Eulerian polynomials and excedance statistics

Abstract

A formula of Stembridge states that the permutation peak polynomials and descent polynomials are connected via a quadratique transformation. The aim of this paper is to establish the cycle analogue of Stembridge's formula by using cycle peaks and excedances of permutations. We prove a series of new general formulae expressing polynomials counting permutations by various excedance statistics in terms of refined Eulerian polynomials. Our formulae are comparable with Zhuang's generalizations (Zhuang 2017 [29]) using descent statistics of permutations. Our methods include permutation enumeration techniques involving variations of classical bijections from permutations to Laguerre histories, explicit continued fraction expansions of combinatorial generating functions in Shin and Zeng (2012) [21] and cycle version of modified Foata-Strehl action. We also prove similar formulae for restricted permutations such as derangements and permutations avoiding certain patterns. Moreover, we provide new combinatorial interpretations for the γ-coefficients of the inversion polynomials restricted on 321-avoiding permutations.

Full text

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS
BIN HAN, JIANXI MAO, AND JIANG ZENG
Abstract. A formula of Stembridge states that the permutation peak polynomials and
descent polynomials are connected via a quadratique transformation. The aim of this
paper is to establish the cycle analogue of Stembridge’s formula by using cycle peaks and
excedances of permutations. We prove a series of new general formulae expressing polyno-
mials counting permutations by various excedance statistics in terms of refined Eulerian
polynomials. Our formulae are comparable with Zhuang’s generalizations [Adv. in Appl.
Math. 90 (2017) 86-144] using descent statistics of permutations. Our methods include
permutation enumeration techniques involving variations of classical bijections from per-
mutations to Laguerre histories, explicit continued fraction expansions of combinatorial
generating functions in Shin and Zeng [European J. Combin. 33 (2012), no. 2, 111–
127] and cycle version of modified Foata-Strehl action. We also prove similar formulae
for restricted permutations such as derangements and permutations avoiding certain pat-
terns. Moreover, we provide new combinatorial interpretations for the γ-coefficients of the
inversion polynomials restricted on 321-avoiding permutations.
Contents
1. Introduction 1
2. Background and preliminaries 5
2.1. Permutation statistics and two bijections 5
2.2. The bijection Φ7
2.3. The bijection Ψ7
2.4. The star variation 8
2.5. Laguerre histories as permutation encodings 11
2.6. Françon-Viennot bijection 12
2.7. Restricted Françon-Viennot bijection 14
2.8. Foata-Zeilberger bijection 15
2.9. Pattern avoidances and 2-Motzkin paths 15
3. Main results 17
4. Proofs using group actions 23
4.1. Proof of Theorem 3.5 23
Date: March 5, 2021.
2010 Mathematics Subject Classification. 05A05, 05A15, 05A19.
Key words and phrases. Eulerian polynomials; Peak polynomials; Gamma-positivity; Derangement
polynomials; q-Narayana polynomials; Continued fractions; Laguerre histories; Françon-Viennot bijection;
Foata-Zeilberger bijection; Cyclic modified Foata-Strehl action.
1

2 B. HAN, J. MAO, AND J. ZENG
4.2. Proof of Theorem 3.6 27
5. Proofs via continued fractions 31
5.1. Some combinatorial continued fractions 31
5.2. Proof of Theorems 3.1, 3.4–3.6, 3.8–3.10 32
5.3. Proof of Theorems 3.12 and 3.13 37
Acknowledgement 38
References 38
1. Introduction
Stieltjes [25] showed that the Eulerian polynomials An(t)can be defined through the
continued fraction (S-fraction) expansion
X
n≥0
An(t)zn=1
1−1·z
1−t·z
1−2·z
1−2t·z
1−. . .
.(1.1)
For an n-permutation σ:= σ(1)σ(2) ···σ(n)of the word 1. . . n, an index i(1≤i≤n−1) is
adescent (resp. excedance) of σif σ(i)> σ(i+ 1) (resp. σ(i)> i). It is well-known [10,19]
that
An(t) = X
σ∈Sn
tdes σ=X
σ∈Sn
texc σ,(1.2)
where Snis the set of n-permutations and des σ(resp. exc σ) denotes the number of
descents (resp. excedances) of σ. The value σ(i)(2≤i≤n−1) is a peak of σif
σ(i−1) < σ(i)> σ(i+ 1) and the peak polynomials are defined by
Ppk
n(x) := X
σ∈Sn
xpk0σ,(1.3)
where pk0σdenotes the number of peaks of σ. The peak polynomials are related to the
Eulerian polynomials by Stembridge’s identity [24, Remark 4.8], see also [3,29],
An(t) = 1 + t
2n−1
Ppk
n4t
(1 + t)2,(1.4)
which can be used to compute the peak polynomials. Obviously Eq. (1.4) is equivalent to
the so-called γ-expansion of Eulerian polynomials
An(t) =
b(n−1)/2c
X
k=0
22k+1−nγn,ktk(1 + t)n−1−2k,(1.5)

