Séminaire Lotharingien de Combinatoire 56 (2007), Article B56d

EXCEDANCE NUMBER FOR INVOLUTIONS IN COMPLEX REFLECTION GROUPS

ELI BAGNO, DAVID GARBER, AND TOUFIK MANSOUR

Abstract. We define the excedance number on the complex reflection groups and compute its multidistribution with the number of fixed points on the set of involutions in these groups. We use some recurrences and generating functions manipulations to obtain our results.

1. Introduction

Let V be a complex vector space of dimension n. A pseudo-reflection on V is a linear transformation on V of finite order which fixes a hyperplane in V pointwise. A complex reflection group on V is a finite subgroup W of GL(V) generated by pseudo-reflections.

Irreducible finite complex reflection groups have been classified by Shephard-Todd [3]. In particular, there is a single infinite family of groups and exactly 34 other “exceptional” complex reflection groups. The infinite family G_{r,p,n}, where r, p, n are positive integers with p | r, consists of the groups of n × n matrices such that:

(1) The entries are either 0 or r^{th} roots of unity;

(2) There is exactly one nonzero entry in each row and each column;

(3) The (r/p)^{th} power of the product of the nonzero entries is 1. The classical Weyl groups appear as special cases: G_{1,1,n} = S_{n} the

symmetric group, G_{2,1,n} = B_{n} the hyperoctahedral group, and G_{2,2,n} = D_{n}, the group of even-signed permutations.

In S_{n} one can define the following well-known parameters: given σ ∈ S_{n}, i ∈ [n] is an excedance of σ if and only if σ(i) > i. The number of excedances is denoted by exc(σ). Another natural parameter on S_{n} is the number of fixed points, denoted by fix(σ).

We say that a permutation π ∈ G_{r,p,n} is an involution if π^{2} = 1. Let I_{r,p,n} be the set of involutions in the complex reflection group G_{r,p,n}.

In this paper we are interested in computing the number of involutions having specific numbers of fixed points and excedances. We do this by producing recurrences, and computing them explicitly by the corresponding generating functions.

Here are our main results (the relevant definitions can be found in Section 3).

Theorem 1.1. (See Corollaries 5.4, 5.5 and 5.6).

(1) The number of involutions π ∈ G_{r,p,n} where r is odd and p | r with exc^{Clr}(π) = m is

∑_{j= \frac{n}{2}}^{n} (n − j)!\binom{n}{n − j, n − j, j − k, k − n + j}(\frac{r}{2})^{n−j}.

(2) The number of involutions π ∈ G_{r,p,n} where r is even and p | \frac{r}{2} with exc^{Clr}(π) = m is

k!\binom{n}{k, k, n − 2k}(\frac{r}{2})^{k},

where k = \frac{m}{r}.

Theorem 1.2. (See Corollary 5.10).

The number of involutions π ∈ G_{r,p,n} (r is even, p ∤ \frac{r}{2}) with exc^{Clr}(π) = m is

\frac{(\frac{m}{r})!}{2^{\frac{m}{r}}} \binom{n}{\frac{m}{r}, \frac{m}{r}, n − 2\frac{m}{r}}(r + 1)^{\frac{m}{r}} .

This paper is organized as follows. In Section 2, we recall some properties of G_{r,p,n}. In Section 3 we define some parameters on G_{r,n} and hence also on G_{r,p,n}. In Section 4 we classify the involutions of G_{r,n} and G_{r,p,n}, and finally in Section 5 we compute the corresponding recurrences and explicit formulas.

2. Preliminaries

2.1. Complex reflection groups.

Definition 2.1. Let r and n be positive integers. The group of colored permutations of n digits with r colors is the wreath product

G_{r,n} = ℤ_{r} ≀ S_{n} = ℤ_{r}^{n} ⋊ S_{n},

consisting of all the pairs (z, τ) where z is an n-tuple of integers between 0 and r − 1 and τ ∈ S_{n}. The multiplication is defined by the following rule: for z = (z_{1}, …, z_{n}) and z′ = (z′_{1}, …, z′_{n})

(z, τ) · (z′, τ′) = ((z_{1} + z′_{τ^{−1}(1)}, …, z_{n} + z′_{τ^{−1}(n)}), τ ∘ τ′)

(here + is taken modulo r).

