We define an analogue of signed Eulerian numbers $f_{n,k}$ for involutions of
the symmetric group and derive some combinatorial properties of this sequence.
In particular, we exhibit both an explicit formula and a recurrence for
$f_{n,k}$ arising from the properties of its generating function.Comment: 10 page

Barnabei, M.

Bonetti, F.

Silimbani, M.

English

arXiv

We define an analogue of signed Eulerian numbers fn;k for involutions

of the symmetric group and derive some combinatorial properties

of this sequence. In particular, we exhibit both an explicit formula and a

recurrence for fn;k arising from the properties of its generating function

Barnabei, Marilena

Bonetti, Flavio

Silimbani, Matteo

AMS Acta

The signed Eulerian numbers on involutions

The signed Eulerian numbers on involutionsMarilena Barnabei, Flavio Bonetti, and Matteo Silimbani ∗Abstract. We define an analogue of signed Eulerian numbers fn,k for in-volutions of the symmetric group and derive some combinatorial propertiesof this sequence. In particular, we exhibit both an explicit formula and arecurrence for fn,k arising from the properties of its generating function.Keywords: involution, descent, signed Eulerian numbers.AMS classification: 05A05, 05A15, 05A19, 05E10.1 IntroductionLet σ be a permutation in Sn. We say that σ has a descent at position iwhenever σ(i) > σ(i+ 1). Analogously, we say that σ has a rise at positioni whenever σ(i) < σ(i+ 1). The number of descents (respectively rises) of apermutation σ is denoted by des(σ) (resp. ris(σ)). The polynomialAn(t) =∑σ∈Sntdes(σ) =n−1∑k=0an,ktk,is known as the Eulerian polynomial, and the integers an,k, i.e., the numberof permutations σ ∈ Sn with des(σ) = k, are called the Eulerian numbers.We recall that the Eulerian numbers satisfy the property an,k = an,n−1−k,that implies ∑σ∈Sntdes(σ) =∑σ∈Sntris(σ). (1)The study of the distribution of the descent statistic has been carried outboth in the case of the symmetric group and of some particular subsets of∗Dipartimento di Matematica - Universita` di Bologna1permutations. For example, Eulerian distribution on the set of involutionsIn ⊆ Sn has been deeply investigated by several authors ([5], [4], [6], and [1]).Loday [8], in his study of the cyclic homology of commutative algebras, intro-duced the sequence (bn,k) of signed Eulerian numbers, namely, the coefficientsof the polynomialBn(t) =∑σ∈Snsgn(σ)tris(σ) =n−1∑k=0bn,ktk.In [3], De´sarme´nien and Foata showed several combinatorial properties satis-fied by the integers bn,k by exploiting their relations with the Eulerian num-bers. More recently, Tanimoto (see, e.g., [11]) developedthe study of signedEulerian numbers from a different point of view.In this paper we study the signed Eulerian numbers on involutions, i.e., thecoefficients of the signed Eulerian polynomial on involutionsFn(t) =∑σ∈Insgnσtris(σ) =n−1∑k=0fn,ktk.To this aim, we exploit a map introduced in [2] that associates an involutionwith a family of generalized involutions, which share the shape of the corre-sponding semistandard Young tableau. This map yields an explicit formulafor the number of involutions in In with a given number of rises. The abovemap yields also an analogue of Worpitzky Identity for the sequence fn,k:s−1∑j=0(n+ jj)fn,s−j−1 =bn2 c∑j=0(−1)j((s+12)+ j − 1j)(sn− 2j)and, hence, an explicit formula for the signed Eulerian numbers on involu-tions.This formula gives the following expression for the generating function of thesigned Eulerian polynomials on involutions:∑n≥0Fn(t)un(1− t)n+1 =∑r≥0tr(1 + u)r+1(1 + u2)(r+22 ).2Applying the Maple package ZeilbergerRecurrence(T, n, k, s, 0..n) to thislast identity, we find a recurrence satisfied by the integers fn,k.Moreover, we exhibit an explicit formula, a recurrence, and a generatingfunction for the sequence Fn(1), with n ∈ N, namely, the difference betweenthe number of even and odd involutions in In.2 The signed Eulerian numbersIn order to investigate the combinatorial properties of the signed Eulerianpolynomial on involutions, we exploit the relations between involutions andparticular biwords, called generalized involutions, introduced in [1].A generalized involution of