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Gessel's fundamental and Stembridge's peak functions are the generatingfunctions for (enriched) $P$-partitions on labelled chains. They are also thebases of two significant subalgebras of formal power series, respectively thering of quasisymmetric functions (QSym) and the algebra of peaks. Hsiaointroduced the monomial peak functions, a basis of the algebra of peaks indexedby odd integer compositions whose relation to peak functions mimics the onebetween the monomial and fundamental bases of QSym. We show that the extensionof monomial peaks to any composition is a new basis of QSym and generaliseHsiao's results including the product rule. To this end we introduce a weightedvariant of posets and study their generating functions.

Weighted posets and the enriched monomial basis of QSym (extended abstract)