## 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.

Original language | English |
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Pages (from-to) | 227-236 |

Number of pages | 10 |

Journal | Order |

Volume | 18 |

Issue number | 3 |

DOIs | |

State | Published - 2001 |

### Bibliographical note

Funding 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.

## Keywords

- Flag operators
- cd-index

## ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology
- Computational Theory and Mathematics