The skew shape poset block-statistic conjecture

Let PP be a skew shape poset of size nn, let Sn(P)S_n(P) denote the set of PP-partitions of the labeling set represented by permutations, and let bl~P(σ)\widetilde{bl}_P(\sigma) be the number of blocks in the valid block decomposition associated with (P,σ)(P,\sigma). Here Ω(P;t)\Omega(P;t) is the order polynomial of PP. Skew shape block-statistic conjecture. For every skew shape poset PP of size nn,

n!Ω(P;t)=σSn(P)tbl~P(σ).n! \, \Omega(P;t)=\sum_{\sigma\in S_n(P)}t^{\widetilde{bl}_P(\sigma)}.

This conjecture proposes a combinatorial interpretation of the coefficients of the order polynomial for skew shape posets, extending the corresponding proved formula for fence posets. Its status is not specified in the source.

Sources & referencesView supporting material

Primary source

Yakob Kahane, “Combinatorial interpretation of the coefficients of the order polynomial of fence posets”, arXiv:2607.11225 (2026).

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