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 3
where γn,k is the number of n-permutations with kpeaks. In the form of (1.5) it is not
difficult to see that Stembridge’s formula (1.4) is actually equivalent to a formula of Foata
and Schüzenberger [10, Théorème 5.6], see also Brändén’s proof using modified Foata-Strehl
action [3]. In the last two decades, many refinements of Stembridge’s identity have been
given by Brändén [3], Petersen [18], Shin and Zeng [21,22], Zhuang [29], Athanasiadis [1]
and others. In particular, Zhuang [29] has proved several identities expressing polynomi-
als counting permutations by various descent statistics in terms of Eulerian polynomials,
extending results of Stembridge, Petersen and Brändén.
By contracting the continued fraction (1.1) starting from the first and second lines (see
Lemma 5.1), respectively, we derive the two Jacobi-type continued fraction (J-fraction)
formulae (cf. [9])
X
n≥0
An+1(t)zn=1
1−(1 + t)·z−1·2·t·z2
1−2(1 + t)·z−2·3·t·z2
1−3(1 + t)·z−3·4·tz2
1− · ··
,(1.6a)
and
X
n≥0
An(t)zn=1
1−(1 + 0 ·t)·z−12·z2
1−(2 + 1 ·t)·z−22·t·z2
1−(3 + 2 ·t)·z−32·t·z2
1− · ··
.
(1.6b)
In view of Flajolet’s combinatorial theory for generic J-type continued fraction expan-
sions [9], Françon-Viennot’s bijection ψF V (resp. its restricted version φFV ) between per-
mutations and Laguerre histories provides a bijective proof of (1.6a) (resp. (1.6b)), while
Foata-Zeilberger’s bijection ψF Z [12] gives a bijective proof of (1.6b). More precisely,
Françon-Viennot [13] set up a bijection (and its restricted version) from permutations to
Laguarre histories using linear statistics of permutation, while Foata-Zeilberger constructed
another bijection [12] using cyclic statistics of permutations. In 1997 Clarke-Steingrímsson-
Zeng [6] constructed a bijection Φon permutations converting statistic des into exc on per-
mutations and linking the restricted Françon-Viennot’s bijection φF V to Foata-Zeilberger
bijection φF Z , see the right diagram in Figure 1. Later, similar to Φ, Shin and Zeng [21]
constructed a bijection Ψon permutations to convert linear statistics to cycle statistics
on permutations corresponding to (1.6a). We will show that the composition ψF V ◦Ψ−1
coincides with a recent bijection of Yan, Zhou and Lin [27], see Figure 1.

4 B. HAN, J. MAO, AND J. ZENG
LHn
Sn+1 Sn+1
ψF V ψY ZL
Ψ
LH∗
n
SnSn
φF V φF Z
Φ
Figure 1. Two factorizations: ψF V =ψY ZL ◦Ψand φFV =φF Z ◦Φ.
The Narayana polynomials Nn(t)can be defined by the S-fraction expansion
X
n≥0
Nn(t)zn=1
1−z
1−t·z
1−z
1−t·z
1− · ··
,(1.7)
see [14]. Note that Nn(1) is the n-th Catalan number Cn=1
n+1 2n
n. Similar to Eulerian
polynomials, by contracting the S-fraction (1.7) we derive immediately the followoing J-
fractions
X
n≥0
Nn+1(t)zn=1
1−(1 + t)·z−t·z2
1−(1 + t)·z−t·z2
1− · ··
(1.8a)
and
X
n≥0
Nn(t)zn=1
1−z−t·z2
1−(1 + t)·z−t·z2
1−(1 + t)·z− · ··
.(1.8b)
Let τ∈S3={123,132,213,231,312,321}. Recall that a permutation σ∈Snis said
to avoid the pattern τif there is no triple of indices i < j < k such that σ(i)σ(j)σ(k)is
order-isomorphic to τ. We shall write Sn(τ)for the set of permutations in Snavoiding
the pattern τ. It is known [14] that the Narayana polynomials have the combinatorial
interpretations
Nn(t) = X
σ∈Sn(231)
tdes σ=X
σ∈Sn(321)
texc σ.(1.9)

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 5
Hence, the Narayana polynomials can be considered as the Eulerian polynomials for re-
stricted permutations. Moreover, they are γ-positive and have the γ-expansion [19, Chapter
4]:
Nn(t) =
(n−1)/2
X
j=0 eγn,jtj(1 + t)n−1−2j,(1.10)
where eγn,j =|{σ∈Sn(231) : des(σ) = pk(σ) = j}|.
In this paper we shall give generalizations of Stembridge’s formula or their γ-analogues
(1.5) and (1.10) using excedance statistics by further exploiting the continued fraction tech-
nique in [20–22]. Indeed, from the observation (cf. [21]) that the gamma-positive formula
of Eulerian polynomials (1.5) is equal to the Jacobi-Rogers polynomial corresponding to
(1.6a), it becomes clear that Flajolet-Viennot’s combinatorial theory of formal continued
fractions could shed more lights on this topic. Our main tool is the combinatorial theory
of continued fractions due to Flajolet [9] and bijections due to Françon-Viennot, Foata-
Zeilberger between permutations and Laguarre histories, see [6,9,12,13,20]. As in [21] this
approach uses both linear and cycle statistics on permutations. There are several well-
known q-Narayana polynomials in the litterature; see [14] and the references therein. As
a follow-up to [14], we shall give more results on q-Narayana polynomials using pattern-
avoiding permutations.
The rest of this paper is organized as follows. In Section 2 after recalling the necessary
definitions and results from [20–22], we link the recent bijection ψY ZL of Yan-Zhou-Lin [27]
to two known bijections; in Section 3 we present our generalized formulae of (1.4) in three
classes:
•Eulerian polynomials for permutations and derangements,
•Eulerian polynomials for pattern-avoiding permutations,
•Eulerian polynomials for signed permutations.
There are two types of proof: group actions of Foata-Strehl’s type and manipulations of
continued fractions. More precisely, we prove Theorems 3.5 and 3.6 using variations of
modified Foata-Strehl action on permutations or Laguerre histories in Section 4; we then
prove these two theorems and the remaining theorems by comparing the continued fraction
expansions of the generating functions in Section 5. In what follows, we shall abbreviate
"generating functions" by "g.f.".
2. Background and preliminaries
2.1. Permutation statistics and two bijections. For σ=σ(1)σ(2) ···σ(n)∈Snwith
convention 0–0, i.e., σ(0) = σ(n+ 1) = 0, a value σ(i)(1≤i≤n) is called
•apeak if σ(i−1) < σ(i)and σ(i)> σ(i+ 1);
•avalley if σ(i−1) > σ(i)and σ(i)< σ(i+ 1);
•adouble ascent if σ(i−1) < σ(i)and σ(i)< σ(i+ 1);
•adouble descent if σ(i−1) > σ(i)and σ(i)> σ(i+ 1).

