## Abstract

We completely characterize 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 | 71-83 |

Number of pages | 13 |

State | Published - 2006 |

Event | 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006 - San Diego, CA, United States Duration: Jun 19 2006 → Jun 23 2006 |

### Conference

Conference | 18th Annual International Conference on Formal Power Series and Algebraic Combinatorics, FPSAC 2006 |
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Country/Territory | United States |

City | San Diego, CA |

Period | 6/19/06 → 6/23/06 |

## Keywords

- Cubical posets
- Factorial function
- Lattices
- Poset structure
- Triangular posets

## ASJC Scopus subject areas

- Algebra and Number Theory