Excedances in classical and affine Weyl groups

Journal of Combinatorial Theory, Series A 120 (2013) 1216–1234

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Journal of Combinatorial Theory,Series A

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Excedances in classical and affine Weyl groups ✩

Pietro Mongelli

Dipartimento di Matematica, Sapienza Università di Roma, P.le Aldo Moro 5, 00185 Rome, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 8 March 2012Available online 21 March 2013

Keywords:Colored excedanceAbsolute excedanceWeyl groups

Based on the notion of colored and absolute excedances introducedby Bagno and Garber we give an analogue of the derangementpolynomials. We derive some basic properties of these polynomials.Moreover, we define an excedance statistic for the affine Weylgroups of type Bn , Cn and Dn and determine the generatingfunctions of their distributions. This paper is inspired to one ofE. Clark, R. Ehrenborg in which the authors introduce the excedancestatistic for the group of affine permutations and ask if this statisticcan be defined for other affine groups.

© 2013 Elsevier Inc. All rights reserved.

1. Introduction

The symmetric group Sn has many interesting permutation statistics. The most important ones areinversions, descents, excedances and the major index. It is well known that the descent statistic andthe excedance statistic are equidistributed, and the same is true for the inversion statistic and themajor index. Some of these statistics, namely inversions and descents, have natural analogues in allCoxeter groups. Generalizing the other two and the corresponding results has been much more diffi-cult and, already for the group of signed permutations S B

n (also known as the hyperoctahedral group)has occupied a number of mathematicians for a number of years (see e.g. [2,6,7,11,12]). The first def-inition of excedance in S B

n can be found in [6]: here Brenti introduced a statistic equidistributed withthe Coxeter descents of S B

n . A good study of it and of its distribution on the set of all derangements inS B

n can be found in [7]. Two other definitions of excedance are given in [2]: here Bagno and Garber in-troduced these statistics for the family of all colored permutation groups (i.e. all the wreath productsZ

nr � Sn , with Zr the cyclic group of order r) which includes S B

n � Zn2 � Sn . The two definitions de-

pend on two orders which are natural if we look at S Bn as the wreath product Zn

2 � Sn . The symmetricgroup Sn and the hyperoctahedral group S B

n are also finite Weyl groups. In this terminology the two

✩ This paper is part of the author’s Ph.D. thesis written under the direction of Prof. F. Brenti at the Univ. “la Sapienza” ofRome, Italy.

E-mail address: [email protected].

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groups are known as An−1 and Bn respectively. They are geometrically viewed as groups generatedby some reflections in R

n . To every finite Weyl group W there is associated an affine Weyl group W ,obtained by adding one more generator which makes the group infinite. Therefore, W is canonicallyembedded in W . In [9] Clark and Ehrenborg define an excedance statistic on the set of all affine per-mutations Sn which is an affine Weyl group of type An−1. The definition (see (6.1)) agrees with theclassical statistic on the subset Sn of Sn .

In this paper we define excedance statistics for all the remaining infinite families of Weyl andaffine Weyl groups and study the distribution of these new statistics (as well as of some of theknown ones) on the whole group and on the set of its derangements. For the affine Weyl groups weare able to obtain affine analogues of two of the excedance statistics that have been introduced so farfor the hyperoctahedral group (but not of the third one). In both cases, these statistics reduce to theknown ones when restricted to Bn .

The organization of this paper is as follows. In the next section we collect several definitions,notation and results that will be used in the sequel. In Sections 3 and 4 we study properties of theexcedance statistics introduced in [2] on S B

n and on the set of all derangements of S Bn . In Section 5 we

investigate the distribution of the excedance statistics on S Dn . In Sections 6 and 7 we introduce the

definitions of excedance for the affine Weyl group SCn and study their generating functions on it and

on the subgroups S Bn , S D

n . In the last section we study some aspects of log-concavity and unimodalityof the generating functions on the groups S B

n , S Dn and their derangements.

2. Notation, definitions and preliminaries

In this section we collect some definitions, notation and results that will be used in the rest ofthis paper. We let P := {1,2,3, . . .}, N := P ∪ {0}, Z be the ring of integers and R be the field of realnumbers. Given n,m ∈ Z, n � m, we let [n,m] := {n,n + 1, . . . ,m}. The cardinality of a set A will bedenoted by |A|. We denote with χ the indicator function.

Given a sequence σ = (a1, . . . ,an) ∈ Zn we say that an index i ∈ [1,n] is an excedance (respectively,

descent) of σ if ai > i (respectively, ai > ai+1). We denote by exc(σ ) (respectively des(σ )) the numberof excedances (respectively descents) of σ . Given a set T we let S(T ) be the set of all bijectionsπ : T → T , and Sn := S([1,n]). If σ ∈ Sn then we write σ = σ1 · · ·σn to mean that σ(i) = σi , fori = 1, . . . ,n. We call such sequence the window notation of σ to distinguish it from the disjoint cyclenotation (see e.g. [14, p. 17]).

We denote by S Bn the group of all bijections σ in S([−n,n] \ {0}) such that σ(−i) = −σ(i) for all

i ∈ [−n,n] \ {0}, with composition as the group operation. This group is usually known as the groupof “signed permutations” on [1,n], or as the hyperoctahedral group of rank n. We have that S B

n is aWeyl group of type Bn (see e.g. [4, Proposition 8.1.3]). If σ ∈ S B

n then we write σ = [a1, . . . ,an] tomean that σ(i) = ai for i = 1, . . . ,n. We refer to it as the window notation of σ . We let

desA(σ ) := des(a1, . . . ,an),

excA(σ ) := exc(a1, . . . ,an),

neg(σ ) := ∣∣{i ∈ [1,n] ∣∣ ai < 0}∣∣. (2.1)

We denote by S Dn the group of all bijections σ ∈ S B

n such that neg(σ ) ≡ 0 (mod 2). It is a Weylgroup of type Dn (see e.g. [4, Proposition 8.2.3]).