6 B. HAN, J. MAO, AND J. ZENG
The set of peaks (resp. valleys, double ascents, double descents) of σis denoted by
Pk σ(resp. Val σ, Da σ, Dd σ).
Let pk σ(resp. val σ,da σ,dd σ) be the number of peaks (resp. valleys, double ascents,
double descents) of σ. For i∈[n] := {1, . . . , n}, we introduce the following statistics:
(31-2)iσ= #{j: 1 < j < i and σ(j)< σ(i)< σ(j−1)}
(2-31)iσ= #{j:i < j < n and σ(j+ 1) < σ(i)< σ(j)}
(2-13)iσ= #{j:i < j < n and σ(j)< σ(i)< σ(j+ 1)}
(13-2)iσ= #{j: 1 < j < i and σ(j−1) < σ(i)< σ(j)}
(2.1)
and define four statistics (see (2.41)):
(31-2) =
n
X
i=1
(31-2)i,(2-31) =
n
X
i=1
(2-31)i,(2-13) =
n
X
i=1
(2-13)i,(13-2) =
n
X
i=1
(13-2)i.
Now, we consider σ∈Snas a bijection i7→ σ(i)for i∈[n], a value x=σ(i)is called
•acyclic peak if i=σ−1(x)< x and x>σ(x);
•acyclic valley if i=σ−1(x)> x and x<σ(x);
•adouble excedance if i=σ−1(x)< x and x<σ(x);
•adouble drop if i=σ−1(x)> x and x>σ(x);
•afixed point if x=σ(x).
We say that i∈[n−1] is an ascent of σif σ(i)< σ(i+ 1) and that i∈[n]is a drop of
σif σ(i)< i. Let
Cpk (resp. Cval,Cda,Cdd,Fix,Drop) (2.2)
be the set of cyclic peaks (resp. cyclic valleys,double excedances,double drops,fixed points,
drops) and denote the corresponding cardinality by cpk (resp. cval,cda,cdd,fix,drop).
Obviously we have
cpk σ=cval σfor σ∈Sn.(2.3)
Moreover, we define
wex σ= #{i:i≤σ(i)}=exc σ+fix σ(2.4a)
crosiσ= #{j:j < i < σ(j)< σ(i)or σ(i)< σ(j)≤i<j},(2.4b)
nestiσ= #{j:j < i < σ(i)< σ(j)or σ(j)< σ(i)≤i<j}.(2.4c)
Let cros =Pn
i=1 crosi,nest =Pn
i=1 nestiand icr σ1=cros σ−1. Note (cf [14, Remark 2.4])
that
nest σ−1=nest σfor σ∈Sn.(2.5)
A pair of integers (i, j)is an inversion of σ∈Snif i < j and σ(i)> σ(j), and σ(i)(resp.
σ(j)) is called inversion top (resp. bottom). Let inv σbe the inverion number of σ.
1Our definition of cros corresponds to icr in [14].

