Coproducts and the cd-Index

  • Authors:

  • Richard Ehrenborg;Margaret Readdy

  • Affiliations:

  • Department of Mathematics, Cornell University, White Hall, Ithaca, NY 14853-7901. Email: jrge@math.cornell.edu;Department of Mathematics, Cornell University, White Hall, Ithaca, NY 14853-7901. Email: readdy@math.cornell.edu

  • Venue:
  • Journal of Algebraic Combinatorics: An International Journal
  • Year:
  • 1998

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Abstract

The linear span of isomorphism classes of posets, {\cal P},has a Newtonian coalgebra structure. We observe thatthe ab-index is a Newtonian coalgebra map from the vector space {\cal P} tothe algebra of polynomials in the noncommutative variables a and b. This enables us to obtain explicit formulas showing how thecd-index of the face lattice of a convex polytope changes when taking the pyramid and the prism of the polytope and the corresponding operations on posets. As a corollary, we have new recursion formulas for the cd-index of the Boolean algebra and the cubical lattice. Moreover, these operationsalso have interpretations for certain classes of permutations,including simsun and signed simsun permutations. We prove an identity for the shelling components of the simplex. Lastly,we show how to compute the ab-index of the Cartesian product of two posetsgiven the ab-indexes of each poset.