We denote by SCn the set of all bijections π ∈ S(Z) such that π(−i) = −π(i) and π(i(2n + 1)) =

π(i) + 2n + 1 for all i ∈ Z, with composition as the group operation. The group SCn is an affine Weyl

group of type C (see e.g. [4, Proposition 8.4.3]). If π ∈ SCn then we write π = [a1, . . . ,an] to mean that

π(i) = ai for i = 1, . . . ,n. We refer to it as the window notation of π . Moreover, sometimes we willwrite π = (rπ ,σπ ), with rπ = (r1, . . . , rn) ∈ Z

n , σπ ∈ S Bn to mean that π(i) = σπ (i) + ri(2n + 1). Note

that any such pair (r, σ ) uniquely determines one element in SCn .

We denote by S Bn the subgroup of SC

n of all bijections π such that |{i � n | π(i) > n}| ≡ 0 (mod 2).The group S B

n is well defined and it is an affine Weyl group of type Bn . Note that π = (rπ ,σπ ) ∈ SCn

is in S Bn if and only if

1218 P. Mongelli / Journal of Combinatorial Theory, Series A 120 (2013) 1216–1234

n∑i=1

ri ≡ 0 (mod 2), (2.2)

as can be proven by inspection. In the following we say that the vector r ∈ Zn is even if it satis-fies (2.2), otherwise we say that r is odd. Finally, we denote by S D

n the subgroup of S Bn of all bijections

π such that |{i > 0 | π(i) < 0}| ≡ 0 (mod 2). The group S Dn is well defined and it is an affine Weyl

group of type Dn . Note that π = (rπ ,σπ ) ∈ S Bn is in S D

n if and only if

σπ ∈ S Dn . (2.3)

The Eulerian polynomials (see e.g. [14, Chapter 1]) are defined by A0(q) = 1 and for n � 1

An(q) :=n−1∑h=0

A(n,h)qh+1 :=∑σ∈Sn

qdes(σ )+1 =∑σ∈Sn

qexc(σ )+1.

The Eulerian polynomials have the following generating function∑n�0

An(q)zn

n! = 1 − q

1 − qe(1−q)z. (2.4)

In what follows we will find it convenient to use the notation

A0 = A0 = 1 and An = An

qfor n � 1. (2.5)

A bijection π : T → T , with T ⊆ Z is a derangement if π(i) �= i for all i ∈ T . The set of all derange-ments on [1,n] is denoted by SDn . Brenti [5] defined the derangement polynomials (to type A) byd0(q) = 1 and

dn(q) :=∑

σ∈SDn

qexc(σ ) for n � 1. (2.6)

Note that dn(q) is symmetric and unimodal for all n (see [5, Corollary 1]) and that it has only realzeros (Canfield, unpublished). The following formula is given in [7, Theorem 1.1] and it is derivedfrom [5].

Theorem 2.1. For n � 0,

dn(q) =n∑

k=0

(−1)n−k(

n

k

)Ak(q).

For all groups S Wn and S W

n (W = C, B, D) we denote with SDWn and SD

Wn respectively, the set

of all their derangements. Finally, given π ∈ S Wn or π ∈ S W

n we denote with fix(π) the number|{i ∈ [1,n] | π(i) = i}| of fixed element in its window notation and we denote |π | be the only per-mutation such that |π |(i) = |π(i)| for all i ∈ [1,n].

3. Colored excedance

In [2] Bagno and Garber introduced a definition of excedance on the set of signed permutations,called colored excedance.

Definition 3.1. For any σ ∈ S Bn we set

excClr(σ ) := 2 excA(σ ) + neg(σ ).


For example, if σ = [4,3,−1,−5,2], then neg(σ ) = 2, excA(σ ) = 2 and excClr(σ ) = 6. The coloredexcedance statistic is also noted as flag-excedance statistic (see e.g. [10–12]). In [2, Lemma 3.5] isgiven the following result which motivates the coefficient 2 in the above definition.

Lemma 3.2. Let σ ∈ S Bn and consider the color order on [−n,n] \ {0}

−1 < −2 < · · · < −n < 1 < 2 < · · · < n.

Then

excClr(σ ) = ∣∣{i ∈ [−n,n] \ {0} ∣∣ σ(i) > i}∣∣

where the comparison is with respect to the color order.

For example, if we write σ = (4,3,−1,−5,2) in an extended form:(−1 −2 −3 −4 −5 1 2 3 4 5−4 −3 1 5 −2 4 3 −1 −5 2

)then we have the six excedances corresponding to columns −1,−2,−3,−4,1,2.

Let Bn(q) be the generating function of the colored descent statistic on S Bn , i.e.

Bn(q) :=∑σ∈S B

n

qexcClr(σ ).

Proposition 3.3. Let n ∈ P. Then∑σ∈S B

n

q2 excA(σ )tneg(σ ) = An

(q2 + t

1 + t

)(1 + t)n.

In particular,

Bn(q) = An(q)(1 + q)n. (3.1)

Proof. Let σ ∈ S Bn be a signed permutation and let |σ | be the corresponding permutation in Sn . Then

the window notation of σ and |σ | differs in neg(σ ) entries. Note that putting a negative sign in anyentry i such that |σi| > i decreases the number of (classical) excedances and obviously increases thestatistic neg. The number of classical excedances does not change if |σi | < i. Let h be the number ofexcedances of |σ |. With previous remarks we have

∑σ∈S B

n

q2 excA(σ )tneg(σ ) =n−1∑h=0

∑|σ |∈Sn

exc(|σ |)=h

∑k1,k2

q2k1tn−k1−k2

(h

k1

)(n − h

k2

)

=n∑

h=0

A(n,h)(q2 + t

)h(1 + t)n−h

= An

(q2 + t

1 + t

)(1 + t)n,

where the indices k1,k2 denote the number of elements i with σ(i) > 0 in the set of all indices with|σ(i)| > i and |σ(i)| � i, respectively.

The last part of the statement follows by letting t = q. �Note that Eq. (3.1) can also be derived by [12, Theorem 5.3].


We now give an analogue formula for derangements in S Bn . Let dB

n (q) be the generating functionof the colored excedances on the set SDB

n , i.e.

dBn (q) :=

∑σ∈SDB

n

qexcClr(σ ).

Proposition 3.4. Let n ∈ P. Then

∑σ∈SDB

n

q2 excA(σ )tneg(σ ) = tnn∑

k=0

(n

k

)(t + 1

t

)k

dk

(q2 + t

t + 1

).

In particular,

dBn (q) = qn

n∑k=0

(n

k

)(q + 1

q

)k

dk(q).