length n is defined to be a biword:α =(xy)=(x1 x2 · · · xny1 y2 · · · yn),such that:• for every 1 ≤ i ≤ n, there exists an index j with xi = yj and yi = xj,• xi ≤ xi+1,• xi = xi+1 =⇒ yi ≥ yi+1.We say that an integer a is a repetition of multiplicity r for the generalizedinvolution α ifxi = yi = xi+1 = yi+1 = · · · = xi+r−1 = yi+r−1 = a.In [2], the authors introduced a map Π from the set of generalized involutionsof length n to the set of involutions In defined as follows: ifα =(x1 x2 · · · xny1 y2 · · · yn),then Π(α) is the involution σσ =(1 2 · · · ny′1 y′2 · · · y′n),3where y′i = 1 if yi is the least symbol occurring in the word y, y′j = 2 if yj isthe second least symbol in y and so on. In the case yi = yj, with i > j, weconsider yi to be less then yj. We call the involution σ = Π(α) the polariza-tion of α.We denote by Genm(σ) the set of generalized involutions of length n, withsymbols taken from [m], whose polarization is σ. Then, we have the followingresult proved in [1]:Proposition 1 Let σ ∈ In be an involution with t rises. Then,|Genm(σ)| =(n+m− t− 1n). (2)¦We recall that the sign of an involution σ ∈ In is determined by the numberfix(σ) of fixed points of σ. More precisely:sgn(σ) = (−1)n−fix(σ)2 .It is well known [9] that the integer fix(σ) depends only on the shape ofthe tableau Tσ. On the other hand, it is immediately seen that the semis-tandard tableau associated with any generalized involution in Genm(σ) bythe Robinson-Schensted-Knuth algorithm has the same shape as the stan-dard tableau Tσ associated with σ. We are interested in counting separatelythe number of even and odd involutions with a given number of rises. Con-sequently, we can define an even generalized involution to be a generalizedinvolution whose polarization is an even involution.Remark that the sign of a generalized involution depends only on the multi-plicity of its repetitions. In fact, given a generalized involution α, we definegfix(α) to be the number of repetitions of odd multiplicity of α (note that,if α ∈ In, then gfix(α) = fix(α)). It is easy to verify that a generalizedinvolution α of length n is even if and only ifn− gfix(σ)2is even. This remark allows to compute explicitly the number a+n,m of gener-alized involutions of length n over [m]:4Proposition 2 We have:a+n,m =bn4 c∑h=0(mn− 4h)((m+12)+ 2h− 12h).Proof Let α be a generalized involution. We remarked that α is evenwhenever n−gfix(σ)2is even. We want to count generalized involutions withn−gfix(σ)2= 2h with fixed h. We can choose the n − 4h fixed points of α in(mn−4h)ways. Then, we choose 2h biletters (i, j) in the set B = {(i, j)|1 ≤ i ≤j ≤ m}, whose cardinality is (m+12). We complete the generalized involutionby adding a biletter (j, i) for every biletter (i, j) previously chosen, hencegetting an even generalized involution.¦On the other hand, recalling that the total number of generalized involutionsof length n over [m] is (see, e.g., [1])an,m =bn2 c∑h=0(m+ n− 2h− 1n− 2h)((m2)+ h− 1h),we get an explicit formula for the number a−n,m of odd generalized involutionsof length n over [m]:a−n,m =bn2 c∑h=0(m+ n− 2h− 1n− 2h)((m2)+ h− 1h)−bn4 c∑h=0(mn− 4h)((m+12)+ 2h− 12h).We remark that, given σ ∈ In, all generalized involutions in Genm(σ) havethe same sign as σ. Hence, Proposition 1 implies thata+n,m =m−1∑k=0(n+ kk)f+n,m−k−1 (3)a−n,m =m−1∑k=0(n+ kk)f−n,m−k−1, (4)where f+n,k and f−n,k denote the number of of positive and negative involutionsin In with k rises, respectively. These identities allow to state the following:5Theorem 3 We have:fn,k =k+1∑m=0(−1)k−m+1(n+ 1k −m+ 1) bn2 c∑j=0(−1)j((m+12)+ j − 1j)(mn− 2j).(5)Proof Set aˆn,m = a+n,m − a−n,m. Then, Identities (3) and (4) implyaˆm,n =m−1∑k=0(n+ kk)fn,m−k−1. (6)By inversion, we have:fn,k =k+1∑m=0(−1)k−m+1(n+ 1k −m+ 1)aˆn,m ==k+1∑m=0(−1)k−m+1(n+ 1k −m+ 1)2 bn4 c∑j=0(mn− 4j)((m+12)+ 2j − 12j)−bn2 c∑j=0(m+ n− 2j − 1n− 2j)((m2)+ j − 1j)that is equivalent to (5).