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 7
For σ∈Snwith convention 0–∞, i.e., σ(0) = 0 and σ(n+ 1) = ∞, let
Lpk (resp. Lval,Lda,Ldd)
be the set of peaks (resp. valleys,double ascents and double decents) and denote the cor-
responding cardinality by lpk (resp. lval,lda and ldd). For i∈[n], the value σ(i)is called
aleft-to-right maximum (resp. right-to-left minimum) if σ(i) = max {σ(1), σ(2), . . . , σ(i)}
(resp. σ(i) = min {σ(i), . . . , σ(n−1), σ(n)}). Similarly, we define left-to-right minimum
(resp. right-to-left maximum).
A double ascent σ(i) (i= 1, . . . , n)is called a foremaximum (resp. afterminimum) of σ
if it is at the same time a left-to-right maximum (resp. right-to-left minimum). Denote the
number of foremaxima (resp. afterminima) of σby fmax σ(resp. amin σ). Note that for
the peak number pk0in (1.3) we have following identities :
pk0=val =pk −1and lval =lpk.(2.6)
Now we recall two bijections Φand Ψdue to Clarke et al. [6] and Shin-Zeng [21], respec-
tively.
2.2. The bijection Φ.Let σ=σ(1) . . . σ(n)∈Sn, an inversion top number (resp. inver-
sion bottom number) of a letter x:= σ(i)in the word σis the number of occurrences of
inversions of form (i, j)(resp (j, i)). A letter σ(i)is a descent top (resp. descent bottom)
if σ(i)> σ(i+ 1) (resp. σ(i−1) > σ(i)). Given a permutation σ, we first construct two
biwords, f
f0and g
g0, where f(resp. g) is the subword of descent bottoms (resp. nonde-
scent bottoms) in σordered increasingly, and f0(resp. g0) is the permutation of descent
tops (resp. nondescent tops) in σsuch that the inversion bottom (resp. top) number of
each letter x:= σ(i)in f0(resp. g0) is (2-31)xσ, and then form the biword w=f
f0
g
g0by
concatenating fand g, and f0and g0, respectively. Rearranging the columns of w, so that
the bottom row is in increasing order, we obtain the permutation τ= Φ(σ)as the top row
of the rearranged bi-word.
The following result can be found in [21, Theorem 12] and its proof.
Lemma 2.1 (Shin-Zeng ).For σ∈Sn, we have
(2-31,31-2,des,asc,lda −fmax,ldd,lval,lpk,fmax)σ
=(nest,icr,drop,exc +fix,cda,cdd,cval,cpk,fix)Φ(σ)(2.7a)
=(nest,cros,exc,drop +fix,cdd,cda,cval,cpk,fix)(Φ(σ))−1,
(Lval,Lpk,Lda,Ldd)σ= (Cval,Cpk,Cda ∪Fix,Cdd)Φ(σ),(2.7b)
and
(2-31)iσ=nestiΦ(σ)∀i= 1, . . . , n. (2.7c)

8 B. HAN, J. MAO, AND J. ZENG
2.3. The bijection Ψ.Given a permutation σ∈Sn, let
ˆσ=1 2 . . . n n + 1
σ(1) + 1 σ(2) + 1 . . . σ(n) + 1 1 ,(2.8)
and τ:= Φ(ˆσ)∈Sn+1. Since the last element of ˆσis 1, the first element of τshould be
n+ 1. Define the bijection Ψ : Sn→Snby
Ψ(σ) := τ(2) . . . τ (n+ 1) ∈Sn.(2.9)
Example 2.2. If σ=412796583, then ˆσ=52381076941, and reading
from left to right, we obtain the corresponding numbers (2-31)iˆσ: 1,1,1,2,0,1,1,0,0,0for
i= 5,2,...,1, andf
f0=1
4
2
9
4
5
6
7
7
10,g
g0=3
2
5
3
8
8
9
6
10
1.
Hence
w=f
f0
g
g0=1
4
2
9
4
5
6
7
7
10
3
2
5
3
8
8
9
6
10
1→10
1
3
2
5
3
1
4
4
5
9
6
6
7
8
8
2
9
7
10.
Thus τ= Φ(ˆσ)=10351496827, and Ψ(σ) = τ(2) . . . τ (10) = 3 5 1 4 9 6 8 2 7.
Lemma 2.3. For i∈[n], we have
(2-31)i+1 ˆσ=((2-13)iσ+ 1 if i+ 1 ∈Lval ˆσ∪Lda ˆσ,
(2-13)iσif i+ 1 ∈Lpk ˆσ∪Ldd ˆσ.
Proof. An increasing (resp. decreasing) run of σis a maximum consecutive increasing
(resp. decreasing) subsequence R:= σ(i)σ(i+ 1) . . . σ(j)of σsuch that σ(i−1) > σ(i)and
σ(j)> σ(j+ 1) (resp. σ(i−1) < σ(i)and σ(j)< σ(j+ 1)) with 1≤i≤j≤n. For any
i∈[n], as ˆσ(n+ 1) = 1, there is a unique way to write
ˆσ=(w1(i+ 1)u1d2. . . uk−1dkif i+ 1 ∈Lval ˆσ∪Lda ˆσ,
w1(i+ 1)d1u2d2. . . ukdkif i+ 1 ∈Lpk ˆσ∪Ldd ˆσ,
where ui(resp. di) is an increasing (resp. decreasing) run, and (i+ 1)u1(resp. (i+ 1)d1)
is an increasing (resp. decreasing) sequence. We say that a run Rcovers iif iis bounded
by max(R)and min(R). It is not hard to show that
#{j≥2 : ujcovers i+ 1}=(#{j≥2 : djcovers i+ 1}+ 1 if i+ 1 ∈Lval ˆσ∪Lda ˆσ,
#{j≥2 : djcovers i+ 1}if i+ 1 ∈Lpk ˆσ∪Ldd ˆσ.
Since (2-13)i(resp. (2-31)i) is the number of increasing (resp. decreasing) runs covering i
to the right of i, we are done. 
We use the aforementioned statistics to define variant boundary conditions. Given a
permutation σ∈Snwith convention ∞ − 0, the number of corresponding peaks, valleys,
double ascents, and double descents of permutation σ∈Snis denoted by rpk σ,rval σ,rda σ,
rdd σrespectively. A double descent σ(i)is called a aftermaximum (resp. foreminimum) of