Proof. In this proof, let SDBn denote the set of all permutations σ in S B

n with |σ(i)| �= i for all i ∈ [1,n].By using the same arguments in the proof of Proposition 3.3, we have

∑σ∈SDB

n

q2 excA(σ )tneg(σ ) =n−1∑h=0

∑σ∈SDn

exc(σ )=h

∑k1,k2

q2k1tn−k1−k2

(h

k1

)(n − h

k2

)

=n−1∑h=0

[qh]dn(q)

(q2 + t

)h(t + 1)n−h

= dn

(q2 + t

t + 1

)(t + 1)n,

where [qh]dn(q) denotes the coefficient of qh in dn(q). Now we consider all permutations inSDB

n \ SDBn . It is clear that for any permutation σ in this set there exists at least one index i such

that σ(i) = −i. By choosing all possible sets of such indices, we get

∑σ∈SDB

n

q2 excA(σ )tneg(σ ) =n∑

k=0

(n

k

) ∑σ∈SDB

k

q2 excA(σ )tneg(σ )tn−k

= tnn∑

k=0

(n

k

)(t + 1

t

)k

dk

(q2 + t

t + 1

).

The last part of the statement follows by letting t = q. �By Proposition 3.4, it is easy to compute the polynomials dB

n (q). Below are the polynomials dBn (q)

for n � 4

dB1 (q) = q,

dB2 (q) = q + 3q2 + q3,

dB3 (q) = q + 7q2 + 13q3 + 7q4 + q5,

dB4 (q) = q + 15q2 + 57q3 + 87q4 + 57q5 + 15q6 + q7.

All previous polynomials are symmetric. This property is true for all n � 1.


Proposition 3.5. For all n � 1 the polynomial dBn (q) is symmetric.

Proof. It suffices to prove that

q2ndBn

(1

q

)= dB

n (q).

Apply Proposition 3.4 and the property that dn(q) is symmetric for n � 1 (see [5, Corollary 1]). �Generating functions of the colored excedance on S B

n and SDBn are strictly related one to the other

as shown by the following easy formula

Bn(q) =n∑

k=0

(n

k

)dB

k (q). (3.2)

In fact, suppose that a signed permutation σ has n − k fixed points. Then by removing the fixedpoints and reducing the remaining elements to [−k,k] \ {0} by keeping the relative order, we get aderangement on [−k,k] \ {0}.

By (3.2) and [12, Theorem 5.2] we have the following result.

Corollary 3.6. We have

ez∑n�0

dBn (q)

zn

n! =∑n�0

Bn(q)zn

n! = q − 1

q − e(q2−1)z.

In [1, Section 4], Adin, Brenti and Roichman defined on S Bn the flag descent number by setting for

each σ ∈ S Bn

fdes(σ ) := 2 desA(σ ) + ε1(σ )

where desA is defined in (2.1) and ε1(σ ) := 1 if σ(1) < 0 and ε1(σ ) := 0 otherwise. It is simple toverify that fdes(σ ) = des(σ (−n), . . . , σ (−1),σ (1), . . . , σ (n)). In [1, Theorem 4.4] it is proved that∑

σ∈S Bn

qfdes(σ ) = (1 + q)n An(q).

By virtue of Proposition 3.3 we have the following already known fact (see e.g. [12, Theorem 1.4]).

Corollary 3.7. The flag descent numbers and the colored excedance numbers are equidistributed, i.e.∑σ∈S B

n

qfdes(σ ) =∑σ∈S B

n

qexcClr(σ ). (3.3)

Next we give a combinatorial interpretation of identity (3.3), by giving a bijection from S Bn onto

itself such that the excedance number of a signed permutation is equal to the flag descent number ofits image.

Combinatorial proof of Corollary 3.7. Let σ ∈ S Bn be a signed permutation and express it as a product

of disjoint signed cycles (see e.g. Brenti [6] and Chen [8]). We write each cycle with its minimumelement first and the cycles are in decreasing (natural) order of their first elements. For example,if σ = [4,−2,6,−5,−1,3] then we can write σ in the cycle form (3,6)(−2)(−5,−1,4). It is clearby definition that the number of colored excedances of σ is the sum of the colored excedances ofeach its cycles. We now give an algorithm that manipulates each cycle, by preserving the excedance


statistic. Let ρ be a cycle of σ whose first element is negative. Then decompose ρ into more cy-cles, just writing a closed and an opened bracket before each negative element different from thefirst. For example, if ρ is the cycle (−4,1,−2,−3,5,6) then we consider the three (ordered) cycles(−4,1), (−2), (−3,5,6). Since σ maps the rightmost element of each new cycle to a negative ele-ment, the excedance number is the same. Note that this operation is invertible: since all cycles areordered by their first elements in decreasing order, we can recognize what cycles are obtained byprevious algorithm. For example, the ordered sequence (−3,6)(−5)(−1,4)(−2) corresponds to thesigned permutation (−3,6)(−5,−1,4,−2) since −1,−2 > −5.

We now consider all cycles without negative elements. We associate to all these cycles the se-quence obtained just writing their elements without brackets. For example, consider the cycles(3,5,4)(1,6) then we associate to them the ordered sequence (35416), which we will consider asa unique cycle.

Now we define the map ψ : S Bn → S B

n such that excClr(σ ) = fdes(ψ(σ )) for all σ ∈ S Bn . Let σ ∈ S B

n .Write σ in cycles as done in the previous algorithm. Then we write a sequence according to thefollows rules: let nc be the number of the cycles and let δ = 1 if the first cycle has only positiveelements, δ = −1 otherwise. If nc is even and δ = 1 or nc is odd and δ = −1 then we write allelements of the first cycle with negative signs and in the same order; if nc is even and δ = −1 or ncis odd and δ = 1 then we write all elements of the first cycle with positive signs and in the reverseorder. In the next steps we write the elements of all other cycles with the opposite sign respect to theprevious cycle: when the sign is positive we reverse the order of the elements in the cycle. We havetherefore a sequence of n elements in [−n,n]\{0}. Let σ ′ be the signed permutation associated to thissequence in window notation and set ψ(σ ) = σ ′ . It is easy to verify that excClr(σ ) = fdes(ψ(σ )). �

For example, if σ = [2,3,−5,−1,4,6] then we rewrite σ = (6)(−5,4)(−1,2,3). Then ψ(σ ) =[6,−5,−4,3,2,1]. We have excClr(σ ) = 6 = fdes(ψ(σ )). If σ = [−2,1,−3,6,−4,5], then we haveσ = (−2,1)(−3)(−4,6,5), ψ(σ ) = [−2,−1,3, −4,−6,−5] and excClr(σ ) = fdes(σ ′) = 5.