¦Remark that Identity (6) gives the following analogue of the Worpitzky Iden-tity:s−1∑j=0(n+ jj)fn,s−j−1 =bn2 c∑j=0(−1)j((s+12)+ j − 1j)(sn− 2j).63 The signed Eulerian polynomial on involu-tionsWe define the n-th signed Eulerian polynomial for involutions to be the poly-nomial:Fn(t) =∑σ∈Insgn(σ)tris(σ) =n−1∑k=0fn,ktk.As shown in the previous section, the relation between involutions and gen-eralized involutions is crucial in our analysis. We denote byRn(t) =∑m≥0aˆn,mtmCm(t) =∑n≥0aˆn,mtnthe row and column generating functions of the array aˆn,m. As seen in theproof of Theorem 3, we haveaˆn,m =bn2 c∑j=0(−1)j((m+12)+ j − 1j)(mn− 2j).and henceCm(t) =(1 + u)m(1 + u2)(m+12 ).Theorem 4 The polynomial Fn(t) satisfies the identity∑n≥0Fn(t)un(1− t)n+1 =∑m≥0tm(1 + u)m+1(1 + u2)(m+22 ). (7)Proof The binomial relation between the sequences fn,k and aˆn,m yields thefollowing identity:tFn(t)(1− t)n+1 = Rn(t).Hence: ∑n≥0Fn(t)un(1− t)n+1 =∑n≥0∑m≥0aˆn,m+1tmun =7=∑m≥0Cm+1tm =∑m≥0tm(1 + u)m+1(1 + u2)(m+22 )as desired.¦We apply the Maple package ZeilbergerRecurrence(T, n, k, s, 0..n) to thesequence fn,k, following along the lines of [6], and we deduce by Identity (7)the following recurrence formula:nfn,k = (2 + k − n)fn−1,k + (2n− k − 1)fn−1,k−1 − (n+ 3k + k2)fn−2,k++(−2 + 4k + 2k2 − 2kn)fn−2,k−1 + (2− k − k2 + 2kn− n2)fn−2,k−2++(−n− k2 − 2k + 2)fn−3,k + (−7 + 4k + 3k2 + 2n− 2kn)fn−3,k−1++(8− 2k− 3k2− 2n+4kn− n2)fn−3,k−2+ (−3+ k2+ n− 2kn+ n2)fn−3,k−3.In conclusion, we study the combinatorial properties of the sequence Fn(1),with n ∈ N. Obviously, Fn(1) is the difference between the number i+n ofeven involutions on n objects and the number i−n of odd involutions. First ofall, we have:Proposition 5 The evaluation of the polynomial Fn(t) at 1 isFn(1) = 2bn4 c∑h=0n!(2h)!(n− 4h)!22h −bn2 c∑h=0n!(n− 2h)!2h .Proof Fix an integer h ≤ bn4c. We count the number of involutions whose cy-cle decomposition consists of 2h transpositions. Choose a word w = w1 · · ·wnconsisting of distinct letters taken from [n]. We have n! choices for w. Thisword correspond to a unique even involution τ with n−4h fixed points definedby the following conditions:τ(w1) = w2 . . . τ(w4h−1) = w4h;τ(w4h+j) = w4h+j8with 0 < j ≤ n − 4h. It is easily checked that the involution τ arises from(n− 4h)!(2h)!22h different words w. These considerations imply thati+n =bn4 c∑h=0n!(2h)!(n− 4h)!22h .On the other hand, it is well known that|In| =bn2 c∑h=0n!(n− 2h)!2h .Hence, since Fn(1) = i+n − i−n = 2i+n − |In|, we get the assertion.¦Proposition 6 The sequence Fn(1) satisfies the recurrenceFn(1) = Fn−1(1)− (n− 1)Fn−2(1).Proof Consider an even involution τ ∈ In. If τ(1) = 1, the restriction of τon the set {2, . . . , n} is an even involution on n− 1 objects. If τ(1) = j 6= 1,the restriction of τ on {2, . . . , n} \ {j} is an odd involution on n− 2 objects.This implies that:i+n = i+n−1 + (n− 1)i−n−2and, analogously,i−n = i−n−1 + (n− 1)i+n−2.These identities gives immediately the assertion.¦These results allow to deduce the following expression for the exponentialgenerating function of the sequence Fn(1):∑n≥0Fn(1)n!tn = et−t22 .9References[1] M.Barnabei, F.Bonetti, M.Silimbani, The descent statistic on involu-tions is not log-concave, submitted[2] M.Barnabei, F.Bonetti, M.Silimbani, The Eulerian distribution on selfevacuated involutions, submitted[3] J.De´sarme´nien, D.Foata, The signed Eulerian numbers, Discr. Math. 99(1992), no. 1-3, 44-58.[4] J.De´sarme´nien, D.Foata, Fonctions syme´triques et se´ries hy-perge´ome´triques basiques multivarie´es, Bull. Soc. Math. France113 (1985), 3-22.[5] I.M.Gessel, C.Reutenauer, Counting permutations with a given cyclestructure and descent set, J. Combin. Theory Ser. A 13 (1972), 135-139.[6] V.J.Guo, J.Zeng, The Eulerian distribution on involutions is indeed uni-modal, J. Combin. Theory Ser. A 113 (2006), no. 6, 1061–1071.[7] D.E.Knuth, Permutations, Matrices and Generalized Young Tableaux,Pacific J. Math. 34 (1970), 709-727.[8] J.L.Loday, Ope´ration sur l’homologie cyclique des alge`bres commuta-tives, Invent. Math. 96 (1989), no. 1, 205-230.[9] M.P.Schu¨tzenberger, Quelques Remarques sur une Construction deSchensted, Math. Scand. 12 (1963), 117-128.[10] R.P.Stanley, Enumerative Combinatorics, Vol. II, Cambridge Studiesin Advanced Mathematics 62. Cambridge University Press, Cambridge(1999).[11] S.Tanimoto, A study of Eulerian numbers for permutations in the alter-nating group, Integers 6, A31 (2006) (electronic).10

