l−1} is split into two subsets I, J such that 0∈I and the order has the properties: (a) (j, i)<(j0, i0) when i &gt

A WREATH PRODUCT MULTIVARIABLE EXTENSION Dominique Foata and Guo-Niu Han

This Note is a complement to paper [6], especially Theorem 1.3, when the total ordering imposed on the wreath product C_l oS_n is defined as follows: the set {0,1, . . . , l−1} is split into two subsets I, J such that 0∈I and the order has the properties:

(a) (j, i)<(j⁰, i⁰) when i > i⁰ for all j, j⁰; (b) (1, i)<(2, i)<· · ·<(n, i) when i∈I; (c) (n, i)<· · ·<(2, i)<(1, i) when i∈J.

When J = ∅, we recover the total order used in [6]. For l = 2, I ={0},J ={1}, we get the natural order defined on the hyperoctahedral group B_n, namely, −n < · · · < −1 < 1 < · · · < n with the convention:

−j ≡(j,1), j ≡(j,0) for all j = 1,2, . . . , n.

We keep the notations used in [6]. When using the above new order, determined by the pair (I, J), the polynomial defined in [6], (1.12) will be redefined as

W_n(s, t, q,(Yi),(Zi), I, J)

:= X

(w,)∈CloS_n

s^fexc(w,)t^fdes(w,)q^fmaj(w,) Y

0≤i≤l−1

Z_i^||iY_i^fixi(w,),

to emphasize the presence of the partition (I, J). The computation made in [6], Section 3 can be reproduced verbatim. Instead of (3.3) and (3.2) we get

X

n≥0

(1 +t+· · ·+t^l−1)W_n(s, t, q,(Y_i),(Z_i), I, J) uⁿ (t^l;q^l)n+1

=X

r≥0

t^rG_r(u;s, q,(Y_i),(Z_i), I, J), where

G_r(u;s, q,(Y_i),(Z_i), I, J) := X

(c,w,)∈^WP(r,I,J)

u^λws^fexc(w,)q^totc Y

0≤i≤l−1

Y_i^fixi(w,)Z_i^||i.

The only difference is the following. The last sum is over the set^WP(r, I, J) of all wreathed permutations (c, w, ) that can be expressed as juxtaposi- tion products



 c w



=



 c⁽¹⁾ w⁽¹⁾

⁽¹⁾







 c⁽²⁾ w⁽²⁾

⁽²⁾



· · ·



 c^(k) w^(k)

^(k)



,

where each c^(j) is of the form a^m_j ^j (1≤ j ≤k), m₁ +m₂ +· · ·+m_k = n and a₁ > a₂ > · · · > a_k. Also, ^(j) = a^m_j ^j, where a_j is the residue of a_j mod l. Then, each word w^(j) is increasing (resp. decreasing) with respect of natural order of the integers, if and only ifj ∈I (resp. j ∈J).

Example. Let l = 3, I ={0,2}, J ={1}. Then





 Id

c w





=







1 2 3 4 5 6 7 8 9 10 11 12 13 12 12 11 9 9 9 7 7 7 7 4 4 2 1 2 13 5 7 12 10 8 4 3 11 9 6 0 0 2 0 0 0 1 1 1 1 1 1 2





,

is such a wreathed permutation. The factors ofwabove the factors 1,1,1,1 and 1,1 of are decreasing.

The important property on such wreathed permutations (which appears to be instrumental in the construction of the next bijection) is the fact each decreasing factor w^(j) such that ^(j) = a^m_j ^j and a_j ∈ J has at most onefixed point of (w, ). In the previous example there is one fixed point in each of the two factorsw^(j) such that a_j = 1, namely 8 and 11.

Let WP⁰_n(r, I, J) denote the subset of WP_n(r, I, J) of all wreathed permutations having no i-fixed points when i ∈ J and let ^WP0(r, I, J) be the union of all ^WP0_n(r, I, J). Also, let ^DW(i)n (r) be the subset of ^NIW_n(r) of allstrictly decreasingwords, whose letters are congruent toi modl and

DW⁽ⁱ⁾(r) be the union of such DW⁽ⁱ⁾_n (r).

Start with a wreathed permutation (c, w, ) from ^WP(r, I, J) and for each i ∈ J let j₁ < j₂ < · · · < j_h(i) be the increasing sequence of all integers j such that j is an i-fixed point: x_j = j and _j = i; define v⁽ⁱ⁾:=c_j1c_j2· · ·c_jh(i). As the wordcis monotonic non-increasing and since there is at most one i-fixed point in each factorw^(j) of w wheni∈J, the word v⁽ⁱ⁾ is strictly decreasing. More essentially,

v⁽ⁱ⁾ ∈^DW(i)(r) if (c, w, )∈^WP(r, I, J).

