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<p>Gessel’s fundamental and Stembridge’s peak functions are the generating functions for (enriched) <i>P</i>-partitions on labeled chains. They are also the bases of two significant subalgebras of formal power series, respectively, the ring of quasisymmetric functions (QSym) and the algebra of peaks. Hsiao introduced the monomial peak functions, a&#xa0;basis of the algebra of peaks indexed by odd integer compositions whose relation to peak functions mimics the one between the monomial and fundamental bases of QSym. We show that the extension of monomial peaks to any composition is a&#xa0;new basis of QSym and generalize Hsiao’s results including the product rule. To this end, we introduce a&#xa0;weighted variant of posets and study their generating functions.</p>

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WEIGHTED POSETS AND THE ENRICHED MONOMIAL BASIS OF \(\varvec{\mathrm{QSym}}\)

  • 期刊论文

  • 发表日期：

    2026年1月1日

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  • E. A. Vassilieva,
  • D. Grinberg

摘要

Gessel’s fundamental and Stembridge’s peak functions are the generating functions for (enriched) P-partitions on labeled chains. They are also the bases of two significant subalgebras of formal power series, respectively, the ring of quasisymmetric functions (QSym) and the algebra of peaks. Hsiao introduced the monomial peak functions, a basis of the algebra of peaks indexed by odd integer compositions whose relation to peak functions mimics the one between the monomial and fundamental bases of QSym. We show that the extension of monomial peaks to any composition is a new basis of QSym and generalize Hsiao’s results including the product rule. To this end, we introduce a weighted variant of posets and study their generating functions.