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Linear inequalities for enumerating chains in partially ordered sets. (English) Zbl 1231.06003
Summary: We characterize the linear inequalities satisfied by flag $$f$$-vectors of all finite bounded posets. We do the same for semipure posets. In particular, the closed convex cone generated by flag $$f$$-vectors of bounded posets of fixed rank is shown to be simplicial, and the closed cone generated by flag $$f$$-vectors of semipure posets of fixed rank is shown to be polyhedral. The extreme rays of both of these cones are described explicitly in terms of quasisymmetric functions. The extreme rays of the first cone are then used to define a new basis for the algebra of quasisymmetric functions. This basis has nonnegative structure constants, for which a combinatorial interpretation involving lattice paths is given.
##### MSC:
 06A07 Combinatorics of partially ordered sets 05A15 Exact enumeration problems, generating functions 05E05 Symmetric functions and generalizations 16T30 Connections of Hopf algebras with combinatorics
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