For all i ∈ J remove all the columns ^cjxj

j

such that j is an i- fixed point and form the family (v⁽ⁱ⁾)_i∈J. After removal the remaining

columns form a wreathed permutation on a subset K of [1, n]. Using the unique increasing bijection φ of K onto the interval [1,|K|], next obtain a wreathed permutation of order |K|. Each factor w^(j) of the original wreathed permutation (c, w, ) is then transformed into a word φ(w^(j)).

The final step consists of replacing each φ(w^(j)) into its mirror-image, whenever i ∈ J. Let (c⁰, w⁰, ⁰) be the wreathed permutation ultimately obtained. Then

(c⁰, w⁰, ⁰)∈WP⁰(r, I, J).

In the previous example the removals of columns 8 and 11 yield







1 2 3 4 5 6 7 9 10 12 13 12 12 11 9 9 9 7 7 7 4 2 1 2 13 5 7 12 10 4 3 9 6 0 0 2 0 0 0 1 1 1 1 2





, and v⁽¹⁾ :=c₈c₁₁ = 7,4.

After reduction we obtain







1 2 3 4 5 6 7 8 9 10 11 12 12 11 9 9 9 7 7 7 4 2 1 2 11 5 7 10 9 4 3 8 6 0 0 2 0 0 0 1 1 1 1 2





.

and, finally,





 Id

c⁰ w⁰

⁰





=







1 2 3 4 5 6 7 8 9 10 11 12 12 11 9 9 9 7 7 7 4 2 1 2 11 5 7 10 3 4 9 8 6 0 0 2 0 0 0 1 1 1 1 2





.

Proposition. The mapping (c, w, ) 7→ ((c⁰, w⁰, ⁰),(v⁽ⁱ⁾)_i∈J) is a bijec- tion ofWP(r, I, J)onto WP⁰(r, I, J)× Q

i∈J

DW⁽ⁱ⁾(r)having the properties:

(1) λw =λw⁰+ P

i∈J

λv⁽ⁱ⁾; (2) totc= totc⁰+ P

i∈J

totv⁽ⁱ⁾;

(3) ||_i =|⁰|_i for i∈I; ||_i =|⁰|_i+λv⁽ⁱ⁾ for i∈J;

(4) fix_i(w, ) = fix_i(w⁰, ⁰) for i∈I; (5) fix_i(w, ) =λv⁽ⁱ⁾ for i∈J; (6) exc(w, ) = exc(w⁰, ⁰);

(7) fexc(w, ) = fexc(w⁰, ⁰) + P

i∈J

i·λv⁽ⁱ⁾.

The proof of the Proposition requires no further techniques.

Now, by using the identity

(−u;q)_N+1 = X

n≥0

X

v∈^DWn(r)

q^totv, we have:

X

n≥0

uⁿ X

v∈^DW(i)n (r)

q^totv = X

n≥0

(uqⁱ)ⁿ X

v⁰∈^DWn(b(r−i)/lc)

q^ltotv0

= (−uqⁱ;q^l)b(r−i)/lc+1.

LetA(r, J) be the set of all sets (w⁽ⁱ⁾)_i∈J, wherew⁽ⁱ⁾∈^DW(i)(r) (i∈J).

Summing over all pairs ((w⁽ⁱ⁾)_i∈J,(c, w, )) from A(r, J)×WP(r, I, J) we get:

X

(w⁽ⁱ⁾)i∈J,(c,w,)

Y

i∈J

(usⁱZ_i)^λw(i)q^totw(i)

×u^λ(c)q^totcs^fexc(w,) Y

i∈I+J

Z_i^||iY

i∈I

Y_i^fixi(w,)Y

i∈J

Y_i^fixi(w,)

=Y

i∈J

(−usⁱqⁱZ_i;q^l)b(r−i)/lc+1

× X

(c,w,)∈^WP(r,I,J)

u^λ(c)q^totcs^fexc(w,) Y

i∈I+J

Z_i^||iY

i∈I

Y_i^fixi(w,)Y

i∈J

Y_i^fixi(w,)

=Y

i∈J

(−usⁱqⁱZ_i;q^l)b(r−i)/lc+1 ×G_r(u;s, q,(Y_i),(Z_i), I, J).

As fexc(w, ) = fexc(w⁰, ⁰) +P

i∈Ji λv⁽ⁱ⁾, the initial expression can also be summed over triples ((w⁽ⁱ⁾)i∈J,(c⁰, w⁰, ⁰),(v⁽ⁱ⁾)i∈J) from A(r, J) ×

WP⁰(r, I, J)×A(r, J) in the form X

(w⁽ⁱ⁾)i∈J,(c⁰,w⁰,⁰),(v⁽ⁱ⁾)i∈J

u^{λ c0}q^totc0s^fexc(w0,0) Y

i∈I+J

Z^|

0|i

i

Y

i∈I

Y^fixi(w

0,⁰) i

×Y

i∈J

(usⁱY_iZ_i)^{λ v(i)}q^totv(i)×Y

i∈J

(usⁱZ_i)^λw(i)q^totw(i)

