k-Eulerian Posets

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.