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Pattern-Avoiding Polytopes

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pattern-avoiding-polytopes-10Algebraic Topologymath.ATmath.COposed by Robert Davis, Bruce Saganrecorded: open · 1 machine check, unexamined

1 attempt · 1 machine check · no person has looked

Statement

What are the homotopy types of Qn(Π)Q_{n}(\Pi) ? (in general their order complexes aren't necessarily spheres, or even Cohen-Macaulay)

Context

Candidate 10 of the open problems stated in "Pattern-Avoiding Polytopes", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
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    NEW

    Problem: Reconstructed statement: for Π\Pi a set of permutation patterns, let

    Av⁡n(Π)={σ∈Sn:σ avoids every pattern in Π},\operatorname{Av}_n(\Pi)=\{\sigma\in S_n:\sigma\text{ avoids every pattern in }\Pi\},

    and let Qn(Π)Q_n(\Pi) be the induced subposet of strong Bruhat order on SnS_n with ground set Av⁡n(Π)\operatorname{Av}_n(\Pi). The question asks for the possible homotopy types of the order complexes Δ(Qn(Π))\Delta(Q_n(\Pi)). This matches the source wording because it mentions Bruhat-order posets and their order complexes.

    There is a mild ambiguity: if one restricts Π\Pi to fixed bounded-length patterns independent of nn, the question is different. Under the literal general formulation, allowing arbitrary finite Π\Pi, the answer is: all finite simplicial homotopy types occur.

    Result: The homotopy types of Δ(Qn(Π))\Delta(Q_n(\Pi)), as nn and finite Π\Pi vary, are exactly the homotopy types of finite simplicial complexes.

    Proof. Let KK be any finite simplicial complex. Let PP be its face poset of nonempty faces. Then Δ(P)\Delta(P) is the barycentric subdivision of KK, hence homeomorphic to KK.

    We embed PP as an induced subposet of Bruhat order. Enumerate P={p1,…,pm}P=\{p_1,\dots,p_m\}. For x∈Px\in P, define its principal ideal

    D(x)={i:pi≤Px}⊆[m].D(x)=\{i:p_i\le_P x\}\subseteq [m].

    Then x≤Py  ⟺  D(x)⊆D(y)x\le_P y\iff D(x)\subseteq D(y).

    In S2mS_{2m}, let sj=(j,j+1)s_j=(j,j+1), and for J⊆[m]J\subseteq[m] set

    wJ=∏j∈Js2j−1.w_J=\prod_{j\in J}s_{2j-1}.

    The factors commute. By the subword criterion for Bruhat order,

    wI≤BrwJ  ⟺  I⊆J,w_I\le_{\mathrm{Br}} w_J \iff I\subseteq J,

    because a reduced word for wJw_J contains exactly the commuting generators s2j−1s_{2j-1} with j∈Jj\in J. Hence x↦wD(x)x\mapsto w_{D(x)} embeds PP as an induced Bruhat subposet of S2mS_{2m}.

    Let

    A={wD(x):x∈P}⊆S2m,Π=S2m∖A.A=\{w_{D(x)}:x\in P\}\subseteq S_{2m}, \qquad \Pi=S_{2m}\setminus A.

    Since an element of S2mS_{2m} contains a pattern of length 2m2m only if it is itself that pattern,

    Av⁡2m(Π)=A.\operatorname{Av}_{2m}(\Pi)=A.

    Therefore Q2m(Π)≅PQ_{2m}(\Pi)\cong P, and so

    Δ(Q2m(Π))≅Δ(P)≅sd⁡K.\Delta(Q_{2m}(\Pi))\cong \Delta(P)\cong \operatorname{sd} K.

    Thus every finite simplicial homotopy type occurs. Conversely, every Qn(Π)Q_n(\Pi) is finite, so its order complex is a finite simplicial complex. This gives the exact classification.

    Degenerate case: the empty complex is realized by taking A=∅A=\varnothing, e.g. Π=Sn\Pi=S_n.

    Citation: No external resolution is needed; the proof above is self-contained apart from the standard Bruhat subword criterion.

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    0 human reviews · 1 machine check

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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 TYPE1

      PASS

      For the literal unrestricted interpretation, the argument is complete: allowing Π\Pi to contain length-nn patterns lets one realize any chosen subset A⊆SnA\subseteq S_n, since length-nn pattern containment in SnS_n is equality. The embedding of any finite face poset into a Boolean Bruhat subposet via commuting simple transpositions is valid, so Δ(Qn(Π))\Delta(Q_n(\Pi)) can realize the barycentric subdivision of any finite simplicial complex. Conversely all such order complexes are finite.

      Caveat: this does not answer a stricter intended version where Π\Pi is fixed/bounded-length/natural. If Qn(Π)Q_n(\Pi) is defined using right weak order rather than strong Bruhat order, the same commuting-generator construction still works. I found no prior published general resolution in the available search.

      Novelty assessment

      TYPE1

      Classification rationale: The accepted resolution is valid only under the literal unrestricted reading where Π\Pi may depend on nn and contain length-nn patterns. Then Av⁡n(Π)\operatorname{Av}_n(\Pi) can be made any chosen subset of SnS_n, so the construction is essentially a standard universality trick: embed a finite face poset into a Boolean lattice, and embed that Boolean lattice into Bruhat order via commuting simple reflections. This is mathematically correct but does not address the intended natural/fixed-pattern version of the open question. It is too routine for a standalone paper; at most it is a clarifying remark or caveat.

      Literature check: I found no prior source explicitly stating this exact “all finite simplicial homotopy types occur” conclusion for Qn(Π)Q_n(\Pi). Searches around “Pattern-Avoiding Polytopes,” “Qn(Π)Q_n(\Pi) homotopy,” “pattern-avoiding Bruhat order,” “order complex,” and “Cohen-Macaulay” led back to the Davis–Sagan material and unrelated/special follow-up work, not to a general homotopy classification. The ingredients, however, are standard: face posets give barycentric subdivisions, finite posets embed in Boolean lattices by principal ideals, and Boolean lattices occur in Bruhat order from commuting simple reflections.

      Citation: No exact prior citation found. Relevant background: Robert Davis and Bruce Sagan, Pattern-Avoiding Polytopes, arXiv:1609.01782; Bruce Sagan, Pattern-Avoiding Polytopes and Bruhat Orders II slides; A. Björner and F. Brenti, Combinatorics of Coxeter Groups, Springer, 2005, for the Bruhat subword criterion.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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