Abstract
We prove various formulas which express exponential generating functions counting permutations by the peak number, valley number, double ascent number, and double descent number statistics in terms of the exponential generating function for Chebyshev polynomials, as well as cyclic analogues of these formulas for derangements. We give several applications of these results, including formulas for the (-1)-evaluation of some of these distributions. Our proofs are combinatorial and involve the use of monomino-domino tilings, the modified Foata-Strehl action (a.k.a. valley-hopping), and a cyclic analogue of this action due to Sun and Wang.
Highlights
Let π = π1π2 · · · πn be a permutation in Sn, the set of permutations of [n] = {1, 2, . . . , n}
Π∈Sn encodes the distribution of the descent number des over Sn, and the nth peak polynomial
Π∈Sn is the analogous polynomial for the peak number pk
Summary
Tpk(π) π∈Sn is the analogous polynomial for the peak number pk It is well-known [9, Theoreme 5.6] that the (−1)-evaluation of the Eulerian distribution is given by the formula. The exponential generating functions for the Eulerian and peak polynomials evaluated at t = −1 can be expressed as the logarithmic derivative of F (x) and G(x), respectively. Using a variant of valley-hopping due to Sun and Wang [21] for derangements, we prove a cyclic analogue of Theorem 3 and use it to derive formulas relating exponential generating functions counting derangements by cyclic statistics with the exponential generating function for our Chebyshev polynomials We use this to prove a result similar to Theorem 1 for the excedance and cyclic peak distributions over derangements, and to prove that the (−1)-evaluation of cyclic double descent distributions over derangements yields the secant numbers
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