EXCEDANCES FOR INVOLUTIONS IN G_{r,p,n} 3

We use some conventions along this paper. For an element π = (z, τ) ∈ G_{r,n} with z = (z_{1}, …, z_{n}) we write z_{i}(π) = z_{i}. For π = (z, τ), we denote |π| = (0, τ), (0 ∈ ℤ_{r}^{n}).

A much more natural way to present G_{r,n} is the following: consider the alphabet Σ = {1^{[0]}, …, n^{[0]}, 1^{[1]}, …, n^{[1]}, …, 1^{[r−1]}, …, n^{[r−1]}} as the set [n] = {1, …, n} colored by the colors 0, …, r − 1. Then, an element of G_{r,n} is a colored permutation, i.e., a bijection π : Σ → Σ such that if π(i) = k^{[t]} then π(i^{[j]}) = k^{[t+j]} where 0 ≤ j ≤ r −1 and the addition is taken modulo r. Occasionally, we write j bars over a digit i instead of i^{[j]}. For example, an element (z, τ) = ((1, 0, 3, 2), (2, 1, 4, 3)) ∈ G_{3,4} will be written as (¯21¯¯¯4¯¯3).

For each p | r we define the complex reflection group

(2.1)  G_{r,p,n} := {g ∈ G_{r,n} | csum(g) ≡ 0 mod p},

where

csum(σ) = ∑_{i=1}^{n} z_{i}(σ).

3. Statistics on G_{r,n} and its subgroups

In this section we define some parameters on G_{r,n}. The complex reflection group G_{r,p,n} inherits all of them. Given any ordered alphabet Σ′, we recall the definition of the excedance set of a permutation π on Σ′,

Exc(π) = {i ∈ Σ′ | π(i) > i},

and the excedance number is defined to be exc(π) = |Exc(π)|.

Definition 3.1. We define the color order on the set

Σ = {1, …, n, ¯1, …, n̄, …, 1^{[r−1]}, …, n^{[r−1]}}

by

1^{[r−1]} < ⋯ < n^{[r−1]} < 1^{[r−2]} < 2^{[r−2]} < ⋯ < n^{[r−2]} < ⋯

< 1^{[0]} < ⋯ < n^{[0]}.

We note that there are some other possible ways of defining orders on Σ, some of them lead to other versions of the excedance number, see for example [1].

Example 3.2. Given the color order

¯¯1 < ¯ ¯2 < ¯ ¯3 < ¯1 < ¯2 < ¯3 < 1 < 2 < 3,

we write σ = (3¯1¯¯2) ∈ G_{3,3} in an extended form:

(\begin{matrix}¯¯1 & ¯¯2 & ¯¯3 & ¯1 & ¯2 & ¯3 & 1 & 2 & 3 \\ ¯¯3 & 1 & ¯2 & ¯3 & ¯¯1 & 2 & 3 & ¯1 & ¯¯2. \end{matrix})

We calculate: Exc(σ) = {¯¯1, ¯ ¯2, ¯ ¯3, ¯1, ¯3, 1} and exc(σ) = 6.

Let σ ∈ G_{r,n}. Recall the definition of the parameter csum from the end of Section 2 as

csum(σ) = ∑_{i=1}^{n} z_{i}(σ),

and define

Exc_{A}(σ) = {i ∈ [n − 1] | σ(i) > i},

where the comparison is with respect to the color order.

Define also

exc_{A}(σ) = |Exc_{A}(σ)|.

Example 3.3. Take σ = (¯1¯¯34¯2) ∈ G_{3,4}. Then csum(σ) = 4, Exc_{A}(σ) = {3} and hence exc_{A}(σ) = 1.

Let σ ∈ G_{r,n}. Recall that for σ = (z, τ) ∈ G_{r,n}, |σ| is the permutation of [n] satisfying |σ|(i) = τ(i). For example, if σ = (¯2¯¯31¯4) then |σ| = (2314).

Now we can define the colored excedance number for G_{r,n}.

Definition 3.4. Define

exc^{Clr}(σ) = r · exc_{A}(σ) + csum(σ).

One can view exc^{Clr}(σ) in a different way (see [1]).

Lemma 3.5. Let σ ∈ G_{r,n}. Consider the set Σ ordered by the color order. Then

exc(σ) = exc^{Clr}(σ).

