Noncommutative Enumeration in Graded Posets

Authors:
Louis J. Billera;Niandong Liu
Affiliations:
Cornell University, Ithaca, NY 14853-4201, USA;Cornell University, Ithaca, NY 14853-4201, USA
Venue:
Journal of Algebraic Combinatorics: An International Journal
Year:
2000

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Abstract

We define a noncommutative algebra of flag-enumeration functionals on graded posets and show it to be isomorphic to the free associative algebra on countably many generators. Restricted to Eulerian posets, this ring has a particularly appealing presentation with kernel generated by Euler relations. A consequence is that even on Eulerian posets, the algebra is free, with generators corresponding to odd jumps in flags. In this context, the coefficients of the cd-index provide a graded basis.