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 9
σif it is at the same time a right-to-left maximum (resp. left-to-right minimum). Denote
the number of aftermaxima (resp. foreminimum) of σby amax σ(resp. fmin σ). For
σ=σ(1)σ(2) ···σ(n)∈Sn, we define two permutations σcand σrby
σc(i) = n+ 1 −σ(i)and σr(i) = σ(n+ 1 −i)for i∈[n].(2.10)
It is not difficult to verify the following properties
(2-31,31-2,des,lda −fmax,ldd,lval,fmax)σ(2.11a)
=(13-2,2-13,asc,rdd −amax,rda,rval,amax)σr(2.11b)
=(31-2,2-31,des,lda −amin,ldd,lpk,amin)σr◦c(2.11c)
=(2-13,13-2,asc,rdd −fmin,rda,rval,fmin)σr◦c◦r,(2.11d)
where σr◦c= (σr)cand σr◦c◦r= (σr◦c)r= (σr)c◦r.
2.4. The star variation. For σ=σ(1) ··· σ(n)∈Sn, we define its star companion σ∗as
a permutation of {0, . . . , n}by
σ∗=0 1 2 . . . n
n σ(1) −1σ(2) −1. . . σ(n)−1.(2.12)
We define the following sets of cyclic star statistics for σ:
Cpk∗σ={i∈[n−1] : (σ∗)−1(i)< i > σ∗(i)},(2.13a)
Cval∗σ={i∈[n−1] : (σ∗)−1(i)> i < σ∗(i)},(2.13b)
Cda∗σ={i∈[n−1] : (σ∗)−1(i)< i < σ∗(i)},(2.13c)
Cdd∗σ={i∈[n−1] : (σ∗)−1(i)> i > σ∗(i)},(2.13d)
Fix∗σ={i∈[n−1] : i=σ∗(i)},(2.13e)
Wex∗σ={i∈[n−1] : i≤σ∗(i)},(2.13f)
Drop∗σ={i∈[n] : i>σ∗(i)}.(2.13g)
The corresponding cardinalties are denoted by cpk∗,cval∗,cda∗,cdd∗,fix∗,wex∗and drop∗,
respectively. By (2.13a), (2.13d) and (2.13g), we have drop∗−1 = cdd∗+cpk∗.Let cyc σ
be the number of cycles of σand cyc∗σ:= cycσ∗. For example, for σ= 3762154, we have
σ∗= 72651043, which has two cycles 1→2→6→4→1and 7→3→5→0→7. Thus
cyc∗σ= 2.
For any subsebt S⊂N, we write S+ 1 := {s+ 1 : s∈S}.
Theorem 2.4. For σ∈Sn, we have
(Val,Pkn,Da,Dd)σ= (Cval∗,Cpk∗,Cda∗∪Fix∗,Cdd∗)Ψ(σ)(2.14)
with Pknσ= Pk σ\ {n}and
((2-13)i,(31-2)i)σ= (nesti,crosi)Ψ(σ)for i∈[n].(2.15)

10 B. HAN, J. MAO, AND J. ZENG
Proof. For σ∈Sn, by definition (2.8), (2.9) and (2.12), we have
τ(i+ 1) = (Ψ(σ))∗(i)+1.(2.16)
and
((Val σ+ 1) ∪ {1},Pk σ+ 1,Da σ+ 1,Dd σ+ 1) = (Lval,Lpk,Lda,Ldd)ˆσ. (2.17)
For 2≤i≤n, by (2.2) and (2.16) we have the following equivalences:
i∈Cval τ⇐⇒ i−1∈Cval∗(Ψ(σ))
and
i<τ(i)and i<τ−1(i)⇐⇒ i < Ψ(σ)(i−1) and i−1<(Ψ(σ))−1(i).
Thus, by (2.8) and (2.9),
(Cval∗Ψ(σ) + 1) ∪ {1}= Cval Φ(ˆσ).(2.18)
In the same vein, we have
Cpk∗Ψ(σ) + 1 = Cpk Φ(ˆσ)\ {n+ 1},
Cda∗Ψ(σ)∪Fix∗Ψ(σ) + 1 = Cda Φ(ˆσ)∪Fix Φ(ˆσ),(2.19)
Cdd∗Ψ(σ) + 1 = Cdd Φ(ˆσ).
Comparing (2.17) and (2.18)-(2.19) and using (2.7b) we derive (2.14).
Next, for (2.15), we only prove nestiΨ(σ) = (2-13)iσand leave crosiΨ(σ) = (31-2)iσto
the interested reader. By Lemma 2.3 we have
(2-31)i+1 ˆσ=((2-13)iσ+ 1 if i+ 1 ∈Lval ˆσ∪Lda ˆσ,
(2-13)iσif i+ 1 ∈Lpk ˆσ∪Ldd ˆσ. (2.20)
So, if we show that
(nest)i+1τ=((nest)iΨ(σ)+1 if i+ 1 ∈Cval τ∪Cda τ∪Fix τ,
(nest)iΨ(σ)if i+ 1 ∈Cpk τ∪Cdd τ , (2.21)
as τ= Φ(ˆσ)and by (2.7b),
(Lval,Lpk,Lda,Ldd)ˆσ= (Cval,Cpk,Cda ∪Fix,Cdd)Φ(ˆσ),(2.22)
the result follows from the identity nestiτ= (2-31)iˆσ(see (2.7c)).
Now we prove (2.21). By (2.4c) the index nestiσ(i∈[n]) can be characterized in terms
of σ∗(see (2.12)) as follows:
nestiσ= #{j∈[n] : j < i ≤σ∗(i)< σ∗(j)or σ∗(j)< σ∗(i)< i < j}.(2.23)
We consider three cases of i+ 1.

