{occ}_{E_\sigma}(\pi),

    where exactly one term is nonzero, namely σ=π\sigma=\pi. Hence mesh-pattern occurrences compute all Betti numbers in the literal sense.

    Similarly, let

    P={Eσ:Y(σ) is not contractible}.\mathcal P=\{E_\sigma : Y(\sigma)\text{ is not contractible}\}.

    Then π\pi avoids P\mathcal P iff Y(π)Y(\pi) is contractible, because occ⁡Eπ(π)=1\operatorname{occ}_{E_\pi}(\pi)=1. Hence such a set of mesh patterns exists.

    This resolves only the unrestricted literal question; it is tautological and does not settle any stronger intended version requiring finite, bounded-size, effective, or structurally natural patterns.

    Citation: No prior source is needed for the above literal resolution. The question is from Dejan Govc and Jason P. Smith, “Asymptotic Behaviour of the Containment of Certain Mesh Patterns,” Discrete Mathematics 345 (2022), 112813, Question 5.2.

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    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification KNOWN

      PASS

      The argument is mathematically valid for the literal unrestricted wording. A fully shaded mesh pattern EσE_\sigma occurs in π\pi exactly when π=σ\pi=\sigma, so these occurrence counts form singleton indicators for permutations. Hence any permutation statistic, including Betti numbers of the permutation complex, is expressible from such counts, and the proposed set P\mathcal P exactly excludes noncontractible cases. This is tautological and does not address any intended finite/natural/effective version, but no such restriction appears in the supplied statement.

      Novelty assessment

      KNOWN

      Classification rationale: The main “mesh-pattern occurrences compute Betti numbers” part is already subsumed by a stronger published theorem: every permutation statistic can be expressed as a possibly infinite linear combination of classical pattern occurrence counts, and classical patterns are mesh patterns with no shaded boxes. Thus the Betti numbers of permutation complexes are included automatically. The fully shaded singleton-pattern argument for the contractibility-avoidance part is tautological and not a publishable new contribution.

      Literature check: The key existing reference is Brändén–Claesson, whose abstract explicitly states that any permutation statistic f:Sym→Cf:\mathrm{Sym}\to\mathbb C has a unique expansion as a possibly infinite linear combination of classical permutation patterns. This is stronger than the first half of Govc–Smith Question 5.2 under its literal wording. Searches for the exact Betti/permutation-complex phrasing did not reveal a later dedicated solution, but the general theorem already covers it.

      Citation: Petter Brändén and Anders Claesson, “Mesh patterns and the expansion of permutation statistics as sums of permutation patterns,” arXiv:1102.4226; Electron. J. Combin. 18(2) (2011), P5.

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