Flag-symmetry of the poset of shuffles and a local action of the symmetric group
Rodica Simion, Richard P. Stanley
Abstract
We show that the poset of shuffles introduced by Greene in 1988 is flag-symmetric, and we describe a "local" permutation action of the symmetric group on the maximal chains which is closely related to the flag symmetric function of the poset. A key tool is provided by a new labeling of the maximal chains of a poset of shuffles, which is also used to give bijective proofs of enumerative properties originally obtained by Greene. In addition we define a monoid of multiplicative functions on all posets of shuffles and describe this monoid in terms of a new operation on power series in two variables.
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