AMS Acta - Alm@DL - Università di Bologna

The signed Eulerian numbers on involutions
Marilena Barnabei, Flavio Bonetti, and Matteo Silimbani ∗
Abstract. We define an analogue of signed Eulerian numbers fn,k for in-
volutions of the symmetric group and derive some combinatorial properties
of this sequence. In particular, we exhibit both an explicit formula and a
recurrence for fn,k arising from the properties of its generating function.
Keywords: involution, descent, signed Eulerian numbers.
AMS classification: 05A05, 05A15, 05A19, 05E10.
1 Introduction
Let σ be a permutation in Sn. We say that σ has a descent at position i
whenever σ(i) > σ(i+ 1). Analogously, we say that σ has a rise at position
i whenever σ(i) < σ(i+ 1). The number of descents (respectively rises) of a
permutation σ is denoted by des(σ) (resp. ris(σ)). The polynomial
An(t) =
∑
σ∈Sn
tdes(σ) =
n−1∑
k=0
an,kt
k,
is known as the Eulerian polynomial, and the integers an,k, i.e., the number
of permutations σ ∈ Sn with des(σ) = k, are called the Eulerian numbers.
We recall that the Eulerian numbers satisfy the property an,k = an,n−1−k,
that implies ∑
σ∈Sn
tdes(σ) =
∑
σ∈Sn
tris(σ). (1)
The study of the distribution of the descent statistic has been carried out
both in the case of the symmetric group and of some particular subsets of
∗Dipartimento di Matematica - Universita` di Bologna
1
permutations. For example, Eulerian distribution on the set of involutions
In ⊆ Sn has been deeply investigated by several authors ([5], [4], [6], and [1]).
Loday [8], in his study of the cyclic homology of commutative algebras, intro-
duced the sequence (bn,k) of signed Eulerian numbers, namely, the coefficients
of the polynomial
Bn(t) =
∑
σ∈Sn
sgn(σ)tris(σ) =
n−1∑
k=0
bn,kt
k.
In [3], De´sarme´nien and Foata showed several combinatorial properties satis-
fied by the integers bn,k by exploiting their relations with the Eulerian num-
bers. More recently, Tanimoto (see, e.g., [11]) developedthe study of signed
Eulerian numbers from a different point of view.
In this paper we study the signed Eulerian numbers on involutions, i.e., the
coefficients of the signed Eulerian polynomial on involutions
Fn(t) =
∑
σ∈In
sgnσtris(σ) =
n−1∑
k=0
fn,kt
k.
To this aim, we exploit a map introduced in [2] that associates an involution
with a family of generalized involutions, which share the shape of the corre-
sponding semistandard Young tableau. This map yields an explicit formula
for the number of involutions in In with a given number of rises. The above
map yields also an analogue of Worpitzky Identity for the sequence fn,k:
s−1∑
j=0
(
n+ j
j
)
fn,s−j−1 =
bn2 c∑
j=0
(−1)j
((s+1
2
)
+ j − 1
j
)(
s
n− 2j
)
and, hence, an explicit formula for the signed Eulerian numbers on involu-
tions.
This formula gives the following expression for the generating function of the
signed Eulerian polynomials on involutions:∑
n≥0
Fn(t)
un
(1− t)n+1 =
∑
r≥0
tr
(1 + u)r+1
(1 + u2)(
r+2
2 )
.
2
Applying the Maple package ZeilbergerRecurrence(T, n, k, s, 0..n) to this
last identity, we find a recurrence satisfied by the integers fn,k.
