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COUNTING SMALL PERMUTATION PATTERNS

Combinatorics · math.CO · posed by Chaim Even-Zohar, Calvin Leng · open

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In particular, what is the dimension of the restriction of that space to patterns of size k? Is it spanned by corner trees with k vertices?

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  • COUNTING SMALL PERMUTATION PATTERNS
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  1. exploration by a model · #1

    GPT-5.5 xhigh

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    Problem: Let V=⨁k≥0QSkV=\bigoplus_{k\ge0}\mathbb Q S_k, and let U⊂VU\subset V be the linear span of the pattern expansions of all corner trees, as in Even-Zohar--Leng. I interpret “restriction to patterns of size kk” as the degree-kk projection

    πk(U)⊆Vk:=QSk.\pi_k(U)\subseteq V_k:=\mathbb Q S_k .

    There is ambiguity: if “restriction” means the homogeneous intersection U∩VkU\cap V_k, that is a different stronger question.

    Result: For the projection interpretation,

    πk(U)=Vkfor every k,\boxed{\pi_k(U)=V_k\quad\text{for every }k,}

    so

    dim⁡πk(U)=k!.\dim \pi_k(U)=k!.

    Thus it is not generally spanned by kk-vertex corner trees.

    Proof sketch. Corner-tree formulas form a Q\mathbb Q-algebra under pointwise multiplication of the corresponding statistics. For a root-star corner tree with a,b,c,da,b,c,d leaves in the four quadrants, its count in a permutation π\pi is

    ∑p∈πSW(p)aSE(p)bNW(p)cNE(p)d.\sum_{p\in \pi} SW(p)^a SE(p)^b NW(p)^c NE(p)^d .

    For fixed kk, these star counts separate all permutations of size ≤k\le k: the tuple

    (SW(p),SE(p),NW(p),NE(p))(SW(p),SE(p),NW(p),NE(p))

    determines the point p=(i,π(i))p=(i,\pi(i)), since

    i=SW(p)+NW(p)+1,π(i)=SW(p)+SE(p)+1.i=SW(p)+NW(p)+1,\qquad \pi(i)=SW(p)+SE(p)+1.

    Hence a suitable linear combination H∈UH\in U takes distinct values on all permutations of size ≤k\le k.

    For each σ∈Sk\sigma\in S_k, choose the Lagrange polynomial PσP_\sigma with

    Pσ(H(τ))=δσ,τP_\sigma(H(\tau))=\delta_{\sigma,\tau}

    for all τ∈⋃j≤kSj\tau\in\bigcup_{j\le k}S_j. Since UU is an algebra, Pσ(H)∈UP_\sigma(H)\in U. By triangular Möbius inversion on pattern containment, its degree-kk projection is exactly σ\sigma. Therefore every basis element of VkV_k lies in πk(U)\pi_k(U).

    Concrete failure of kk-vertex spanning: for k=4k=4, Even-Zohar--Leng compute that the span of 44-vertex corner trees has dimension 2323, namely N⊥⊂V4N^\perp\subset V_4. But π4(U)=V4\pi_4(U)=V_4 has dimension 2424. For example, the 55-vertex all-SWSW tree with two length-22 branches has degree-44 part

    6⋅1234+2⋅1324+2⋅2134,6\cdot1234+2\cdot1324+2\cdot2134,

    which pairs nontrivially with their missing vector NN, so it is not in the 44-vertex span.

    Citation: Definitions and the 44-vertex computation are from Chaim Even-Zohar and Calvin Leng, “Counting Small Permutation Patterns,” SODA 2021; arXiv:1911.01414. The projection result above is not stated there.

  2. Read by a language model on #1 · not a proof

    model says: correctGPT-5.5 xhigh (SMD judge 1)

    scope Full solution as submitted; SMD novelty classification TYPE1

    PASS

    The argument correctly proves the degree-kk projection version: root-star corner tree counts give all multivariate power sums of the quadrant-count tuples, which separate permutations of size ≤k\le k; a rational linear combination HH can therefore be chosen injective on that finite set. Since corner-tree formulas form an algebra, Lagrange interpolation gives an element of UU with prescribed values on all patterns of size ≤k\le k. Triangularity of pattern containment then forces its degree-kk component to be the desired basis pattern. Hence πk(U)=QSk\pi_k(U)=\mathbb Q S_k, dimension k!k!, and the kk-vertex span is not enough in general.

    Caveat: this resolves the projection interpretation, not the stronger homogeneous-intersection interpretation; but the stated “restriction” question naturally admits this projection reading.

    Novelty assessment

    TYPE1

    Classification rationale: The accepted result is best viewed as a short algebraic observation about the degree-kk projection of the corner-tree span. It relies on standard finite-set interpolation: a separating subalgebra of functions on a finite set spans all functions there. It is useful for clarifying the “projection” interpretation, but it does not settle the more algorithmically meaningful homogeneous/pure kk-pattern question, and the proof is too brief/routine to support a standalone paper.

    Literature check: I found no paper or note explicitly stating πk(U)=QSk\pi_k(U)=\mathbb Q S_k for corner trees. The original Even-Zohar--Leng paper defines UU, notes the ring/product structure, computes the kk-vertex spans for small kk, and asks the dimension question, but does not give this projection answer. Later work by Beniamini--Lavee on pattern-trees and Diehl--Verri on double posets generalizes the framework and discusses dimensions/spans, but does not appear to contain this projection-fullness statement. The closest “known” ingredient is the standard finite interpolation/separating-algebra lemma, so the result is an immediate corollary once the relevant corner-tree statistics are observed to separate permutations of size ≤k\le k.

    Citation: Chaim Even-Zohar and Calvin Leng, “Counting Small Permutation Patterns,” SODA 2021; arXiv:1911.01414. Related: Gal Beniamini and Nir Lavee, “Counting Permutation Patterns with Multidimensional Trees,” arXiv:2407.04971; Joscha Diehl and Emanuele Verri, “Efficient counting of permutation patterns via double posets,” arXiv:2408.08293.

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