有色多重集欧拉多项式的变体及其应用
Variations of colored multiset Eulerian polynomials and applications
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中文总结 AI 辅助
本文引入有色多重集欧拉多项式的上升类似物，推导其生成函数恒等式，证明其为半开格多面体的h^*-多项式且满足自交错性，结合递推关系证明两多项式均为实根，所得恒等式用于解释三类多面体的h^*-多项式。
中文摘要 AI 辅助
Deligeorgaki、Han和Solus引入了有色多重集欧拉多项式，推导了一个推广MacMahon恒等式的生成函数恒等式，证明了在适当参数条件下它们具有自交错性，并将这些多项式识别为膨胀格单形直积的h^*-多项式。本文中，我们引入了有色多重集欧拉多项式的一个上升类似物，推导了该多项式的显式生成函数恒等式，证明它等于一族半开格多面体的h^*-多项式，并验证该上升多项式在相同参数条件下也满足自交错性。通过建立递推关系，我们证明这两个多项式对于所有正整数参数都是实根的。所得恒等式进一步应用于从偏序集定义的Pitman–Stanley多面体、合成多面体和一族自反格多面体的h^*-多项式的组合解释。
英文摘要
Deligeorgaki, Han and Solus introduced colored multiset Eulerian polynomials, derived a generating function identity which generalizes MacMahon's identity, proved their self-interlacing under suitable parameter conditions, and identified these polynomials as the h^*-polynomials of direct products of dilated lattice simplices. In this paper, we introduce an ascent analogue of the colored multiset Eulerian polynomial, derive an explicit generating function identity for this polynomial, show that it is equal to the h^*-polynomial of a family of half-open lattice polytopes, and verify that this ascent polynomial also satisfies self-interlacing under the same parameter conditions. By establishing recurrence relations, we prove that both polynomials are real-rooted for all positive integer parameters. The obtained identities are further applied to interpret combinatorially the h^*-polynomials of Pitman--Stanley polytopes, composition polytopes and a family of reflexive lattice polytopes defined from preorders.
