Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

The f-Vector of the Descent Polytope

  • 118 Accesses

  • 3 Citations

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 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.

References

  1. 1.

    Chebikin, D.: Polytopes, generating functions, and new statistics related to descents and inversions in permutations. Doctoral dissertation, Massachusetts Institute of Technology (2008)

  2. 2.

    de Bruijn, N.G.: Permutations with given ups and downs. Nieuw Arch. Wiskd. 18, 61–65 (1970)

  3. 3.

    Ehrenborg, R., Mahajan, S.: Maximizing the descent statistic. Ann. Comb. 2, 111–129 (1998)

  4. 4.

    Ehrenborg, R., Levin, M., Readdy, M.: A probabilistic approach to the descent statistic. J. Combin. Theory Ser. A 98, 150–162 (2002)

  5. 5.

    Ehrenborg, R., Kitaev, S., Perry, P.: A spectral approach to pattern avoiding permutations, preprint (2010), arXiv: 1009.2119

  6. 6.

    Niven, I.: A combinatorial problem on finite sequences. Nieuw Arch. Wiskd. 16, 116–123 (1968)

  7. 7.

    Readdy, M.A.: Extremal problems for the Möbius function in the face lattice of the n-octahedron. Discrete Math., 139, 361–380 (1995). Special issue on Algebraic Combinatorics

  8. 8.

    Sagan, B.E., Yeh, Y.-N., Ziegler, G.: Maximizing Möbius functions on subsets of Boolean algebras. Discrete Math. 126, 293–311 (1994)

  9. 9.

    Stanley, R.P.: Two poset polytopes. Discrete Comput. Geom. 1, 9–23 (1986)

  10. 10.

    Stanley, R.P.: Enumerative Combinatorics, vol. I. Wadsworth/Brooks/Cole, Pacific Grove (1986)

  11. 11.

    The Online Encyclopedia of Integer Sequences http://oeis.org/

  12. 12.

    Viennot, G.: Permutations ayant une forme donnée. Discrete Math. 26, 279–284 (1979)

Download references

Author information

Correspondence to Richard Ehrenborg.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

Keywords

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