## Abstract

We give a complete classification of the factorial functions of Eulerian binomial posets. The factorial function B (n) either coincides with n!, the factorial function of the infinite Boolean algebra, or 2^{n - 1}, the factorial function of the infinite butterfly poset. We also classify the factorial functions for Eulerian Sheffer posets. An Eulerian Sheffer poset with binomial factorial function B (n) = n ! has Sheffer factorial function D (n) identical to that of the infinite Boolean algebra, the infinite Boolean algebra with two new coatoms inserted, or the infinite cubical poset. Moreover, we are able to classify the Sheffer factorial functions of Eulerian Sheffer posets with binomial factorial function B (n) = 2^{n - 1} as the doubling of an upside-down tree with ranks 1 and 2 modified. When we impose the further condition that a given Eulerian binomial or Eulerian Sheffer poset is a lattice, this forces the poset to be the infinite Boolean algebra B_{X} or the infinite cubical lattice C_{X}^{< ∞}. We also include several poset constructions that have the same factorial functions as the infinite cubical poset, demonstrating that classifying Eulerian Sheffer posets is a difficult problem.

Original language | English |
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Pages (from-to) | 339-359 |

Number of pages | 21 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 114 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2007 |

### Bibliographical note

Funding Information:The first author was partially supported by National Science Foundation grant 0200624 and by a University of Kentucky College of Arts & Sciences Faculty Research Fellowship. The second author was partially supported by a University of Kentucky College of Arts & Sciences Research Grant. Both authors thank Gábor Hetyei for inspiring them to study Eulerian binomial posets, the Banff International Research Station where some of the ideas for this paper were developed, and the Mittag-Leffler Institute where this paper was completed. Both authors gratefully acknowledge the careful and thoughtful comments made by one of the anonymous referees.

## Keywords

- Infinite Boolean algebra
- Infinite butterfly poset
- Infinite cubical poset and lattice
- Upper binomial poset

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics