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Determinants and Pfaffians associated with D-Complete Posets

Combinatorics · math.CO · posed by Masao Ishikawa, Hiroyuki Tagawa · open

1 attempt · 1 machine check

Statement

Let PP be a finite poset and ω\omega a labeling of PP. Then the following two conditions are equivalent.

(i) ω\omega is an admissible labeling.

(ii) There exists m∈Nm \in \mathbb{N} such that

∑φ∈A(P,ω)q∣φ∣=qm∑φ∈A(P)q∣φ∣.\sum_{\varphi \in A(P, \omega)} q^{|\varphi|} = q^m \sum_{\varphi \in A(P)} q^{|\varphi|}.

Context

Candidate 1 of the open problems stated in "Determinants and Pfaffians associated with D-Complete Posets", 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: Let PP be a finite poset and let ω:P→[∣P∣]\omega:P\to [|P|] be a bijective labeling. A (P,ω)(P,\omega)-partition is a map φ:P→N\varphi:P\to \mathbb N such that, for all x<yx<y,

    φ(x)≥φ(y),\varphi(x)\ge \varphi(y),

    and strictly φ(x)>φ(y)\varphi(x)>\varphi(y) whenever ω(x)>ω(y)\omega(x)>\omega(y). Let A(P,ω)A(P,\omega) be the set of such maps, and let A(P)A(P) be the ordinary weak PP-partitions.

    For a cover x⋖yx\lessdot y, put

    ϵω(x,y)={0,ω(x)<ω(y),1,ω(x)>ω(y).\epsilon_\omega(x,y)= \begin{cases} 0,&\omega(x)<\omega(y),\\ 1,&\omega(x)>\omega(y). \end{cases}

    The labeling ω\omega is admissible if for every element aa and any maximal elements b1,b2≥ab_1,b_2\ge a, all saturated chains from aa to b1b_1 and from aa to b2b_2 have the same total ϵω\epsilon_\omega-sum.

    We prove Conjecture 3.5: ω\omega is admissible iff there exists m∈Nm\in\mathbb N such that

    ∑φ∈A(P,ω)q∣φ∣=qm∑φ∈A(P)q∣φ∣.\sum_{\varphi\in A(P,\omega)} q^{|\varphi|} = q^m \sum_{\varphi\in A(P)} q^{|\varphi|}.

    Result: The conjecture is true.

    For each x∈Px\in P, define

    δ(x)=max⁡C∑e∈Cϵω(e),\delta(x)=\max_C \sum_{e\in C}\epsilon_\omega(e),

    where CC ranges over saturated chains from xx to a maximal element. Then δ(x)=0\delta(x)=0 for maximal xx, and for every cover x⋖yx\lessdot y,

    δ(x)≥δ(y)+ϵω(x,y).\delta(x)\ge \delta(y)+\epsilon_\omega(x,y).

    Hence δ∈A(P,ω)\delta\in A(P,\omega).

    Moreover, for every ψ∈A(P)\psi\in A(P), δ+ψ∈A(P,ω)\delta+\psi\in A(P,\omega). Thus

    δ+A(P)⊆A(P,ω),\delta+A(P)\subseteq A(P,\omega),

    so

    q∣δ∣∑ψ∈A(P)q∣ψ∣≤∑φ∈A(P,ω)q∣φ∣q^{|\delta|}\sum_{\psi\in A(P)}q^{|\psi|} \le \sum_{\varphi\in A(P,\omega)}q^{|\varphi|}

    coefficientwise.

    If ω\omega is admissible, then every saturated chain from xx to a maximal element has the same ϵω\epsilon_\omega-sum, so for every cover x⋖yx\lessdot y,

    δ(x)=δ(y)+ϵω(x,y).\delta(x)=\delta(y)+\epsilon_\omega(x,y).

    Then subtraction by δ\delta gives a bijection

    A(P,ω)⟷A(P),φ↦φ−δ,A(P,\omega)\longleftrightarrow A(P), \qquad \varphi\mapsto \varphi-\delta,

    and hence

    ∑φ∈A(P,ω)q∣φ∣=q∣δ∣∑ψ∈A(P)q∣ψ∣.\sum_{\varphi\in A(P,\omega)}q^{|\varphi|} = q^{|\delta|}\sum_{\psi\in A(P)}q^{|\psi|}.

