Department of Mathematics

Combinatorics and Graph Theory

  •  Descent and peak polynomials
  •  10/04/2016
  •  4:10 PM - 5:00 PM
  •  C304 Wells Hall
  •  Bruce Sagan, MSU

Let p = p_1 ... p_n be a permutation in the symmetric group S_n written as a sequence. The descent set of p is the set of indices i such that p_i > p_{i+1}. A classic result of MacMahon states the the number of permutations in S_n with a given descent set is a polynomial in n. But little work seems to have been done concerning the properties of these polynomials. The peak set of p is the set of indices i such that p_{i+1} < p_i > p_{i+1}. Recently Billey, Burdzy, and Sagan proved that the number of permutations in S_n with given peak set is a polynomial in n times a power of two. I will survey what is known about these two polynomials, including their degrees, roots, coefficients, and analogues for other Coxeter groups.

 

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Michigan State University
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