We say that i ∈ [n] is an absolute fixed point of σ ∈ G_{r,n} if |σ|(i) = i.

4. Involutions in G_{r,p,n}

As was already mentioned, we say that σ is an involution if σ^{2} = 1. In this section we classify the involutions of G_{r,p,n}. Note that each

involution of G_{r,p,n} can be decomposed into a product of ‘atomic’ involutions of two types: absolute fixed points and 2-cycles.

We start with the absolute fixed points. In the case p = 1, i.e., ℤ_{r} ≀ S_{n} = G_{r,n}, we distinguish between two subcases according to the parity of r. In the case of even r, an absolute fixed point can be one of the following two kinds: π(i) = i or π(i) = i^{[^{\frac{r}{2}}]}. If r is odd, an absolute fixed point can be only of the first kind.

If p > 1 and r is odd, we have the same absolute fixed points as in the case p = 1. On the other hand, if r is even, then we have to distinguish again between two subcases. If p | \frac{r}{2}, then the absolute fixed points in I_{r,p,n} are exactly as those of I_{r,1,n}. If p ∤ \frac{r}{2}, then an element of G_{r,p,n} with an odd number of fixed points of the form π(i) = i^{[^{\frac{r}{2}}]} is not an element of I_{r,p,n} and thus the only absolute fixed points are of the form π(i) = i or pairs of absolute fixed points of the form π(i) = i^{[^{\frac{r}{2}}]}; π(j) = j^{[^{\frac{r}{2}}]}.

In all cases, the 2-cycles have the form π(i) = j^{[k]}; π(j) = i^{[r−k]} where 0 ≤ k ≤ r − 1.

We conclude this section with an example.

Example 4.1. Let r = 18, p = 6, n = 7 and let

π = ( \begin{matrix}1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 7^{[2]} & 3 & 2 & 4^{[9]} & 5^{[9]} & 6 & 1^{[16]} \end{matrix} ) ∈ I_{18,6,7}.

Then π can be decomposed into the absolute fixed points ( \begin{matrix}6 \\ 6 \end{matrix} ), the pair

of absolute fixed points ( \begin{matrix}4 & 5 \\ 4^{[9]} & 5^{[9]} \end{matrix} ) and the following two 2-cycles:

( \begin{matrix}1 & 7 \\ 7^{[2]} & 1^{[16]} \end{matrix} ) and ( \begin{matrix}2 & 3 \\ 3 & 2 \end{matrix} ).

5. Recurrences and explicit formulas

In this section, we compute recurrences and explicit formulas for

f_{r,p,n}(u, v, w) = ∑_{π∈I_{r,p,n}} u^{fix(π)}v^{exc_{A}(π)}w^{csum(π)}

for all r and p where p | r.

5.1. Recurrences for G_{r,n} = G_{r,1,n}. Let π be any colored involution in I_{r,n} = I_{r,1,n}. Then we have either π(n) = n^{[j]} or π(n) = k^{[j]} with k < n.

If π(n) = n^{[j]}, then we distinguish between two subcases according to the parity of r, as we have seen in Section 4. If r is even we have j = 0 or j = \frac{r}{2}. If r is odd then j = 0.

For π ∈ I_{r,1,n} such that π(n) = n^{[j]}, define π′ ∈ I_{r,1,n−1} by ignoring the last digit of π. For π ∈ I_{r,1,n} with π(n) = k^{[j]} and π(k) = n^{[r−j]}, define π′′ ∈ I_{r,1,n−2} in the following way: write π in its complete notation, i.e., as a matrix of two rows, as in Example 4.1. The first row of π′′ is (1, 2, …, n − 2) while the second row is obtained from the second row of π by ignoring the digits n and k and the other digits are placed in an order preserving way with respect to the second row of π. Here is an explicit formula for the map π ↦ π′′:

π′′(i) = \begin{cases}π(i) & if 1 ≤ i < k and π(i) < k, \\ π(i) − 1 & if 1 ≤ i < k and π(i) > k, \\ π(i − 1) & if k ≤ i < n and π(i) < k, \\ π(i − 1) − 1 & if k ≤ i < n and π(i) > k. \end{cases}

Note that the map π ↦ π′ is a bijection from the set

{π ∈ I_{r,1,n} | π(n) = n^{[j]}} (j fixed)

to I_{r,1,n−1}, while π ↦ π′′ is a bijection from the set

{π ∈ I_{r,1,n} | π(n) = k^{[j]}} (j fixed)

to I_{r,1,n−2}.