Remark 3.8. The previous algorithm used to create the bijection ψ , allows us to give another proof ofCorollary 3.6. By writing each signed permutation σ ∈ S B

n in cycles after our algorithm, we have thatthe excedance number of σ is the sum of the excedances of its cycles. Therefore, we have to compute∑

m�1

∑l1,...,lm�1

(l1 + · · · + lm

l1, . . . , lm

) m∏j=1

Al j

(q2)(ql−1 + ql) zl1+···+lm

(l1 + · · · + lm)! .

The multinomial coefficient gives us the number of all partitions of l1 + · · · + lm in m blocks (i.e. theelements in the cycles); each factor Al j (q

2) gives the statistic∑

q2 excA(σ ) for all possible rearrange-

ments of the elements in the j-th block; ql + ql−1 gives the number of negative signs respectivelywhen the first block has or has not negative elements. Then, by applying (2.4), we get the same resultof Corollary 3.6.

To conclude this section, we give a recursion formula for the generating function of the derange-ments. In [3, Proposition 5.1] there is a recurrence relation for the generating function of all signedpermutations.

Proposition 3.9. For n � 2 the generating function

GFDBn (q, t) :=

∑σ∈SDB

n

q2 excA(σ )tneg(σ )

satisfies the following recursion:

GFDBn (q, t) = (

q2(n − 1) + nt)

GFDBn−1(q, t) + (

q2 + t)1 − q2

2q

∂

∂qGFDB

n−1(q, t)

+ (q2 + t

)(n − 1)GFDB

n−2(q, t).


Proof. Let σ ∈ S Bn−1. The permutation σ can be identified with σ ∈ S B

n having σ (i) = σ(i) for alli � n − 1 and σ(n) = n. Fix an index i � n − 1 and consider the two transpositions (i,n) and (i,−n).Then it is clear that

excA(σi(i,n)

) = 1 + excA(σ ) − χ(σ(i) > i

), neg

(σi(i,n)

) = neg(σ ),

excA(σi(i,−n)

) = excA(σ ) − χ(σ(i) > i

), neg

(σi(i,−n)

) = neg(σ ) + 1.

Therefore, we have

GFDBn (q, t) = t GFDB

n−1(q, t) +n−1∑i=1

∑σ ′∈SDB

n−2

q2 excA(σ ′)tneg(σ ′)(q2 + t)

+n−1∑i=1

∑σ ′∈SDB

n−1

q2 excA(σ ′)−2χ(σ ′(i)>i)tneg(σ ′)(q2 + t),

where t GFDn−1(q, t) corresponds to all elements σ ∈ SDBn having a cycle (−n), the first sum to all

elements having a cycle (i,±n) and the second sum to all other elements. Then,

GFDBn (q, t) = t GFDB

n−1(q, t) + (n − 1)(q2 + t

)GFDB

n−2(q, t)

+ (q2 + t

) ∑σ ′∈SDB

n−1

(excA

(σ ′)q2 excA(σ ′)−2tneg(σ ′)

+ (n − 1 − excA

(σ ′))q2 excA(σ ′)tneg(σ ′))

= t GFDBn−1(q, t) + (

q2 + t)(n − 1)GFDB

n−2(q, t)

+ (q2 + t

)(1 − q2

2q

∂

∂qGFDB

n−1(q, t) + (n − 1)GFDBn−1(q, t)

). �

4. Other excedance statistics

In [2] another excedance statistic was introduced. Given σ ∈ S Bn recall that |σ | is the permutation

in Sn such that |σ |(i) = |σ(i)| for all i ∈ [1,n].

Definition 4.1. For each σ ∈ S Bn , the absolute excedance number of σ is the number

excAbs(σ ) = exc(|σ |) + neg(σ ).

It is easy to verify that∑σ∈S B

n

qexc(|σ |)tneg(σ ) = (1 + t)n An(q). (4.1)

In particular, if we set t = q, by Proposition 3.3 we get∑σ∈S B

n

qexcAbs(σ ) =∑σ∈S B

n

qexcClr(σ ). (4.2)

We give a bijection which proves (4.2) combinatorially. Let σ ∈ S Bn and define φ : S B

n → S Bn by set-

ting φ(σ )(i) = −σ(i) if |σ(i)| > i, and φ(σ )(i) = σ(i) otherwise. It is easy to verify that excClr(σ ) =


excAbs(φ(σ )). For example, let σ = [4,3,−2,−6,5,−1], then φ(σ ) = [−4,−3,−2,6,5,−1] andexcClr(σ ) = 7 = excAbs(φ(σ )). Note that since φ(SDB

n ) = SDBn , it follows that∑

σ∈SDBn

qexcAbs(σ ) =∑

σ∈SDBn

qexcClr(σ ). (4.3)

With the same techniques used in the previous section, it is possible to verify that

∑σ∈SDB

n

qexc(|σ |)tneg(σ ) =n∑

h=0

(n

h

)dh(q)(t + 1)htn−h. (4.4)

By (4.1) and (4.3), many of results given for the colored excedance can be extended for the absoluteexcedance. Here we summarize them.

Corollary 4.2. Let n ∈ P then

∑σ∈SDB

n

qexcAbs(σ ) = dBn (σ ) =

n∑k=0

(n

h

)qn−k(q + 1)kdk(q).

Moreover the previous polynomial is symmetric and the exponential generating function is∑n�0

∑σ∈SDB

n

qexcAbs(σ ) zn

n! = q − 1

qez − eq2 .

Another immediate consequence is the following corollary.

Corollary 4.3. The flag descent numbers and the absolute excedance numbers are equidistributed, i.e.∑σ∈S B

n

qfdes(σ ) =∑σ∈S B

n

qexcAbs(σ )

and the map ψ ◦ φ−1 : S Bn → S B

n is a bijection which proves combinatorially this identity.

In the literature there exists another kind of excedance. Brenti [6] introduced the type B excedanceas follows.