Moreover, we exhibit an explicit formula, a recurrence, and a generating
function for the sequence Fn(1), with n ∈ N, namely, the difference between
the number of even and odd involutions in In.
2 The signed Eulerian numbers
In order to investigate the combinatorial properties of the signed Eulerian
polynomial on involutions, we exploit the relations between involutions and
particular biwords, called generalized involutions, introduced in [1].
A generalized involution of length n is defined to be a biword:
α =
(
x
y
)
=
(
x1 x2 · · · xn
y1 y2 · · · yn
)
,
such that:
• for every 1 ≤ i ≤ n, there exists an index j with xi = yj and yi = xj,
• xi ≤ xi+1,
• xi = xi+1 =⇒ yi ≥ yi+1.
We say that an integer a is a repetition of multiplicity r for the generalized
involution α if
xi = yi = xi+1 = yi+1 = · · · = xi+r−1 = yi+r−1 = a.
In [2], the authors introduced a map Π from the set of generalized involutions
of length n to the set of involutions In defined as follows: if
α =
(
x1 x2 · · · xn
y1 y2 · · · yn
)
,
then Π(α) is the involution σ
σ =
(
1 2 · · · n
y′1 y
′
2 · · · y′n
)
,
3
where y′i = 1 if yi is the least symbol occurring in the word y, y
′
j = 2 if yj is
the second least symbol in y and so on. In the case yi = yj, with i > j, we
consider yi to be less then yj. We call the involution σ = Π(α) the polariza-
tion of α.
We denote by Genm(σ) the set of generalized involutions of length n, with
symbols taken from [m], whose polarization is σ. Then, we have the following
result proved in [1]:
Proposition 1 Let σ ∈ In be an involution with t rises. Then,
|Genm(σ)| =
(
n+m− t− 1
n
)
. (2)
¦
We recall that the sign of an involution σ ∈ In is determined by the number
fix(σ) of fixed points of σ. More precisely:
sgn(σ) = (−1)n−fix(σ)2 .
It is well known [9] that the integer fix(σ) depends only on the shape of
the tableau Tσ. On the other hand, it is immediately seen that the semis-
tandard tableau associated with any generalized involution in Genm(σ) by
the Robinson-Schensted-Knuth algorithm has the same shape as the stan-
dard tableau Tσ associated with σ. We are interested in counting separately
the number of even and odd involutions with a given number of rises. Con-
sequently, we can define an even generalized involution to be a generalized
involution whose polarization is an even involution.
Remark that the sign of a generalized involution depends only on the multi-
plicity of its repetitions. In fact, given a generalized involution α, we define
gfix(α) to be the number of repetitions of odd multiplicity of α (note that,
if α ∈ In, then gfix(α) = fix(α)). It is easy to verify that a generalized
involution α of length n is even if and only if
n− gfix(σ)
2
is even. This remark allows to compute explicitly the number a+n,m of gener-
alized involutions of length n over [m]:
4
Proposition 2 We have:
a+n,m =
bn4 c∑
h=0
(
m
n− 4h
)((m+1
2
)
+ 2h− 1
2h
)
.
Proof Let α be a generalized involution. We remarked that α is even
whenever n−gfix(σ)
2
is even. We want to count generalized involutions with
n−gfix(σ)
2
= 2h with fixed h. We can choose the n − 4h fixed points of α in(
