Flag-symmetry of the poset of shuffles and a local action of the symmetric group

1998/06/10 by Rodica Simion, Richard P. Stanley, Simion, Rodica +1
Mathematics · #05A15 (Primary) 05E05 #05E25 (Secondary) #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A15 #msc:05E05 #msc:05E25

paper · pdf · doi:10.48550/arxiv.math/9806057

34 pages, 6 figures

arxiv created 1998/06/10 · openalex publication_date 1998/06/10 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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