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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Linear inequalities for enumerating chains in partially ordered sets


Authors: Samuel K. Hsiao, Lauren Rose, Rachel Stahl and Ezra M. Winston
Journal: Trans. Amer. Math. Soc. 363 (2011), 2715-2732
MSC (2010): Primary 06A07, 05E99, 16T30
Published electronically: November 30, 2010
MathSciNet review: 2763734
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Abstract: 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.


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Additional Information

Samuel K. Hsiao
Affiliation: Mathematics Program, Bard College, Annandale-on-Hudson, New York 12504

Lauren Rose
Affiliation: Mathematics Program, Bard College, Annandale-on-Hudson, New York 12504

Rachel Stahl
Affiliation: Mathematics Program, Bard College, Annandale-on-Hudson, New York 12504

Ezra M. Winston
Affiliation: Mathematics Program, Bard College, Annandale-on-Hudson, New York 12504

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05253-5
PII: S 0002-9947(2010)05253-5
Keywords: Flag $f$-vector, bounded poset, semipure poset, quasisymmetric function
Received by editor(s): April 10, 2009
Received by editor(s) in revised form: October 19, 2009
Published electronically: November 30, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.