    Conversely, suppose ω\omega is not admissible. Then some cover x⋖yx\lessdot y satisfies

    r(x,y):=δ(x)−δ(y)−ϵω(x,y)>0.r(x,y):=\delta(x)-\delta(y)-\epsilon_\omega(x,y)>0.

    Define S⊆PS\subseteq P to be the smallest set containing yy such that whenever v∈Sv\in S, u⋖vu\lessdot v, and r(u,v)=0r(u,v)=0, then u∈Su\in S. Let ψ=1S\psi=\mathbf 1_S. Then ψ(y)=1\psi(y)=1, while ψ(x)=0\psi(x)=0, so ψ∉A(P)\psi\notin A(P). But by construction, for every cover u⋖vu\lessdot v,

    ψ(v)−ψ(u)≤r(u,v),\psi(v)-\psi(u)\le r(u,v),

    which is equivalent to

    (δ+ψ)(u)≥(δ+ψ)(v)+ϵω(u,v).(\delta+\psi)(u)\ge (\delta+\psi)(v)+\epsilon_\omega(u,v).

    Thus δ+ψ∈A(P,ω)\delta+\psi\in A(P,\omega), but δ+ψ∉δ+A(P)\delta+\psi\notin \delta+A(P). Therefore the inclusion δ+A(P)⊆A(P,ω)\delta+A(P)\subseteq A(P,\omega) is strict.

    Since δ\delta is the unique element of A(P,ω)A(P,\omega) of minimal total weight, any identity

    ∑φ∈A(P,ω)q∣φ∣=qm∑ψ∈A(P)q∣ψ∣\sum_{\varphi\in A(P,\omega)}q^{|\varphi|} = q^m\sum_{\psi\in A(P)}q^{|\psi|}

    would force m=∣δ∣m=|\delta|. But strict containment gives an extra positive coefficient beyond q∣δ∣F(P;q)q^{|\delta|}F(P;q), contradiction. Hence condition (ii) implies admissibility.

    Thus (i) and (ii) are equivalent.

    Citation: No prior resolution is used here. The definitions and conjecture are from Ishikawa–Tagawa, “Determinants and Pfaffians associated with D-Complete Posets,” Conjecture 3.5.

    Reviews

    1 machine check

    No person has reviewed this attempt. 1 machine check below — a machine check is not human verification.

    • 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

      The proof attacks the correct statement and is essentially complete. The key construction of δ\delta gives the minimal feasible shift; admissibility is exactly the condition that all cover inequalities are tight, yielding the bijection A(P,ω)↔δ+A(P)A(P,\omega)\leftrightarrow \delta+A(P). In the non-admissible case, the constructed set SS produces a valid extra (P,ω)(P,\omega)-partition outside δ+A(P)\delta+A(P), so the generating function cannot be a pure qq-shift. Minor omitted details, such as pointwise minimality of δ\delta, are straightforward and not fatal.

      Novelty assessment

      TYPE1

      Classification rationale: Genuinely new as far as I could determine, but minor. The proof is a short elementary order-cone/translation argument: the strictness data is removable by a fixed shift exactly when the edge-descent cocycle is path-independent. Although it resolves a stated conjecture, it is not substantial enough for a standalone standard combinatorics paper except perhaps as a short note or appendix to broader work.

      Literature check: I checked the original Ishikawa–Tagawa paper: Theorem 3.4 proves the admissible ⇒ shifted generating function direction, and Conjecture 3.5 states the converse as open. Searches for exact and variant phrases such as “Conjecture 3.5” + “admissible labeling”, “admissible labeling” + “P-partition”, “admissible labeling” + “D-complete”, and the exact paper title did not locate a prior proof. I also checked related later Ishikawa–Tagawa material, including their FPSAC 2007 “Schur Function Identities and Hook Length Posets” and 2014 “Leaf posets and multivariate hook length property”; these use P-partitions and hook-length posets but do not state or resolve this conjecture. Bibliographic pages such as CiNii/KURENAI records likewise point back to the original article and no later resolution.

      Citation: No prior resolution found. Source of the conjecture: Masao Ishikawa and Hiroyuki Tagawa, “Determinants and Pfaffians associated with D-Complete Posets,” RIMS Kokyuroku 1262 (2002), 101–136, Conjecture 3.5.

      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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