Definition 4.4. Given σ ∈ S Bn , an element i ∈ [1,n] is a type B excedance of σ if σ(i) = −i or

σ(|σ(i)|) > σ(i).

Such statistic has the same distribution of the descent statistic in the Coxeter group theoretic sense(see [6] for more details). For this reason we denote this excedance number of a signed permutationσ ∈ S B

n with excCox(σ ). For a good exposition about the excedance of type B and derangements withrespect to these statistics see the paper of Chen, Tang and Zhao [7].

5. Excedance on the Weyl group S Dn

Since S Dn is a subgroup of S B

n we can define the excedance statistics introduced in previous sectionson it. In particular, since each permutation in S D

n has an even number of negative elements in itswindow notation, it is simple to compute the generating function for the excedance statistics on S D

n .


Corollary 5.1. For all n ∈ P we get

∑σ∈S D

n

qexcClr(σ ) = 1

2

(An(q)(1 + q)n + An(−q)(1 − q)n),

and ∑σ∈SDD

n

qexcClr(σ ) = 1

2

(dB

n (q) + dBn (−q)

)

=n∑

k=0

qn−k

2

(n

k

)((1 + q)kdk(q) + (−1)n−k(1 − q)kdk(−q)

).

It suffices to take from the generating functions on S Bn only the even powers of q.

It is also simple to compute the absolute excedance statistic.

Corollary 5.2. For all n ∈ P we get

∑σ∈S D

n

qexcAbs(σ ) = 1

2An(q)

((1 + q)n + (1 − q)n)

and

∑σ∈SDB

n

qexcAbs(σ ) = 1

2

n∑h=0

(n

h

)dh(q)

((1 + q)hqn−h + (1 − q)h(−q)n−h).

It suffices to take only even powers of the indeterminate t in (4.1) and in (4.4).For the excedance statistic of type B it suffices to note that if σ ∈ S B

n is such that |σ | �= (1,2, . . . ,n),let m be the greatest index such that σ(m) �= ±m then excCox(σm) = excCox(σ ) where σm is obtainedfrom σ just by changing the sign of m in its window notation. For example excCox([1,3,−4,2,−5]) =excCox([1,3,4,2,−5]) = 3. Therefore, except for the elements with only cycles of length 1, the distri-bution of the excedance is the same for permutations with even negative elements or odd negativeelements. If we denote with BCox

n (q) the generating function BCoxn (q) = ∑

σ∈S Bn

qexcCox(σ ) and with

dCoxn (q) the generating function of the derangements with respect to the excedances of type B we

have the following.

Proposition 5.3. For all n ∈ P

∑σ∈S D

n

qexcCox(σ ) = BCoxn (q) − (1 + q)n

2+ (1 + q)n + (1 − q)n

2= BCox

n (q) + (1 − q)n

2,

and ∑σ∈SDB

n

qexcCox(σ ) = 1

2dCox

n (q) + ε(n)

2qn,

where ε(n) = 1 if n is even, ε(n) = 0 if n is odd.


6. Colored excedance on the affine Weyl groups

In [9] an excedance statistic was introduced on the affine Weyl group Sn of all bijections π : Z → Z

satisfying the two conditions

π(i + n) = π(i) + n for all in∑

i=1

(π(i) − i

) = 0.

It is well known that Sn is an affine Weyl group of type An−1. The excedance is defined as follows:given π ∈ Sn ,

exc(π) =n∑

i=1

∣∣∣∣⌈π(i) − i

n

⌉∣∣∣∣ (6.1)

and when π is a bijection on [1,n] the definition agrees with the classical notion. The following resultis given in [9, Theorem 6.5].

Theorem 6.1. The generating function for affine excedances is given by

∑π∈ Sn

qexc(π) = 1

(1 − q2)n−1

n−1∑k=0

A(n,k + 1)

n−1−k∑i=0

(n − 1 − k

i

)(n − 1 + k

n − 1 − i

)q2i+k.

In the following we give the definitions of the affine excedance for the other infinite families ofaffine Weyl groups and compute their generating functions.

Definition 6.2. The colored affine excedance for any permutation π ∈ SCn is given by

excClr(π) = 2n∑

i=1

∣∣∣∣⌈π(i) − i

2n + 1

⌉∣∣∣∣ + neg(σπ ).

We recall that σπ is the element in S Bn defined by σπ (i) := π(i) (mod 2n + 1). For exam-

ple, let π ∈ SC5 be the permutation given by π = [−3,−1,4,2,16] in window notation. Then

σπ = (−3,−1,4,2,5) and excClr(π) = 2(0 + 0 + 1 + 0 + 1) + 2 = 6.It is easy to verify that if π is a bijection on [−n,n] then this definition agrees with Definition 3.1.

In the following we identify each element π ∈ SCn with the pair (rπ ,σπ ) as defined in Section 2.

Lemma 6.3. For all π ∈ SCn , let pσπ

∈ {−1,0}n be the vector with pσπ (i) = −1 if σπ (i) > i and 0 otherwise.Then the excedance statistic is given by

excClr(π) = 2‖rπ − pσπ ‖ + neg(σπ )

where ‖ · ‖ denotes the 1-norm in the Euclidian space Rn.

Proof. It suffices to note that∣∣∣∣⌈π(i) − i

2n + 1

⌉∣∣∣∣ =∣∣∣∣⌈ (2n + 1)ri + σπ (i) − i

2n + 1

⌉∣∣∣∣ ={ |ri| if i � σπ (i),

|ri + 1| if i < σπ(i).

By summing over all i the lemma follows. �Here we give a useful easy result.


Lemma 6.4. Let a,b ∈ N then∑r∈Za×(Z∗)b

q‖r‖ = 2bqb(1 + q)a

(1 − q)a+b.

Proof. It suffices to compute (1 + 2q + 2q2 + · · ·)a(2q + 2q2 + · · ·)b . �Theorem 6.5. The generating function for the colored affine excedance of SC

n is given by

GFClrCn

(q) :=∑π∈ SC

n

qexcClr(π) = n!(

1 + q2

1 − q

)n

.

Proof. By Lemma 6.3 we have that

GFClrCn

(q) =∑σ∈S B

n

qneg(σ )

( ∑r∈Zn

q2‖r−pσ ‖)

.

