A poset P is called k-Eulerian if every interval of rank k is Eulerian. The class of k-Eulerian posets interpolates between graded posets and Eulerian posets. It is a straightforward observation that a 2k-Eulerian poset is also (2k+1)-Eulerian. We prove that the ab-index of a (2k+1)-Eulerian poset can be expressed in terms of c = a + b, d = ab + ba and e2k+1 = (a - b)2k+1. The proof relies upon the algebraic approaches of Billera-Liu and Ehrenborg-Readdy. We extend the Billera-Liu flag algebra to a Newtonian coalgebra. This flag Newtonian coalgebra forms a Laplace pairing with the Newtonian coalgebra k(a, b) studied by Ehrenborg-Readdy. The ideal of flag operators that vanish on (2k + 1)-Eulerian posets is also a coideal. Hence, the Laplace pairing implies that the dual of the coideal is the desired subalgebra of k(a, b). As a corollary we obtain a proof of the existence of the cd-index which does not use induction.
|Number of pages||10|
|State||Published - 2001|
Bibliographical noteFunding Information:
The author thanks Gábor Hetyei and Margaret Readdy for inspiring discussions. The author was supported by National Science Foundation, DMS 97-29992, and NEC Research Institute, Inc. while at the Institute for Advanced Study and partially supported by Swedish Natural Science Research Council DNR 702-238/98. This paper was completed at Cornell University under support by National Science Foundation grants DMS 98-00910 and DMS 99-83660.
- Flag operators
ASJC Scopus subject areas
- Algebra and Number Theory
- Geometry and Topology
- Computational Theory and Mathematics