## Abstract

For a positive integer n and a subset S⊆[n-1], the descent polytope DP _{S} is the set of points (x_{1},...,x_{n}) in the n-dimensional unit cube [0,1]^{n} such that x_{i}≥x_{i+1} if i∈S and x_{i}≤x_{i+1} otherwise. First, we express the f-vector as a sum over all subsets of [n-1]. Second, we use certain factorizations of the associated word over a two-letter alphabet to describe the f-vector. We show that the f-vector is maximized when the set S is the alternating set {1,3,5,...}∩[n-1]. We derive a generating function for F_{S}(t), written as a formal power series in two non-commuting variables with coefficients in ℤ[t]. We also obtain the generating function for the Ehrhart polynomials of the descent polytopes.

Original language | English |
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Pages (from-to) | 410-424 |

Number of pages | 15 |

Journal | Discrete and Computational Geometry |

Volume | 45 |

Issue number | 3 |

DOIs | |

State | Published - Apr 2011 |

## Keywords

- Alternating set
- Descent set statistics
- Ehrhart polynomial
- Maximizing inequalities
- Non-commutative rational generating function

## ASJC Scopus subject areas

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