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Rationality, irrationality, and Wilf equivalence in generalized factor order

Combinatorics · math.CO · posed by Sergey Kitaev, Jeffrey Liese, Jeffrey Remmel, Bruce Sagan · open

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Statement

What about Wilf equivalence in [m]∗[m]^* where [m]={1,2,…,m}[m] = \{1, 2, \dots, m\}? ... Is it true that u∼mvu \sim_m v if and only if u∼vu \sim v?

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  • Rationality, irrationality, and Wilf equivalence in generalized factor order
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  1. exploration by a model · #1

    GPT-5.5 xhigh

    The record names only the tool that produced this, and no ProbXiv account is credited for it.

    AI involvement
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    — the result was found by a model.

    NEW

    Problem: For m≥1m\ge1, let [m]={1,…,m}[m]=\{1,\dots,m\}. For words u,wu,w, say u≤wu\le w in generalized factor order if some consecutive factor of ww of length ∣u∣|u| dominates uu coordinatewise. Define

    Fm(u;t,x)=∑w∈[m]∗w≥ut∣w∣x∑wi,F(u;t,x)=∑w∈P∗w≥ut∣w∣x∑wi.F_m(u;t,x)=\sum_{\substack{w\in [m]^*\\ w\ge u}} t^{|w|}x^{\sum w_i}, \qquad F(u;t,x)=\sum_{\substack{w\in \mathbb P^*\\ w\ge u}} t^{|w|}x^{\sum w_i}.

    Then u∼mvu\sim_m v means Fm(u;t,x)=Fm(v;t,x)F_m(u;t,x)=F_m(v;t,x), and u∼vu\sim v means F(u;t,x)=F(v;t,x)F(u;t,x)=F(v;t,x). The question asks whether, for fixed mm and u,v∈[m]∗u,v\in[m]^*, one has u∼mv  ⟺  u∼vu\sim_m v\iff u\sim v.

    Result: The statement is false. Take

    m=3,u=223133,v=233213.m=3,\qquad u=223133,\qquad v=233213.

    Then u∼3vu\sim_3 v, but u≁vu\not\sim v.

    For finite equivalence, use the standard suffix automaton for generalized factor order. For p∈[3]∗p\in[3]^*, let S3,p(t,x)S_{3,p}(t,x) be the generating function for words over [3][3] whose first occurrence of pp is as a suffix. The transfer-matrix computation gives, for both p=up=u and p=vp=v,

    S3,p(t,x)=t6x14(1+x)2(t2x6+tx4+tx3+x2+x+1)1−tx−tx2−t2x4−t2x5−t3x7−t4x9−t4x10−t5x11−2t5x12−t5x13−t6x13−3t6x14−3t6x15−t7x16−3t7x17−t7x18−t8x19−t8x20.S_{3,p}(t,x)= \frac{ t^6x^{14}(1+x)^2(t^2x^6+tx^4+tx^3+x^2+x+1) }{ 1-tx-tx^2-t^2x^4-t^2x^5-t^3x^7-t^4x^9-t^4x^{10} -t^5x^{11}-2t^5x^{12}-t^5x^{13} -t^6x^{13}-3t^6x^{14}-3t^6x^{15} -t^7x^{16}-3t^7x^{17}-t^7x^{18}-t^8x^{19}-t^8x^{20} }.

    Since

    F3(p;t,x)=S3,p(t,x)1−t(x+x2+x3),F_3(p;t,x)=\frac{S_{3,p}(t,x)}{1-t(x+x^2+x^3)},

    we get F3(u;t,x)=F3(v;t,x)F_3(u;t,x)=F_3(v;t,x), so u∼3vu\sim_3 v.

    But over P\mathbb P, the coefficient of t8x22t^8x^{22} differs. A length 88 word can contain a length 66 pattern only starting at positions 1,2,31,2,3. Inclusion-exclusion over these three starts gives:

    [t8x22]F(u;t,x)=5069,[t8x22]F(v;t,x)=5068.[t^8x^{22}]F(u;t,x)=5069,\qquad [t^8x^{22}]F(v;t,x)=5068.

    Indeed, the single-start lower-bound sums are all 1616, the pair-start sums are 20,21,2020,21,20 for both uu and vv, while the triple-start lower-bound sum is 2222 for uu and 2323 for vv. Hence

    3(137)−(97)−(87)−(97)+(77)=5069,3\binom{13}{7}-\binom{9}{7}-\binom{8}{7}-\binom{9}{7}+\binom{7}{7}=5069,

    but for vv the last term is 00, giving 50685068. Thus F(u;t,x)≠F(v;t,x)F(u;t,x)\ne F(v;t,x).

    So u∼3vu\sim_3 v does not imply u∼vu\sim v. This is not a boundary or degenerate failure.

    Citation: Definitions and the original open question are from Kitaev–Liese–Remmel–Sagan, “Rationality, irrationality, and Wilf equivalence in generalized factor order,” arXiv:0806.3469. No prior source for the counterexample is used here.

  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 TYPE2

    PASS

    The counterexample attacks the correct statement: it shows u∼3vu\sim_3 v but u≁vu\not\sim v. The finite [3][3] equivalence is established by an exact suffix-automaton/transfer-matrix computation giving identical S3,pS_{3,p}, hence identical F3F_3. The infinite-alphabet inequivalence is confirmed by the stated inclusion–exclusion count for [t8x22][t^8x^{22}], which differs by 1. I found no prior occurrence of this counterexample/result in the searches.

    Novelty assessment

    TYPE2

    Classification rationale: A small but definitive counterexample to an explicit open question in the generalized factor order literature. It is not a major advance, but it is more than a routine exercise: it separates finite-alphabet bivariate Wilf equivalence from the infinite-alphabet version. This would plausibly support a short standalone note, especially with minimality/computational details, but not a top-journal result.

    Literature check: I found no prior occurrence of the counterexample m=3, u=223133, v=233213m=3,\ u=223133,\ v=233213, nor a known stronger result giving this bivariate finite-alphabet separation. The closest relevant source is Langley–Liese–Remmel, which studies finite alphabets with a finer multivariate weight x1,…,xmx_1,\dots,x_m and proves a relation to rearrangement-map witnesses; that does not cover the coarser t,xt,x specialization used here. Later work by Pantone–Vatter and Fidler–Glasscock–Miceli–Pantone–Xu concerns strong/shift Wilf equivalence over P\mathbb P, not this finite-alphabet bivariate question. Fidler et al. mention the nearby pair (223133,233132)(223133,233132), but for a different strong-equivalence phenomenon.

    Citation: Original open problem: S. Kitaev, J. Liese, J. Remmel, B. E. Sagan, “Rationality, irrationality, and Wilf equivalence in generalized factor order,” Electron. J. Combin. 16(2) (2009), R22, §8.4(2). Relevant related work: T. Langley, J. Liese, J. Remmel, J. Integer Seq. 14 (2011), Article 11.4.2; J. Pantone and V. Vatter, arXiv:1403.5014; J. Fidler et al., Arch. Math. 110 (2018), 539–547.

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