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The enriched q-monomial basis of the quasisymmetric functions
Journal article   Open access   Peer reviewed

The enriched q-monomial basis of the quasisymmetric functions

Darij Grinberg and Ekaterina A. Vassilieva
The Electronic journal of combinatorics, v 31(4), 12409
18 Oct 2024
DOI: https://doi.org/10.37236/12409
url
https://doi.org/10.37236/12409View
Published, Version of Record (VoR)Open Access (License Unspecified),  Open

Abstract

Mathematics Mathematics, Applied Physical Sciences Science & Technology
We construct a new family (eta((q))(alpha))(alpha is an element of Comp) of quasisymmetric functions for each element q of the base ring. We call them the "enriched q-monomial quasisymmetric functions". When r := q + 1 is invertible, this family is a basis of QSym. It generalizes Hoffman's "essential quasi-symmetric functions" (obtained for q = 0) and Hsiao's "monomial peak functions" (obtained for q = 1), but also includes the monomial quasisymmetric functions as a limiting case. We describe these functions eta((q))(alpha) by several formulas, and compute their products, coproducts and antipodes. The product expansion is given by an exotic variant of the shuffle product which we call the "stufufuffle product" due to its ability to pick several consecutive entries from each composition. This "stufufuffle product" has previously appeared in recent work by Bouillot, Novelli and Thibon, generalizing the "block shuffle product" from the theory of multizeta values.

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Mathematics, Applied
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