For any r, if π(n) = n^{[j]} then

fix(π) = fix(π′) + 1,
exc_{A}(π) = exc_{A}(π′),
csum(π) = csum(π′) + j.

If π(n) = t^{[j]}, then the parameters satisfy

fix(π) = fix(π′′),
exc_{A}(π) = exc_{A}(π′′) + δ_{j,0},
csum(π) = csum(π′′) + r(1 − δ_{j,0}),

where δ_{i,j} is the Kronecker delta defined by

δ_{i,j} = \begin{cases}1 & if i = j, \\ 0 & if i ≠ j. \end{cases}

The above consideration gives the following recurrences, where we define μ_{r} = 1 + w^{\frac{r}{2}} for even r, and μ_{r} = 1 otherwise:

f_{r,1,n}(u, v, w) = uμ_{r}f_{r,1,n−1}(u, v, w) + (n − 1)(v + (r − 1)w^{r})f_{r,1,n−2}(u, v, w), n ≥ 1.

5.2. Explicit formulas for G_{r,n} = G_{r,1,n}. We turn now to the explicit formula. Define

F_{r,p}(x; u, v, w) = ∑_{n≥0} f_{r,p,n}(u, v, w)\frac{x^{n}}{n!}

= ∑_{n≥0} ∑_{π∈I_{r,p,n}} (u^{fix(π)}v^{exc_{A}(π)}w^{csum(π)}) \frac{x^{n}}{n!} .

Rewriting the recurrence in terms of generating functions, we obtain that

x\frac{∂}{∂x}F_{r,1}(x; u, v, w) = ∑_{n≥1} \frac{f_{r,1,n}(u, v, w)}{(n − 1)!}x^{n}

= uxμ_{r} ∑_{n≥1} \frac{x^{n−1}}{(n − 1)!}f_{r,1,n−1}(u, v, w)+

+ x^{2}(v + (r − 1)w^{r}) ∑_{n≥2} \frac{x^{n−2}}{(n − 2)!}f_{r,1,n−2}(u, v, w)

= uxμ_{r}F_{r,1}(x; u, v, w)+

+ x^{2}(v + (r − 1)w^{r})F_{r,1}(x; u, v, w).

Thus, the generating function F_{r,1}(x; u, v, w) satisfies

\frac{\frac{∂}{∂x}F_{r,1}(x; u, v, w)}{F_{r,1}(x; u, v, w)} = uμ_{r} + x(v + (r − 1)w^{r}).

Integrating with respect to x on both sides of the above differential equation, using the fact that F_{r,1}(0; u, v, w) = 1, we obtain the following proposition.

Proposition 5.1. Let r ≥ 1. The generating function F_{r,1}(x; u, v, w) is given by

e^{uxμ_{r}+ \frac{1}{2} x^{2}(v+(r−1)w^{r})}.

We are looking for an explicit expression for the polynomial f_{r,1,n}(u, v, w). From the definitions we have that \frac{f_{r,1,n}(u,v,w)}{n!} is the coefficient of x^{n} in F_{r,1}(x; u, v, w), namely [x^{n}]F_{r,1}(x; u, v, w). Computing the coefficient of x^{n} in the Maclaurin series of F_{r,1}(x; u, v, w) one gets

f_{r,1,n}(u, v, w) = n![x^{n}] ∑_{j≥0} \frac{1}{j!} (uxμ_{r} + \frac{1}{2}x^{2}(v + (r − 1)w^{r}))^{j}

= n![x^{n}] ∑_{j≥0} ∑_{i=0}^{j} \binom{j}{i}\frac{x^{i+j}u^{j−i}(v + (r − 1)w^{r})^{i}}{j!2^{i}} μ_{r}^{j−i}

= n! ∑_{j=n/2}^{n} \binom{j}{n − j}\frac{u^{2j−n}(v + (r − 1)w^{r})^{n−j}}{j!2^{n−j}} μ_{r}^{2j−n}.

Hence, we have the following corollary.

Corollary 5.2. The polynomial f_{r,1,n}(u, v, w) is given by (5.1)

∑_{j=n/2}^{n}(n − j)!\binom{n}{n − j, n − j, 2j − n}\frac{u^{2j−n}(v + (r − 1)w^{r})^{n−j}}{2^{n−j}} μ_{r}^{2j−n}.