m
n−4h
)
ways. Then, we choose 2h biletters (i, j) in the set B = {(i, j)|1 ≤ i ≤
j ≤ m}, whose cardinality is (m+1
2
)
. We complete the generalized involution
by adding a biletter (j, i) for every biletter (i, j) previously chosen, hence
getting an even generalized involution.
¦
On the other hand, recalling that the total number of generalized involutions
of length n over [m] is (see, e.g., [1])
an,m =
bn2 c∑
h=0
(
m+ n− 2h− 1
n− 2h
)((m
2
)
+ h− 1
h
)
,
we get an explicit formula for the number a−n,m of odd generalized involutions
of length n over [m]:
a−n,m =
bn2 c∑
h=0
(
m+ n− 2h− 1
n− 2h
)((m
2
)
+ h− 1
h
)
−
bn4 c∑
h=0
(
m
n− 4h
)((m+1
2
)
+ 2h− 1
2h
)
.
We remark that, given σ ∈ In, all generalized involutions in Genm(σ) have
the same sign as σ. Hence, Proposition 1 implies that
a+n,m =
m−1∑
k=0
(
n+ k
k
)
f+n,m−k−1 (3)
a−n,m =
m−1∑
k=0
(
n+ k
k
)
f−n,m−k−1, (4)
where f+n,k and f
−
n,k denote the number of of positive and negative involutions
in In with k rises, respectively. These identities allow to state the following:
5
Theorem 3 We have:
fn,k =
k+1∑
m=0
(−1)k−m+1
(
n+ 1
k −m+ 1
) bn2 c∑
j=0
(−1)j
((m+1
2
)
+ j − 1
j
)(
m
n− 2j
)
.
(5)
Proof Set aˆn,m = a
+
n,m − a−n,m. Then, Identities (3) and (4) imply
aˆm,n =
m−1∑
k=0
(
n+ k
k
)
fn,m−k−1. (6)
By inversion, we have:
fn,k =
k+1∑
m=0
(−1)k−m+1
(
n+ 1
k −m+ 1
)
aˆn,m =
=
k+1∑
m=0
(−1)k−m+1
(
n+ 1
k −m+ 1
)2 bn4 c∑
j=0
(
m
n− 4j
)((m+1
2
)
+ 2j − 1
2j
)
−
bn2 c∑
j=0
(
m+ n− 2j − 1
n− 2j
)((m
2
)
+ j − 1
j
)
that is equivalent to (5).
¦
Remark that Identity (6) gives the following analogue of the Worpitzky Iden-
tity:
s−1∑
j=0
(
n+ j
j
)
fn,s−j−1 =
bn2 c∑
j=0
(−1)j
((s+1
2
)
+ j − 1
j
)(
s
n− 2j
)
.
6
3 The signed Eulerian polynomial on involu-
tions
We define the n-th signed Eulerian polynomial for involutions to be the poly-
nomial:
Fn(t) =
∑
σ∈In
sgn(σ)tris(σ) =
n−1∑
k=0
fn,kt
k.
As shown in the previous section, the relation between involutions and gen-
eralized involutions is crucial in our analysis. We denote by
Rn(t) =
∑
m≥0
aˆn,mt
m
Cm(t) =
∑
n≥0
aˆn,mt
n
the row and column generating functions of the array aˆn,m. As seen in the
proof of Theorem 3, we have
aˆn,m =
bn2 c∑
j=0
(−1)j
((m+1
2
)
+ j − 1
j
)(
m
n− 2j
)
.
and hence
Cm(t) =
(1 + u)m
(1 + u2)(
m+1
2 )
.
Theorem 4 The polynomial Fn(t) satisfies the identity∑
n≥0
Fn(t)
un
(1− t)n+1 =
∑
m≥0
tm
(1 + u)m+1
(1 + u2)(
m+2
2 )
. (7)
Proof The binomial relation between the sequences fn,k and aˆn,m yields the
following identity:
tFn(t)
(1− t)n+1 = Rn(t).
Hence: ∑
n≥0
Fn(t)
un
(1− t)n+1 =
∑
n≥0
∑
m≥0
aˆn,m+1t
mun =
7
=
∑
m≥0
Cm+1t
m =
∑
m≥0
tm
(1 + u)m+1
(1 + u2)(
m+2
2 )
as desired.
¦
We apply the Maple package ZeilbergerRecurrence(T, n, k, s, 0..n) to the
sequence fn,k, following along the lines of [6], and we deduce by Identity (7)
the following recurrence formula:
nfn,k = (2 + k − n)fn−1,k + (2n− k − 1)fn−1,k−1 − (n+ 3k + k2)fn−2,k+
+(−2 + 4k + 2k2 − 2kn)fn−2,k−1 + (2− k − k2 + 2kn− n2)fn−2,k−2+
+(−n− k2 − 2k + 2)fn−3,k + (−7 + 4k + 3k2 + 2n− 2kn)fn−3,k−1+
+(8− 2k− 3k2− 2n+4kn− n2)fn−3,k−2+ (−3+ k2+ n− 2kn+ n2)fn−3,k−3.