Since for any fixed σ , ‖r − pσ ‖ assumes the same values as ‖r‖ for r in all Zn , we can conclude byLemma 6.4 that

GFClrCn

(q) =∑σ∈S B

n

qneg(σ )

(1 + q2

1 − q2

)n

= n!(

1 + q2

1 − q

)n

. �

It is easy to compute the generating function for the derangements.

Theorem 6.6. The generating function for the colored affine excedance of SDCn is given by

GFDClrCn

:=(

1 + q2

1 − q

)n n∑i=0

(q(1 + q)

1 + q2

)i

dn−i(1).

Proof. By Lemma 6.3 we have that

GFDClrCn

=n∑

h=0

∑σ∈S B

nfix(σ )=h

qneg(σ )∑

r∈Zn−h×(Z∗)h

q2‖r−pσ ‖,

where fix(σ ) denotes the number of elements i ∈ [1,n] such that σ(i) = i. Note that r ranges inZ

n−h × (Z∗)h since (r, σ ) is a derangement and therefore ri �= 0 for all indices such that σ(i) = i.Note that for any fixed σ , ‖r − pσ ‖ assumes the same values as ‖r‖ for r in Z

n−h × (Z∗)h (notethat when pσ (i) �= 0 then ri could be zero). Then we conclude by Lemma 6.4

GFDClrCn

=n∑

h=0

∑σ∈S B

nfix(σ )=h

qneg(σ )

(2hq2h(1 + q2)n−h

(1 − q2)n

).

Now we have to compute the coefficient∑

σ∈S Bn , fix(σ )=h qneg(σ ) . We first choose h elements which

are fixed by σ . Then we distinguish the indices i ∈ [1,n] such that σ(i) = −i from the others. Then

∑σ∈S B

nfix(σ )=h

qneg(σ ) =(

n

h

) n−h∑k=0

(n − h

k

) ∑σ∈SDn−h−k

qk(1 + q)n−h−k.


Therefore we complete as follows

GFDClrCn

=(

1 + q2

1 − q

)n n∑h=0

(n

h

) n−h∑k=0

(n − h

k

)dn−h−k(1)

qk

(1 + q)h+k

2hq2h

(1 + q2)h

=(

1 + q2

1 − q

)n n∑i=0

i∑h=0

(q

1 + q

)i

dn−i(1)

(n

h

)(n − h

i − h

)(2q

1 + q2

)h

=(

1 + q2

1 − q

)n n∑i=0

(n

i

)(q

1 + q

)i

dn−i(1)

(2q

1 + q2+ 1

)i

,

where in the second equation we introduce the index i = h + k. �We now study the statistic on the group S B

n .

Theorem 6.7. The generating function for the colored affine excedance of S Bn is given by

GFClrBn

(q) :=∑π∈ SC

n

qexcClr(π) = n!2

(1 + q2

1 − q

)n

+(

(1 + q)2(1 − q)

1 + q2

)n

An

(q − 1

q + 1

).

Proof. By definition for any fixed σ ∈ S Bn we have that∑

r∈Zn even

q2‖r−pσ ‖ =∑

r′q2‖r′‖ (6.2)

where the second sum is taken on all even (resp. odd) elements r′ ∈ Zn if pσ is even (resp. odd). In

this proof we denote with R(1) = ∑r even q2‖r‖ and with R(−1) = ∑

r odd q2‖r‖ . We can write

GFClrBn

(q) =n∑

h=0

∑σ∈Sn

exc(σ )=h

∑k1,k2

(h

k1

)(n − h

k2

)qn−k1−k2 R

((−1)k1

). (6.3)

We now justify the previous expression: we obtain all permutations in S Bn by adding negative signs

to all permutations in Sn . We consider a permutation σ ∈ Sn with h excedances (in the classicalsense). Each negative sign added to elements corresponding to an excedance of σ decreases the valueexcA(σ ). The index k1 denotes the number of all positive signs in set of all excedances of σ . Thefactor R((−1)k1 ) follows by (6.2). All the monomials qn−k1−k2 are the contributions of qneg(σ ) in thedefinition of GFClr

Bn(q).

Note that in the first product of (6.3) all the even (resp. odd) powers of q have the same coefficientR((−1)k1 ). Therefore we can write

GFClrBn

(q) = n!2

(1 + q2

1 − q

)n

+(

(1 + q)2(1 − q)

1 + q2

)n

An

(q − 1

q + 1

). �

The formula for the derangements is not so easy.

Theorem 6.8. The generating function for the colored affine excedance of the derangements of S Bn is given by

GFDClrBn

(q) :=∑

σ∈SDBn

qexcClr(σ )

= (1 + q2)n

2(1 − q)n

n∑a=0

(n

a

)((1 + q)q

1 + q2

)a

dn−a(1)

+ (1 + q)2n(1 − q)n

2(1 + q2)n

n∑(n

a

)((1 − 2q − q2)q

(1 + q)2(1 − q)

)a

dn−a

(q − 1

q + 1

).

a=0


Sketch of the proof. By using the same techniques of Theorem 6.7, the generating function GFDClrBn

(q)

is given by

n∑a=0

(n

a

) n−a∑h=0

∑σ∈Dn−a

exc(σ )=h

∑k0,k1,k2

(a

k0

)(h

k1

)(n − a − h

k2

)qa+h−k0−k1+k2 Rk0

((−1)k1

),

where Rk0 (−1) denotes the odd powers of the formal series in Lemma 6.4, with b = k0 and a = n−k0.The index a denotes the fixed elements for a permutation σ ∈ Sn , the index h denotes the ex-

cedances of σ , k0 are the positive signs in the set of fixed element of σ , k1 are the positive signsin the set of excedances of σ , and k2 the negative signs in all other positions. The result follows byalgebraic computation. �

Finally, we consider the group S Dn . By (2.3) we know that the colored affine excedance statistic in

S Dn has only monomials in q with even exponents. Therefore, the following results are trivial.

Corollary 6.9. The generating function for the colored affine excedance of S Dn is given by

GFClrDn

(q) :=∑

π∈ S Dn

qexcClr(σ ) =GFClr

Bn(q) + GFClr

Bn(−q)

2.

The generating function for the colored affine excedance of the derangements of S Dn is given by

GFDClrDn

(q) :=∑

π∈SDDn

qexcClr(σ ) =GFDClr

Bn(q) + GFDClr

Bn(−q)

2.