If we substitute w = 1 and compute the coefficient of u^{m}v^{ℓ} in Formula (5.1), we get the following result.

Corollary 5.3. Let r ≥ 1. The number of colored involutions in G_{r,n} with exactly m absolute fixed points and exc_{A}(π) = ℓ is given by

(\frac{n − m}{2})!(r − 1)^{\frac{n−m}{2} −ℓ}\binom{n}{\frac{n−m}{2}, m, \frac{n−m}{2} − ℓ, ℓ}2^{\frac{m(3−2k)−n}{2}} ,

where k ∈ {0, 1} and k ≡ r (mod 2).

It is easy to see that if r = 1, then 2exc_{A}(π)+fix(π) = n for each involution π of S_{n}. From the above corollary we have then that the number of involutions in S_{n} with exactly ℓ excedances is given by \frac{l!}{2^{l}} \binom{n}{l,l,n−2l}.

We turn now to the computation of the number of involutions with a fixed number of excedances. We do this by substituting u = 1 and v = w^{r} in Formula (5.1).

Corollary 5.4. The number of involutions π ∈ G_{r,n} with exc^{Clr}(π) = m is

k!\binom{n}{k, k, n − 2k}(\frac{r}{2})^{k}  if r ≡ 1 (mod 2),
∑_{j= \frac{n}{2}}^{n} (n − j)!\binom{n}{n − j, n − j, j − k, k − n + j}(\frac{r}{2})^{n−j}
if r ≡ 0 (mod 2),

where k = \frac{m}{r}.

Note that k is an integer, since exc^{Clr}(π) is an integral multiplicity of r, for π ∈ I_{r,n}.

5.3. Recurrences and explicit formulas for G_{r,p,n} where r is odd, p > 1. As we have seen in Section 4, the involutions in this case coincide with the involutions of G_{r,1,n} where r is odd and thus we have the following corollary.

Corollary 5.5. The recurrence for f_{r,p,n}(u, v, w) for odd r is

f_{r,p,n}(u, v, w) = uf_{r,p,n−1}(u, v, w)

+ (n − 1)(v + (r − 1)w^{r})f_{r,p,n−2}(u, v, w), n ≥ 1,

and thus its explicit formula is

f_{r,p,n}(u, v, w)

= ∑_{j=n/2}^{n}(n − j)!\binom{n}{n − j, n − j, 2j − n}\frac{u^{2j−n}(v + (r − 1)w^{r})^{n−j}}{2^{n−j}} .

5.4. Recurrences and explicit formulas for G_{r,p,n} where r is even and p > 1, p | \frac{r}{2}. Also in this case, we have that the involutions coincide with the involutions of G_{r,1,n} where r is even, and thus we have the following corollary.

Corollary 5.6. The recurrence for f_{r,p,n}(u, v, w) for even r and p > 1, p | \frac{r}{2} is

f_{r,p,n}(u, v, w) = u(1 + w^{\frac{r}{2}} )f_{r,p,n−1}(u, v, w)

+ (n − 1)(v + (r − 1)w^{r})f_{r,p,n−2}(u, v, w), n ≥ 1,

and thus its explicit formula is

f_{r,p,n}(u, v, w) = ∑_{j=n/2}^{n}(n − j)!\binom{n}{n − j, n − j, 2j − n}

· \frac{u^{2j−n}(v + (r − 1)w^{r})^{n−j}}{2^{n−j}} (1 + w^{\frac{r}{2}} )^{2j−n}.

5.5. Recurrences and explicit formulas for G_{r,p,n} where r is even and p > 1, p ∤ \frac{r}{2}. Let π be any colored involution in I_{r,p,n}. Then, according to Section 4, we have either π(n) = n^{[j]} (where j = 0 or j = \frac{r}{2}) or π(n) = k^{[j]} with k < n.

We start with the recurrence. Let π be any colored involution in I_{r,p,n}. Then we have several cases:

(1) π(n) = n. In this case define π′ ∈ I_{r,p,n−1} by ignoring the last digit of π. The map π ↦ π′ is a bijection from the set {π ∈ I_{r,p,n} | π(n) = n} to I_{r,p,n−1}.