= X

(v⁽ⁱ⁾)i∈J

Y

i∈J

(usⁱY_iZ_i)^{λ v(i)}q^totv(i)

× X

(c⁰,w⁰,⁰),(w⁽ⁱ⁾)i∈J

u^{λ c0}q^totc0s^fexc(w0,0) Y

i∈I+J

Z^|

0|i

i

Y

i∈I

Y^fixi(w

0,⁰) i

×Y

i∈J

(usⁱZ_i)^{λ w(i)}q^totw(i)

= Y

i∈J

(−usⁱqⁱY_iZ_i;q^l)b(r−i)/lc+1

× X

(c,w,)∈^WP(r,I,J)

u^{λ c}q^totcs^fexc(w,) Y

i∈I+J

Z_i^||iY

i∈I

Y_i^fixi(w,).

But the last sum is also equal to X

(c,w,)∈^WP(r)

u^{λ c}q^totcs^fexc(w,) Y

i∈I+J

Z_i^||iY

i∈I

Y_i^fixi(w,)

=G_r(u;s, q,(Y_i),(Z_i))|_Y

i=1 for all i∈J,

whereG_r(u;s, q,(Y_i),(Z_i)), itself expressed in [6] (3.2), has been proved to be equal to

Q

1≤i≤l

(uqⁱsⁱZ_i;q^l)b(r−i)/lc+1

Q

0≤i≤l−1

(uqⁱsⁱY_iZ_i;q^l)b(r−i)/lc+1

H_r(u;q, s,(Z_i)), where

H_r(u;q, s,(Z_i)) =Z₀(1−q^ls^l)

×

Z₀−Z₀q^ls^l +

l

X

i=1

qⁱsⁱZ_i−

l

X

i=1

qⁱsⁱZ_i(uq^ls^lZ₀;q^l)b(r−i)/lc+1

(uZ₀;q^l)b(r−i)/lc+1

−1

.

Hence

G_r(u;s, q,(Y_i),(Z_i))|_Y

i=1 for all i∈J

= Q

i∈I

(uqⁱsⁱZ_i;q^l)b(r−i)/lc+1

Q

i∈I

(uqⁱsⁱY_iZ_i;q^l)b(r−i)/lc+1

H_r(u;q, s,(Zi)).

From the identity Y

i∈J

(−usⁱqⁱZ_i;q^l)b(r−i)/lc+1×G_r(u;s, q,(Yi),(Zi), I, J)

=Y

i∈J

(−usⁱqⁱY_iZ_i;q^l)b(r−i)/lc+1×G_r((u;s, q,(Y_i),(Z_i))|_Y

i=1 for alli∈J

we deduce:

G_r(u;s, q,(Yi),(Zi), I, J)

= Q

i∈J

(−usⁱqⁱY_iZ_i;q^l)b(r−i)/lc+1

Q

i∈J

(−usⁱqⁱZ_i;q^l)b(r−i)/lc+1

Q

i∈I

(uqⁱsⁱZ_i;q^l)b(r−i)/lc+1

Q

i∈I

(uqⁱsⁱY_iZ_i;q^l)b(r−i)/lc+1

×H_r(u;q, s,(Z_i)).

Consequently, the factorial generating function for the polynomials W_n(s, t, q,(Y_i),(Z_i), I, J) can be expressed in the form:

X

n≥0

(1 +t+· · ·+t^l−1)W_n(s, t, q,(Y_i),(Z_i), I, J) uⁿ (t^l;q^l)_n+1

=X

r≥0

t^rG_r(u;s, q,(Yi),(Zi), I, J).

When l = 2, I = {0}, J = {1}, Z₀ = 1, Z₁ = Z, with the convention

−j ≡(j,1), j ≡(j,0) for all j = 1,2, . . ., we recover the following identity derived in [1] for the hyperoctahedral group:

X

n≥0

(1 +t)Wn(s, t, q, Y0, Y₁, Z) uⁿ (t²;q²)_n+1

=X

r≥0

t^r (u;q²)_br/2c+1 (uY₀;q²)_br/2c+1

(−usqY₁Z;q²)_b(r+1)/2c (−usqZ;q²)_b(r+1)/2c

× (us²q²;q²)_br/2c(1−s²q²) (u;q²)_b(r+1)/2c(u;q²)_br/2c (u;q²)_br/2c+1

(u;q²)_b(r+1)/2c (u;q²)_br/2c−s²q²(us²q²;q²)_br/2c +sqZ(u;q²)_br/2c (u;q²)_b(r+1)/2c−(us²q²;q²)_b(r+1)/2c

.

References

[1] Dominique Foata, Guo-Niu Han. Signed words and permutations, V; a sextuple distribution, Ramanujan J., vol. 19, , p. 29–52.

[6] Dominique Foata; Guo-Niu Han. The decrease value theorem with an application to permutation statistics, 19 p.