Abstract
The diamond product is the poset operation that when applied to the face lattices of two polytopes results in the face lattice of the Cartesian product of the polytopes. Application of the diamond product to two Eulerian posets is a bilinear operation on the cd-indices of the two posets, yielding a product on cd-polynomials. A lattice path interpretation is provided for this product of two cd-monomials.
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References
Bayer M., Billera L.: Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets. Invent. Math. 79(1), 143–157 (1985)
Bayer M., Klapper A.: A new index for polytopes. Discrete Comput. Geom. 6(2), 33–47 (1991)
Ehrenborg R.: On posets and Hopf algebras. Adv. Math. 119(1), 1–25 (1996)
Ehrenborg R., Fox H.: Inequalities for cd-indices of joins and products of polytopes. Combinatorica 23(3), 427–452 (2003)
Ehrenborg R., Readdy M.: Coproducts and the cd-index. J. Algebraic Combin. 8(3), 273–299 (1998)
Ehrenborg R., Readdy M.: The Tchebyshev transforms of the first and second kind. Ann. Combin. 14(2), 211–244 (2010)
Kalai G.: A new basis for polytopes. J. Combin. Theory Ser. A 49(2), 191–209 (1988)
Slone, M.: Homological combinatorics and extensions of the cd-index. Doctoral Dissertation. University of Kentucky, Kentucky (2008)
Stanley, R.P.: Enumerative Combinatorics, Vol. I. Second Edition. Cambridge University Press, Cambridge (2011)
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Fox, N.B. A Lattice Path Interpretation of the Diamond Product. Ann. Comb. 20, 569–586 (2016). https://doi.org/10.1007/s00026-016-0323-z
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DOI: https://doi.org/10.1007/s00026-016-0323-z