We note that in SCn there are more permutations with zero excedances than just the identity. If we

want that the only permutation with zero excedances is the trivial permutation then we can changethe definition as follows.

Definition 6.10. Let π ∈ SCn . Then the variant colored affine excedance of π is given by

excClr 2(π) =n∑

i=1

∣∣∣∣⌈π(i) − i

2n + 1

⌉∣∣∣∣ + excA(σπ ) + neg(σπ ).

For example, let π ∈ SC5 be the permutation given by π = [−3,−1,4,2,16] in window notation.

Then σπ = (−3,−1,4,2,5) and excClr 2(π) = (0 + 0 + 1 + 0 + 1) + 1 + 2 = 5.It is easy to verify that with this definition the only permutation with zero excedances is the

identity. The generating functions can be computed with the same techniques of the previous results,therefore we omit the proofs.

Theorem 6.11. The generating function for the variant colored affine excedance of SCn is given by

GFClr 2Cn

(q) :=∑π∈ SC

n

qexcClr 2(σ ) =(

(1 + q)2

1 − q

)n

An

(2q

1 + q

);

the generating function on its derangement set is given by


GFDClr 2Cn

(q) :=∑

π∈SDCn

qexcClr 2(σ )

=(

(1 + q)2

1 − q

)n n∑a=0

(n

a

)(q(3 + q)

(1 + q)2

)a

dn−a

(2q

1 + q

).

The analogous formulas for the group S Bn are more complicated.

Theorem 6.12. The generating function for the variant colored affine excedance of S Bn is given by

GFClr 2Bn

(q) :=∑π∈ SC

n

qexcClr 2(σ ) = 1

2

((1 + q)2

1 − q

)n

An

(2q

1 + q

)+ (1 − q)n

2;

the generating function on its derangement set is given by

GFDClr 2Bn

(q) :=∑

π∈SDCn

qexcClr 2(σ ) = 1

2

(q(3 + q)

1 − q

)n

+ (−q)n

2

+ 1

2

((1 + q)2

1 − q

)n n−1∑a=0

(n

a

)(q(3 + q)

(1 + q)2

)a

dn−a

(2q

1 + q

).

The analogue of Lemma 6.3 is the following identity for any π = (rπ ,σπ ) ∈ S Bn :

excClr 2(π) = ‖rπ − pσπ ‖ + excA(σπ ) + neg(σπ ).

If r ∈ Zn is even then it is obvious that ‖r − pσπ ‖ + excA(σπ ) is an even number. Therefore the

generating functions on S Dn and its derangements subset are even formal power series.

Corollary 6.13. The generating function for the variant colored affine excedance of S Dn is given by

GFClr 2Dn

(q) :=∑

π∈ S Dn

qexcClr 2(σ ) = 1

2

(GFClr 2

Bn(q) + GFClr 2

Bn(−q)

);the generating function on its derangement set is given by

GFDClr 2Dn

(q) :=∑

π∈SDDn

qexcClr 2(σ ) = 1

2

(GFDClr 2

Bn(q) + GFDClr 2

Bn(−q)

).

7. Absolute excedance for the affine Weyl groups

In this section we introduce the analogue of the absolute excedance for the affine Weyl groups.We follow the same ideas in the previous section.

Definition 7.1. The absolute affine excedance for any permutation π ∈ SCn is given by

excAbs(π) =n∑

i=1

∣∣∣∣⌈ |π(i)| − i

2n + 1

⌉∣∣∣∣ + neg(σπ ),

where σπ is defined in Section 2.


For example, let π ∈ SC5 be the permutation given by π = [−3,−1,4,2,16] in window notation.

Then σπ = (−3,−1,4,2,5) and excAbs(π) = (1 + 0 + 1 + 0 + 1) + 2 = 5.We now want to have a result as in Lemma 6.3 that allows us to compute easily the value of the

absolute affine excedance. There is no an explicit simple formula but we note that for i ∈ [1,n]∣∣∣∣⌈ |π(i)| − i

2n + 1

⌉∣∣∣∣ =⎧⎨⎩

ri + χ(σπ (i) > i) if ri > 0,

χ(|σπ (i)| > i) if ri = 0,

|ri| + χ(σπ (i) < −i) if ri < 0

(7.1)

where π = (rπ ,σπ ) is as usual. In particular, if ri goes in Z, the previous expression assume all valuesin {1,1,2,2,3,3, . . .} if |σπ (i)| > i and in {0,1,1,2,2,3,3, . . .} otherwise. Moreover, if i is such that|σπ (i)| > i and ri goes in Z and it is always even or always odd then the previous expression assumesall values in P.

Now we can describe the generating function.

Theorem 7.2. The generating function for the absolute affine excedance of SCn is given by

GFAbsCn

(q) :=∑π∈ SC

n

qexcAbs(π) =(

(1 + q)2

1 − q

)n

An

(2q

1 + q

).

Proof. By (7.1), we have

GFAbsCn

(q) =n−1∑h=0

∑σ∈Sn

exc(σ )=h

n∑k=0

(n

k

)qk

∑r∈(Z∗)h×Zn−h

q‖r‖,

where h denotes the classical excedances of the permutation σ and k denotes all possible negativesigns to have permutations in S B

n . Now apply Lemma 6.4 and conclude that

GFAbsCn

(q) =n−1∑h=0

∑σ∈Sn

exc(σ )=h

(q + 1)n 2hqh(1 + q)n−h

(1 − q)n

=(

(1 + q)2

1 − q

)n

An

(2q

1 + q

). �

Theorem 7.3. The generating function for the absolute affine excedance of the derangements of SCn is given by

GFDAbsCn

(q) :=∑

π∈SDCn

qexcAbs(π)

=(

(1 + q)2

(1 − q)

)n n∑a=0

(n

a

)(q(3 + q)

(1 + q)2

)a

dn−a

(2q

1 − q

).

Sketch of the proof. Just compute

n∑a=0

(n

a

) n−a∑h=0

∑σ∈Dn−a

exc(σ )=h

(q + 1)n−aa∑

k=0

(a

k

)qa−k

∑r∈(Z∗)h+k×Zn−h−k

q‖r‖. �


Theorem 7.4. The generating function for the colored affine excedance of S Bn is given by

GFAbsBn

(q) :=∑π∈ S B

n

qexcAbs(π)

= 1

2

((1 + q)2

1 − q

)n

An

(2q

1 + q

)+ (1 − q)n

2.

