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The f-Vector of the Descent Polytope

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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 iS 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.


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Correspondence to Richard Ehrenborg.

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Chebikin, D., Ehrenborg, R. The f-Vector of the Descent Polytope. Discrete Comput Geom 45, 410–424 (2011). https://doi.org/10.1007/s00454-010-9316-6

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