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Séminaire Lotharingien de Combinatoire, 78B.33 (2017), 12 pp.

# Darij Grinberg

# Double Posets and the Antipode of QSym

**Abstract.**
A quasisymmetric function is assigned to every double poset (that is,
every finite set endowed with two partial orders) and any weight
function on its ground set. This generalizes well-known objects such
as monomial and fundamental quasisymmetric functions, (skew) Schur
functions, dual immaculate functions, and quasisymmetric
(*P*,ω)-partition enumerators. We prove a formula for the
antipode of this function that holds under certain conditions (which
are satisfied when the second order of the double poset is total, but
also in some other cases); this restates (in a way that to us seems
more natural) a result by Malvenuto and Reutenauer, but our proof is
new and self-contained. We generalize it further to an even more
comprehensive setting, where a group acts on the double poset by
automorphisms.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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