On some Euler-Mahonian distributions

We prove that the pair of statistics (des,maj) on multiset permutations is equidistributed with the pair (stc,inv) on certain quotients of the symmetric group. We define the analogue of the statistic stc on multiset permutations, whose joint distribution with the inversions equals that of (des,maj). We extend the definition of the statistic stc to hyperoctahedral and even hyperoctahedral groups. Such functions, together with the Coxeter length, are equidistributed, respectively, with (ndes,nmaj) and (ddes,dmaj).


introduction
The first result about the enumeration of multiset permutations with respect to statistics now called descent number and major index is due to MacMahon. Let ρ = (ρ 1 , . . . , ρ m ) be a composition of N ∈ N. We denote by S ρ the set of all permutations of the multiset {1 ρ 1 , . . . , m ρm }. The descent set Des(w) of w = w 1 · · · w N ∈ S ρ is Des(w) = {i ∈ [N − 1] | w i > w i+1 }. The descent and major index statistics on S ρ are des(w) = | Des(w)| and maj(w) = i∈Des (w) i.
Then ([9, §462, Vol. 2, Ch. IV, Sect. IX]) where, for n, k ∈ N we put The well known result about the equidistribution, on multiset permutations, of the inversion number with the major index, goes also back to MacMahon; Foata and Schützenberger [6] proved that this equidistribution refines, in the case of the symmetric group, to inverse descent classes. A pair of statistics that is equidistributed with (des, maj) is called Euler-Mahonian.
In [11] Skandera introduced an Eulerian statistic, which he called stc, on the symmetric group, and proved that the pair (stc, inv) is Euler-Mahonian. In this note we prove that the joint distribution of (stc, inv) on certain quotients of the symmetric group is indeed the same as the distribution of (des, maj) on multiset permutations; we use this result to define a statistic mstc that is Eulerian on multiset permutations and that, together with inv is equidistributed with the pair (des, maj).
The Eulerian polynomial is (essentially) the descent polynomial on the symmetric group S n . Frobenius proved (see [7]) that this polynomial has real, simple, negative roots, and that −1 features as a root if and only if n is even. Simion proved later that the descent polynomials of permutations of any multiset are also real rooted, with simple, negative roots (see [10]). We use our first result of equidistribution to show that on the set of permutations of words in the alphabet {1 r , 2 r }, the polynomial of the joint distribution of des and maj admits, for odd r a unique unitary factor. This factorisation, together with the one of Carlitz's q-Eulerian polynomial (the polynomial of the joint distribution of des and maj on the symmetric group) that we show in [4], may be considered a refinement of Frobenius' result, and supports a conjecture we made in [4] and that we translate in Section 2 in terms of the joint distribution of (stc, inv) on quotients of the symmetric group.
Generalisations of MacMahon's result (1.1) to signed permutations were first obtained by Adin, Brenti and Roichman in [1] and to even-signed permutations by Biagioli in [2]. In the last section of this note we define Eulerian statistics nstc and dstc that, together with the length, are equidistributed, respectively, with the Euler-Mahonian pairs (ndes, nmaj) on the hyperoctahedral group and (ddes, dmaj) on the even hyperoctahedral group.