In conclusion, we study the combinatorial properties of the sequence Fn(1),
with n ∈ N. Obviously, Fn(1) is the difference between the number i+n of
even involutions on n objects and the number i−n of odd involutions. First of
all, we have:
Proposition 5 The evaluation of the polynomial Fn(t) at 1 is
Fn(1) = 2
bn4 c∑
h=0
n!
(2h)!(n− 4h)!22h −
bn2 c∑
h=0
n!
(n− 2h)!2h .
Proof Fix an integer h ≤ bn
4
c. We count the number of involutions whose cy-
cle decomposition consists of 2h transpositions. Choose a word w = w1 · · ·wn
consisting of distinct letters taken from [n]. We have n! choices for w. This
word correspond to a unique even involution τ with n−4h fixed points defined
by the following conditions:
τ(w1) = w2 . . . τ(w4h−1) = w4h;
τ(w4h+j) = w4h+j
8
with 0 < j ≤ n − 4h. It is easily checked that the involution τ arises from
(n− 4h)!(2h)!22h different words w. These considerations imply that
i+n =
bn4 c∑
h=0
n!
(2h)!(n− 4h)!22h .
On the other hand, it is well known that
|In| =
bn2 c∑
h=0
n!
(n− 2h)!2h .
Hence, since Fn(1) = i
+
n − i−n = 2i+n − |In|, we get the assertion.
¦
Proposition 6 The sequence Fn(1) satisfies the recurrence
Fn(1) = Fn−1(1)− (n− 1)Fn−2(1).
Proof Consider an even involution τ ∈ In. If τ(1) = 1, the restriction of τ
on the set {2, . . . , n} is an even involution on n− 1 objects. If τ(1) = j 6= 1,
the restriction of τ on {2, . . . , n} \ {j} is an odd involution on n− 2 objects.
This implies that:
i+n = i
+
n−1 + (n− 1)i−n−2
and, analogously,
i−n = i
−
n−1 + (n− 1)i+n−2.
These identities gives immediately the assertion.
¦
These results allow to deduce the following expression for the exponential
generating function of the sequence Fn(1):∑
n≥0
Fn(1)
n!
tn = et−
t2
2 .
9
References
[1] M.Barnabei, F.Bonetti, M.Silimbani, The descent statistic on involu-
tions is not log-concave, submitted
[2] M.Barnabei, F.Bonetti, M.Silimbani, The Eulerian distribution on self
evacuated involutions, submitted
[3] J.De´sarme´nien, D.Foata, The signed Eulerian numbers, Discr. Math. 99
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[4] J.De´sarme´nien, D.Foata, Fonctions syme´triques et se´ries hy-
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[6] V.J.Guo, J.Zeng, The Eulerian distribution on involutions is indeed uni-
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[10] R.P.Stanley, Enumerative Combinatorics, Vol. II, Cambridge Studies
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10


We de ne an analog of signed Eulerian numbers fn;k for involutions of the symmetric group and derive some combinatorial properties of this sequence. In particular, we exhibit both an explicit formula and a recurrence for f_{n,k} arising from the properties of its generating function

BARNABEI, MARILENA

BONETTI, FLAVIO

SILIMBANI, MATTEO

Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna

The signed Eulerian Numbers on Involutions

http://amsacta.unibo.it/2444/1/signedinvo.pdf

The signed Eulerian numbers on involutions

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Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna