Abstract
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 e 2k+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.
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Ehrenborg, R. k-Eulerian Posets. Order 18, 227–236 (2001). https://doi.org/10.1023/A:1012296719116
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DOI: https://doi.org/10.1023/A:1012296719116