A Group Action on Derangements *

In this paper we define a cyclic analogue of the MFS-action on derangements, and give a combinatorial interpretation of the expansion of the n-th derangement polynomial on the basis {q k (1 + q) n−1−2k }, k = 0, 1,. .. , (n − 1)/2.


Introduction
Let [n] denote the set {1, 2, . . ., n} and let S n denote the set of all permutations of [n].For π = π 1 π 2 • • • π n ∈ S n and x ∈ [n], we write π as the concatenation π = w 1 w 2 xw 3 w 4 , where w 2 is the maximal contiguous subword immediately to the left of x whose letters are all smaller than x, and w 3 is the maximal contiguous subword immediately to the right of x whose letters are all smaller than x.Following Foata and Strehl [4,5], this concatenation is called the x-factorization of π.For example, let π = 714358296 and x = 5.Then w 1 = 7, w 2 = 143, w 3 = ∅ and w 4 = 8296.
For any subset S ⊆ [n], define ϕ S (π) = x∈S ϕ x (π).From the definition, if x is a double ascent (double descent, resp.) of π, then x is a double descent (double ascent, resp.) of ϕ x (π).The group Z n 2 acts on S n via the functions ϕ S , S ∈ [n] and call this action the MFS-action.
By the theory of symmetric functions, Brenti [2] showed that derangement polynomials are symmetric and unimodal polynomials.Using the method of continued fractions, Shin and Zeng [7] gave a combinatorial interpretation for coefficients in the expansion of the n-th derangement polynomial on the basis {q k (1 + q) n−1−2k }, k = 0, 1, . . ., (n − 1)/2 .In this note, we define a cyclic analogous of the MFS-action on derangements and give a new proof for the result of Shin and Zeng.

Let
Denote by Exc(π) the set of all excedances in π and let exc(π) = |Exc(π)|.The n-derangement polynomial D n (q) is the generating function of statistic excedance over the set D n , i.e., where Recall that a permutation π ∈ S n may be regarded as a disjoint union of its distinct cycles For a derangement π, each cycle contains at least two elements.The standard cycle representation of π is defined by requiring that (i) each cycle is written with its largest element first, and (ii) the cycles are written in increasing order of their largest elements [8].For example, the standard cycle representation of π = 456321 ∈ D 6 is (52)(6143).Throughout the paper all permutations are written in standard cycle representation.
The map is well-defined.To see this, let and w 1 denotes the maximal contiguous subword immediately to the left of x whose letters are all smaller than where C i = (w 0 xw 1 w 2 ) and w 1 denotes the maximal contiguous subword immediately to the right of x whose letters are all smaller than x.Then , where Ci = (w 0 w 1 xw 2 ).Hence the map θ x is welldefined for all x ∈ [n].Table 1 gives an example of the maps θ x on π = (623)(87514) for all x ∈ [8], where o(π) = 62387514.For π ∈ D n , let Orb c (π) denote the orbit including π under the CMFS-action.There is a unique derangement in Orb c (π), denoted by π, such that π has no cyclic double ascents.The next is the main results of this note.q exc(σ) = q exc(π) (1 + q) n−2exc(π) = q cpeak(π) (1 + q) n−2cpeak(π) .

Table 1 .
The function θ x is an involution and θ x θ y = θ y θ x for all x, y ∈ [n].For any subset S ⊆ [n], define the function θ S (π) : D n → D n by The group Z n 2 acts on D n via the functions θ S , S ∈ [n] and call this action the CMFSaction.