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 11
(a) if i+ 1 < τ (i+ 1), then i < (Ψ(σ))∗(i). By (2.16), we have
#{j∈[n] : j+ 1 < i + 1 < τ (i+ 1) < τ(j+ 1)}(2.24a)
=#{j∈[n] : j < i ≤(Ψ(σ))∗(i)<(Ψ(σ))∗(j)}
and
#{j∈[n] : j+ 1 > i + 1 ≥τ(i+ 1) > τ(j+ 1)}(2.24b)
= #{j∈[n] : j > i > (Ψ(σ))∗(i)>(Ψ(σ))∗(j)}.
Since τ(1) = n+ 1 and 1< i + 1 < τ(i+ 1) < n + 1, by (2.4c), (2.24a) and (2.24b)
we obtain
nesti+1 τ= 1+#{j∈[n] : j+ 1 < i + 1 < τ (i+ 1) < τ(j+ 1)}
+#{j∈[n] : j+ 1 > i + 1 ≥τ(i+ 1) > τ(j+ 1)}
= 1+#{j∈[n] : j < i ≤(Ψ(σ))∗(i)<(Ψ(σ))∗(j)}
+#{j∈[n] : j > i > (Ψ(σ))∗(i)>(Ψ(σ))∗(j)},
which is equal to nestiΨ(σ)+1by (2.23).
(b) if i+ 1 > τ(i+ 1), then i > (Ψ(σ))∗(i). Similarly to (a) we get nesti+1τ=nestiΨ(σ).
(c) if i+ 1 = τ(i+ 1), then i= (Ψ(σ))∗(i). It is easy to see that
#{j∈[n+ 1] : j > i + 1 > τ (j)}= #{j∈[n+ 1] : j < i + 1 < τ (j)}.(2.25)
As τ(1) = n+ 1, we have
#{j∈[n+ 1] : j > i + 1 > τ (j)}=#{j∈[n+ 1] : j < i + 1 < τ (j)}
=#{j∈[n] : j+ 1 < i + 1 < τ (j+ 1)}+ 1
=#{j∈[n] : j < i < (Ψ(σ))∗(j)}+ 1 (by (2.16))
Then, we have nesti+1τ=nestiΨ(σ)+1 by using (2.4c) (resp. (2.23) ) to compute
nesti+1τ(resp. nestiΨ(σ)).

Since asc =val +da,des =pk +dd −1,wex∗=cval∗+cda∗+fix∗,drop∗−1 = cdd∗+cpk∗,
we get the following result in [21, Theorem 12].
Corollary 2.1 (Shin-Zeng).For σ∈Snwe have
(2-13,31-2,des,asc,da,dd,val)σ
=(nest,cros,drop∗−1,wex∗,cda∗+fix∗,cdd∗,cval∗)Ψ(σ).(2.26)
2.5. Laguerre histories as permutation encodings. A 2-Motzkin path is a lattice path
starting and ending on the horizontal axis but never going below it, with possible steps
(1,1),(1,0), and (1,−1), where the level steps (1,0) can be given either of two colors: blue
and red, say. The length of the path is defined to be the number of its steps. For our
purpose it is convenient to identify a 2-Motzkin path of length nas a word s:= s1. . . sn

12 B. HAN, J. MAO, AND J. ZENG
•
•••
•
• • •
•
pi00010110
LbLrLrLb
Figure 2. A Laguerre history (s,p)of lenth 8.
on the alphabet {U,D,Lr,Lb}such that |s1. . . sn|U= [s1. . . sn|Dand the height of the ith
step is nonnegative, i.e.,
hi(s) := |s1. . . si|U−[s1. . . si|D≥0 (i= 1, . . . , n),(2.27)
where |s1. . . si|Uis the number of letters Uin the word s1. . . si. By (1.8a) we see that the
number of 2-Motzkin paths of length nis the Catalan number Cn+1 .
ALaguerre history (resp. restricted Laguerre history) of length nis a pair (s,p), where
sis a 2-Motzkin path s1. . . snand p= (p1, . . . , pn)with 0≤pi≤hi−1(s)(resp. 0≤pi≤
hi−1(s)−1if si=Lror D) with h0(s)=0. Let LHn(resp. LH∗
n) be the set of Laguerre
histories (resp. restricted Laguerre histories) of length n. There are several well-known
bijections between Snand LH∗
nand LHn−1, see [6,8].
2.6. Françon-Viennot bijection. We recall a version of Françon and Viennot’s bijection
ψF V :Sn+1 → LHn. Given σ∈Sn+1, the Laguerre history ψF V (σ)=(s,p)is defined as
follows:
si=








Uif i∈Val σ
Dif i∈Pk σ
Lbif i∈Da σ
Lrif i∈Dd σ
(2.28)
and pi= (2-13)iσfor i= 1, . . . , n.
For example, if σ=412796583∈S9, then
(s,p) = ((U,Lb,Lr,D,U,Lr,Lb,D),(0,0,0,1,0,1,1,0)) ∈ LH8,
which is depicted in Figure 2.
For σ∈Sn+1, we define the following sets
Scval = {i∈[n] : i<σ(i)and i+ 1 ≤σ−1(i+ 1)},(2.29a)
Scpk = {i∈[n] : i≥σ(i)and i+ 1 > σ−1(i+ 1)},(2.29b)
Scda = {i∈[n] : i<σ(i)and i+ 1 > σ−1(i+ 1)},(2.29c)
Scdn = {i∈[n] : i≥σ(i)and i+ 1 ≤σ−1(i+ 1)}.(2.29d)