We have

fix(π) = fix(π′) + 1,
exc_{A}(π) = exc_{A}(π′),
csum(π) = csum(π′).

(2) π(n) = n^{[^{\frac{r}{2}}]} and there exists some k < n such that π(k) = k^{[^{\frac{r}{2}}]}.

Define π′′ ∈ I_{r,p,n−2} as in Section 5.1.

Note that π ↦ π′′ is a bijection from the set

\{π ∈ I_{r,p,n} | π(n) = n^{[^{\frac{r}{2}}]}\}

to I_{r,p,n−2}.

We have

fix(π) = fix(π′′) + 2,
exc_{A}(π) = exc_{A}(π′′),
csum(π) = csum(π′′) + r.

(3) π(n) = k^{[j]} with k < n and we have π(k) = n^{[r−j]}. In this case, we use π′′ ∈ I_{r,p,n−2} as above. Note that in this case π ↦ π′′ is a bijection from the set

{π ∈ I_{r,p,n} | π(n) = k^{[j]}}

to I_{r,p,n−2}. We get in this case

fix(π) = fix(π′′),
exc_{A}(π) = exc_{A}(π′′) + δ_{j,0},
csum(π) = csum(π′′) + r(1 − δ_{j,0}).

The above consideration gives the following recurrence:

f_{r,p,n}(u, v, w) = uf_{r,p,n−1}(u, v, w)

+ (n − 1)(u^{2}w^{r} + (r − 1)w^{r} + v)f_{r,p,n−2}(u, v, w),

where n ≥ 1.

By similar arguments to the ones we have used in Section 5.2, we get the following generating function and explicit formula.

Proposition 5.7. Let r ≥ 1. The generating function F_{r,p}(x; u, v, w) is given by

e^{ux+ \frac{1}{2} x^{2}((u^{2}+(r−1))w^{r}+v)}.

Corollary 5.8. The polynomial f_{r,p,n}(u, v, w) is given by (5.2)

∑_{j=n/2}^{n}(n − j)!\binom{n}{n − j, n − j, 2j − n}\frac{u^{2j−n}(v + (u^{2} + (r − 1))w^{r})^{n−j}}{2^{n−j}} .

If we substitute w = 1 and compute the coefficient of u^{m}v^{ℓ} in Formula (5.2), we get the following result.

Corollary 5.9. Let r ≥ 1. The number of colored involutions in G_{r,p,n} (r is even, p ∤ \frac{r}{2}) with exactly m absolute fixed points and exc_{A}(π) = ℓ is given by

∑_{j= \frac{n}{2}}^{n} \frac{(n − j)!}{2^{n−j}} \binom{n}{n − j, 2j − n, l, \frac{n−m}{2} − l, \frac{m+n}{2} − j}(r − 1)^{\frac{n−m}{2} −l}.

For computing the number of involutions with a fixed number of excedances, we substitute u = 1 and v = w^{r} in Formula (5.2).

Corollary 5.10. The number of involutions π ∈ G_{r,p,n} (r is even, p ∤ \frac{r}{2}) with exc^{Clr}(π) = m is

\frac{(\frac{m}{r})!}{2^{\frac{m}{r}}} \binom{n}{\frac{m}{r}, \frac{m}{r}, n − 2\frac{m}{r}}(r + 1)^{\frac{m}{r}} .

References

[1] E. Bagno and D. Garber, On the excedance number of colored permutation groups, Seminaire Lotharingien Combin. 53 (2006), Art. B53f, 17 pp. (Elec-

  tronic, available at http://igd.univ-lyon1.fr/~slc).

[2] G. Ksavrelof and J. Zeng, Two involutions for signed excedance numbers, Seminaire Lotharingien Combin. 49 (2003), Art. B49e, 8 pp. (Electronic, available

  at http://igd.univ-lyon1.fr/~slc).

[3] G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954), 274–304.

Einstein institute of Mathematics, The Hebrew University, Givat Ram, 91904 Jerusalem, Israel, and The Jerusalem College of Technology, Jerusalem, Israel

E-mail address: bagnoe@math.huji.ac.il,bagnoe@jct.ac.il

School of Sciences, Holon Institute of Technology, PO Box 305, 58102 Holon, Israel

E-mail address: garber@hit.ac.il

Department of Mathematics, University of Haifa, Haifa, Israel

E-mail address: toufik@math.haifa.ac.il