Stc on quotients of the symmetric group and multiset permutations
For n, m ∈ N, m ≤ n we denote with [n] = {1, . . . , n} and [m, n] = {m, m + 1, . . . , n}. For a permutation σ ∈ S n we use the one-line notation or the disjoint cycle notation.
The Coxeter length for σ ∈ S n coincides with the inversion number inv(σ) = |{(i, j) ∈ [n] × [n] | i < j, σ(i) > σ(j)}|. Also, for a (signed) permutation σ ∈ S n (respectively, B n ), we let It is well-known that the symmetric group S n is in bijection with the set of words w = w 1 · · · w n ∈ E n where One of such bijections is the Lehmer code, defined as follows. For The sum of the c i s gives, for each permutation, the inversion number. The statistic stc, that together with the length constitutes an Euler-Mahonian pair equidistributed with (des, maj), is defined as follows (cf. [11, Definition 3.1]): stc(σ) = st(code(σ)), where for a word w ∈ E n that is, the maximal r for which there exists a subword of w of length r elementwise strictly greater than the r-staircase word (r − 1)(r − 2) · · · 1 0.
Given a composition ρ of N , the corresponding set of multiset permutations S ρ is naturally in bijection with certain quotients and inverse descent classes of S N . In particular, for ρ = (ρ 1 , . . . , ρ m ) a composition of N , for i = 1 . . . , m − 1 we let We let S R c N and IS R N denote, respectively, the quotient and the inverse descent class of the symmetric group A natural way to associate a permutation to a multiset permutation is the standardisation. Given ρ a composition of N and a word w in the alphabet {1 ρ 1 , . . . , m ρm }, std(w) is the element of S N obtained by w substituting, in the order of appearance in w from left to right, the ρ 1 1s with the sequence 1 2 . . . ρ 1 , the ρ 2 2s with the sequence ρ 1 + 1 . . . ρ 1 + ρ 2 and so on. So for example if ρ = (2, 3, 2) and w = 1223132 ∈ S ρ , then std(w) = 1346275 ∈ S 7 .
The following result is due to Foata and Han.
Proof. The standardisation std is a bijection between S ρ and IS R N , and preserves des and maj, so By Proposition 2.2 the last term is the desired distribution on S R c N : As an application, we prove a result about the bivariate factorisation of the polynomial C ρ (x, q), that in [4] is used to prove deduce analytic properties of some orbit Dirichlet series. We say that a bivariate polynomial f (x, y) ∈ Z[x, y] is unitary if there exist integers α, β ≥ 0 and g ∈ Z[t] so that f (x, y) = g(x α y β ) and all the complex roots of g lie on the unit circle (see also [4,Remark 2.9]). Proposition 2.4. Let ρ = (r, r) where r ≡ 1 (mod 2). Then where C ρ (x, q) has no unitary factor.
Before we prove Proposition 2.4, we give a nice description of the stc for permutations with at most one descent. Then has at most a descent at ρ 1 , so its code is of the form code(w) = c 1 · · · c ρ 1 0 · · · 0, with 0 ≤ c 1 ≤ . . . ≤ c ρ 1 . The first (possibly) non-zero element of the code is exactly the number of elements of the second block for which the image is in the first block. This number coincides with the length of the longest possible subword of the code which is elementwise greater than a staircase word.
We reformulate [4,Conjecture B] in terms of the bivariate distribution of (stc, ) on quotients of the symmetric group.
for some C ρ (x, q) ∈ Z[x, q] with no unitary factors. Proposition 2.3 suggests a natural extension of the definition of the statistic stc to multipermutations, thus answering a question raised in [11].
For w ∈ S ρ , std(w) ∈ IS R N . So we have a bijection between multiset permutations S ρ and the quotient S R c N istd : S ρ → S R c N , istd(w) = (std(w)) −1 which is inversion preserving: inv(w) = inv(istd(w)).
Definition 2.6. Let ρ be a composition of N . For a multiset permutation w ∈ S ρ the multistc is mstc(w) = stc(istd(w)).
The pair (mstc, inv) is equidistributed with (des, maj) on S ρ , as which together with (1.1) proves the following theorem.

Signed and even-signed permutations
MacMahon's result (1.1) for the symmetric group (i.e. for ρ 1 = . . . ρ m = 1) is often present in the literature as Carlitz's identity, satisfied by Carlitz's q-Eulerian polynomial A n (x, q) = σ∈Sn x des(σ) q maj(σ) . Such result was extended, for suitable statistics, to the groups of signed and even-signed permutations. The major indices so defined are in both cases equidistributed with the Coxeter length . In this section we define type B and type D analogues of the statistic stc, that together with the length satisfy these generalised Carlitz's identities.
Motivated by (3.1) and the well-known fact that the length in type B may be also written as we define the analogue of the statistic stc for signed permutations as follows. Proof. We use essentially the same argument as in the proof of [8,Theorem 3]. There, the following decomposition of B n is used. Every permutation τ ∈ S n is associated with 2 n elements of B n , via the choice of the n signs. More precisely, given a signed permutation σ ∈ B n one can consider the ordinary permutation in which the elements are in the same relative positions as in σ. We write π(σ) = τ . Then where B(τ ) = {σ ∈ B n | π(σ) = τ }. So every σ ∈ B n is uniquely identified by the permutation τ = π(σ) and the choice of signs J(σ) = {σ(j) | j ∈ Neg(σ)}. Clearly, for σ ∈ B n we have I(σ) = I(π(σ)), and thus stc(σ) = stc(π(σ)). So, for τ = π(σ) x nstc(σ) q (σ) = x stc(τ ) q inv(τ ) The claim follows, as (1 + xq i ).

3.2.
Eulerian companion for the length on D n . The even hyperoctahedral group D n is the subgroup of B n of signed permutations for which the negative statistic is even: Also for σ in D n the Coxeter length can be computed in terms of statistics: The problem of finding an analogue, on the group D n of even signed permutations, was solved in [2], where type D statistics des and maj were defined, as follows. For σ ∈ D n where DNeg(σ) = {i ∈ [n]|σ(i) < −1}. The following holds.   We now show that the statistic just defined constitutes an Eulerian partner for the length on D n , that is, the following holds. Proof. We use, as in [2] the following decomposition of D n . Let For α ∈ T n and τ ∈ S n the following holds: (ατ ) = (α)+ (τ ) = nsp(α)+inv(τ ), nsp(ατ ) = nsp(α), dstc(ατ ) = stc(τ )+neg(α)+ε(σ).