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 13
σ∈DD 4,k σ σrΨ(σ)∈SDE 4,k (31-2)σ(2-13)σinvΨ(σ)excΨ(σ)
k= 0 1324 4231 1423 0 1 2 1
1423 3241 1432 1 0 3 1
2314 4132 4123 0 2 3 1
2413 3142 4132 1 1 4 1
k= 1 3412 2143 3214 1 0 3 1
2134 4312 3124 0 1 2 1
3124 4213 4213 1 1 4 1
4123 3214 4231 2 0 5 1
Figure 3. Illustration of Ψon DD 4,k with their statistics.
Yan, Zhou and Lin [27] constructed a bijection ψY Z L :Sn+1 → LHn, which can be
defined as follows. For σ∈Sn+1, let ψY ZL (σ) = (s,p)with
si=








Uif i∈Scval σ,
Dif i∈Scpk σ,
Lbif i∈Scda σ,
Lrif i∈Scdn σ,
(2.30)
and pi=nestiσfor i= 1, . . . , n.
Theorem 2.5. We have ψF V =ψY ZL ◦Ψ.
Proof. Let ψ=ψF V ◦Ψ−1, which is a bijection from Sn+1 to LHn. By Theorem 2.4, for
σ∈Sn+1, we can define ψ(σ)=(s,p)as follows: for i= 1, . . . , n,
si=








Uif i∈Cval∗σ,
Dif i∈Cpk∗σ,
Lbif i∈Cda∗σ∪Fix∗σ,
Lrif i∈Cdd∗σ,
(2.31)
with pi=nestiσ. Comparing (2.30) and (2.31) it suffices to show that for σ∈Sn+1,
(Scval,Scpk,Scdn,Scda)σ= (Cval∗,Cpk∗,Cdd∗,Cda∗∪Fix∗)σ. (2.32)
We just prove Scval σ= Cval∗σand omit the similar proof of other cases. As Cval∗(σ) =
{i:i+ 1 < σ(i), i < σ−1(i+ 1)}, comparing with (2.29a) we need only to show that
Scval σ⊂Cval∗σ. If i∈Scval(σ), then i<σ(i)and i+ 1 ≤σ−1(i+ 1). Suppose
i+ 1 = σ(i), then σ−1(i+ 1) = i, which contradicts the second inequality. So i+ 1 < σ(i),
and i∈Cval∗σ. We are done. 
Given a 2-Motzkin path sof length nwe define θ(s)to be the 2-Motzkin path obtained
by switching all the letters Lbwith Lrin s. By abuse of notation, for a Laguerre history
(s,p)∈ LHnwe define
θ(s,p)=(θ(s),p).(2.33)

14 B. HAN, J. MAO, AND J. ZENG
Corollary 2.2. The two sextuple statistics
(nest,cros,exc,cdd∗,cda∗+fix∗,cpk∗)and (2-13,31-2,des,da,dd,pk −1)
are equidistributed on Sn.
Proof. For σ∈Sn, let τ=ψ−1◦θ◦ψF V (σ). It follows from (2.33), (2.28) and (2.31) that
(Val,Pkn,Dd,Da,(2-13)i)σ
=(Cval∗,Cpk∗,Cda∗∪Fix∗,Cdd∗,nesti)τ, ∀i∈[n].
Let ψF V (σ)=(s,p)and ψ(τ)=(s0,p0). Then hi(s,p) = hi(s0,p0)for all i∈[n]. It is not
difficult to prove by induction that
(2-13)iσ+ (31-2)iσ=hi−1(s,p),(2.34a)
nestiσ+crosiσ=hi−1(s0,p0).(2.34b)
Thus we have (31-2)iσ=crosiτ. As exc =wex∗=cval∗+cda∗+fix∗,des =val +dd,
cpk∗=cval∗, and val =pk −1, the proof is completed. 
For k∈[n]we define the subsets of Sn:
DD n,k :={σ∈Sn:des σ=k, dd σ= 0},(2.35a)
DE ∗
n,k :={σ∈Sn:exc σ=k, cda∗σ+fix∗σ= 0},(2.35b)
SDE n,k :={σ∈Sn:exc σ=k, scda(σ)=0}.(2.35c)
Theorem 2.6. For 0≤k≤(n−1)/2we have
γn,k(q) := X
σ∈DD n,k
q2(31-2)σ+(2-13)σ(2.36a)
=X
σ∈DE ∗
n,k
qinv σ−exc σ(2.36b)
=X
σ∈SDE n,k
qinv σ−exc σ.(2.36c)
Proof. For σ∈Sn, recall that σr:= σ(n)··· σ(2)σ(1) (see (2.10)). By (2.26),
2(31-2) + 2-13σ=2(2-13) + 31-2σr= (2nest +cros)Ψ(σr).
Invoking the following formula for inversion numbers (cf. [20, Eq. (40)])
inv =exc + 2nest +cros,(2.37)
we derive 2(31-2) + (2-13)σ= (inv −exc)Ψ(σr).(2.38)
Besides, by (2.26) and (2.32) we have
(des,dd)σ= (asc,da)σr
= (wex∗,cda∗+fix∗)Ψ(σr)