Proof. We have to consider two cases: first suppose that σ ∈ Sn is a classical permutation with atleast one excedance. Fix an index corresponding to one such excedance; for example, suppose it is 1.Then we choose all possible signs in σ and choose all vectors r in such a way that (r2, . . . , rn) aretaken freely and r1 determines the parity of r. Then when we sum qexcAbs(r,σ ) on all possible valuesof r we have the same result of∑

r∈P×(Z∗)h−1×Zn−h

q‖r‖

as justified by the comments immediately after (7.1). Otherwise suppose that σ is the identity, thenwe have to sum the contributions of (even) r and of all possible signs. Then we have

GFAbsBn

(q) =n∑

h=1

∑σ∈Sn

exc(σ )=h

(q + 1)n∑

r∈P×(Z∗)h−1×Zn−h

q‖r‖ + (q + 1)n∑

r∈Zn even

q‖r‖.

The result follows after algebraic manipulation by Lemma 6.4. �Theorem 7.5. The generating function for the absolute affine excedance of the derangements of S B

n is given by

GFDAbsBn

(q) :=∑

π∈SDBn

qexcClr(π)

= 1

2

((1 + q)2n

(1 − q)n

n−2∑a=0

(n

a

)(q(3 + q)

(1 + q)2

)a

dn−a

(2q

1 + q

))+ (q(3 + q))n

2(1 − q)n+ (−q)n

2.

Sketch of the proof. With the same arguments of Theorem 7.4, it suffices to compute

n−2∑a=0

(n

a

) n−a−1∑h=1

∑σ∈Dn−a

exc(σ )=h

(q + 1)n−aa∑

k=0

(a

k

)qa−k

∑r∈P×(Z∗)h+k−1×Zn−h−k

q‖r‖

+n∑

k=0

(n

k

)qn−k

∑r∈(Z∗)k×Zn−k even

q‖r‖. �

Note that for SCn and S B

n the generating functions of the affine absolute excedance and the vari-ant affine colored excedance are the same. Then we ask if there exists a bijection that shows theseidentities in a combinatorial way. Given π = (rπ ,σπ ) ∈ SC

n and i ∈ [1,n], we say that i satisfies thecondition C with respect to π if both ri � 0 and σ(i) > i or both ri � 0 and σ(i) < −i. We define themap

φ : SCn → SC

n

by setting for all (r, σ ) ∈ SCn , φ((r, σ )) = ((r′, σ ′)) with r′

i = −ri and σ ′(i) = −σ(i) if i satisfies con-dition C with respect to π and r′

i = ri and σ ′(i) = σ(i) otherwise. It is easy to verify by the analogue


of Lemma 6.3 and by (7.1) that excClr 2(π) = excAbs(φ(π)); moreover, φ|S Bn

� φ up to a natural iden-

tification, where φ is defined immediately after (4.2). For example, let π = [13,−14,12,4,16] ∈ SC5 .

Then φ(π) = [−13,14,12,4,16] and excClr 2(π) = 7 = excAbs(φ(π)).Note that this bijection does not preserve the number of negative signs in the window notation of

the permutations. Therefore the generating functions on S Dn of the affine absolute excedance will be

different from the generating functions of the variant affine colored excedance.

Proposition 7.6. The generating function for the affine absolute excedance of S Dn is given by

GFAbsDn

(q) :=∑

π∈ S Dn

qexcAbs(π) =(

1

2+ (1 − q)n

2(1 + q)n

)GFAbs(q)

Bn.

The generating function for the affine absolute excedance of derangements on S Dn is given by

GFDAbsDn

(q) :=∑

π∈SDDn

qexcAbs(π) = 1

2GFDAbs

Bn(q) + 1

4

(q(q − 3)

1 + q

)n

+ qn

4+ (1 + q)n

4

n−1∑a=0

(n

a

)(q

q + 1

)a

dn−a

(2q

1 + q

).

The proof uses the same arguments of those of Theorems 7.2 and 7.3.

8. Final remarks

In this paper we do not generalize the excedance statistic in Definition 4.4 to the affine Weylgroups. Given π = (rπ ,σπ ) one can compute for example∣∣∣∣⌈π |π(i)| − π(i)

2n + 1

⌉∣∣∣∣and note that a negative sign in one entry of rπ significantly changes its value. Moreover, the authorhas not found any such definition with the condition excCox(π) = 0 implies that π is the identity asdone for the variant colored affine excedance and the absolute affine excedance.

In this last part we study unimodality and log-concavity aspects of the generating functions com-puted in this paper. We recall that a sequence of positive integers a0, . . . ,an is unimodal if there existsan index i such that ai � ai+1 if i < i and ai � ai+1 otherwise; it is log-concave if a2

i � ai−1ai+1 forall i ∈ [1,n − 1]. A polynomial with nonnegative coefficients is unimodal (resp. log-concave) if andonly if the sequence of its coefficients is unimodal (resp. log-concave). It is well known that a poly-nomial with nonnegative coefficients and with only real roots is log-concave and that log-concavityimplies unimodality. Moreover the Eulerian polynomials have only real roots and therefore they arelog-concave and unimodal.

By Proposition 3.3, it is easy to verify that the generating function Bn(q) = ∑σ∈S B

nqexcClr =∑

σ∈S Bn

qexcAbshas only real roots and therefore it is log-concave and unimodal. We ask if the same is

true for the generating function on the derangements, but the answer is negative. For n = 5 the poly-nomial dB

5 (q) has only 5 real roots instead of 9. However, with the help of a computer, it is possibleto conjecture the following result.

Conjecture 8.1. Let n � 1. Then the polynomial dBn (q) is log-concave. In particular it is unimodal.

By [13, Theorem 4.5] and Corollaries 5.1 and 5.2 it is possible to prove that the generating func-tions of the colored and absolute excedance statistics on S D

n have both real roots as polynomials in q2.We also conjecture the following result.


Conjecture 8.2. The generating function of the colored excedance on SDDn is log-concave for all n � 1.

The generating function of the absolute excedance on SDDn is log-concave for all n � 10 and unimodal for

all n � 1.

The first part is a consequence of Conjecture 8.1. All previous conjectures are verified by a com-puter for all n � 200.

References

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Excedances in classical and affine Weyl groups