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 15
= (cval∗+cda∗+fix∗,cda∗+fix∗)Ψ(σr).
Hence, if dd(σ)=(cda∗+fix∗)Ψ(σ)=0, from (2.26) we see that σ∈DD n,k if and only if
Ψ(σr)∈DE ∗
n,k. By (2.38) this implies (2.36b). Finally, we derive (2.36c) from (2.32). 
Remark 2.7. Yang-Zhou-Lin [27] proved that
γn,k(q) = X
σ∈DD n,k
q(31-2)σ+2(2-13)σ
which first appeared as the γ-coefficents of the polynomial Pσ∈Sntexcσqinvσ−excσin [22].
2.7. Restricted Françon-Viennot bijection. We recall a restricted version of Françon
and Viennot’s bijection φF V :Sn→ LH∗
n. Given σ∈Sn, the Laguerre history (s,p)is
defined as follows:
si=








Uif i∈Lval σ
Dif i∈Lpk σ
Lbif i∈Lda σ
Lrif i∈Ldd σ
(2.39)
and pi= (2-31)iσfor i= 1, . . . , n.
2.8. Foata-Zeilberger bijection. This bijection φF Z encodes permutations using cyclic
statistics. Given σ∈Sn,φF Z :Sn→ LH∗
nis for i= 1, . . . , n,
si=








Uif i∈Cval σ
Dif i∈Cpk σ
Lbif i∈Cda σ∪Fix σ
Lrif i∈Cdd σ
(2.40)
with pi=nestiσ. By (2.7b) and (2.7c), we can build a comutative diagram, see the right
diagram of Figure 1.
2.9. Pattern avoidances and 2-Motzkin paths. We shall consider the so-called vincu-
lar patterns [2]. The number of occurrences of vincular patterns 31-2,2-31,2-13 and 13-2
in π∈Snare defined (cf. (2.1)) by
(31-2) π= #{(i, j) : i+ 1 < j ≤nand π(i+ 1) < π(j)< π(i)},
(2-31) π= #{(i, j) : j < i < n and π(i+ 1) < π(j)< π(i)},
(2-13) π= #{(i, j) : j < i < n and π(i)< π(j)< π(i+ 1)},
(13-2) π= #{(i, j) : i+ 1 < j ≤nand π(i)< π(j)< π(i+ 1)}.
(2.41)
Similarly, we use Sn(31-2) to denote the set of permutations of length nthat avoid the
vincular pattern 31-2, etc. In order to apply Laguerre history to count pattern-avoiding
permutations, we will need the following results in [14, Lemma 2.8 and 2.9].

16 B. HAN, J. MAO, AND J. ZENG
Lemma 2.8. [14, Lemma 2.8] For any n≥1, we have
Sn(2-13) = Sn(213),Sn(31-2) = Sn(312),(2.42)
Sn(13-2) = Sn(132),Sn(2-31) = Sn(231).(2.43)
Lemma 2.9. [14, Lemma 2.9]
(i) A permutation π∈Snbelongs to Sn(321) if and only if nestπ= 0.
(ii) The mapping Φhas the property that Φ(Sn(231)) = Sn(321).
We use 2-Mnto denote the set of 2-Motzkin paths of length nand 2-M∗
nto denote its
subset that is composed of 2-Motzkin paths without Lr-step at level zero, i.e., if hi−1= 0,
then si6=Lr. Let e
φF V ,e
φF Z ,e
ψF V and e
ψY ZL be the restriction of φF V ,φF Z ,ψF V and ψY ZL
on the sets Sn(231),Sn(321),Sn+1(213) and Sn+1 (321), respectively.
Theorem 2.10. We have
(1) The mapping e
φF V is a bijection from Sn(231) to 2-M∗
n.
(2) The mapping e
φF Z is a bijection from Sn(321) to 2-M∗
n.
(3) The mapping e
ψF V is a bijection from Sn(213) to 2-Mn.
(4) The mapping e
ψY ZL is a bijection from Sn(321) to 2-Mn.
Proof. We just prove (1) and leave the others to the reader. If σ1,σ2∈Sn(231), let
φF V (σi) = (si,pi)for i= 1,2. By definition we have (2-31)σ1= (2-31)σ2= 0, which implies
that p1=p2= (0,0,··· ,0); as φF V is a bijection, we derive that s16=s2. Hence, the
mapping e
φF V is an injection from Sn(231) to 2-M∗
n. Noticing that the g.f. Pn≥0|2-M∗
n|zn
has the continued fraction expansion (1.8b) with t= 1, we derive that |Sn(231)|=|2-M∗
n|=
Cn. Thus, the mapping e
φF V is a bijection.

Theorem 2.11. Let e
Φbe the restriction of Φon Sn(231). Then e
Φis a bijection from
Sn(231) to Sn(321). Moreover, for σ∈Sn(231), we have
(31-2,des,asc,lda −fmax,ldd,lval,lpk,fmax)σ
=(icr,drop,exc +fix,cda,cdd,cval,cpk,fix)e
Φ(σ)(2.44)
=(cros,exc,drop +fix,cdd,cda,cval,cpk,fix)(e
Φ(σ))−1.
Proof. For σ∈Sn(231), we have (2-31)i= 0 for i∈[n]. So the inversion bottom (resp.
top) number of each letter in f0(resp. g0) equals 0. Let τ=e
Φ(σ). By definition of Φ(cf.
Section 2.2) the letters in f0(resp. g0) are in increasing order. It is not hard to verify that
nesti(τ)=0for each i∈[n]. By Lemma 2.9, we derive that τ∈Sn(321). For σ1, σ2∈
Sn(231), since Φis a bijection, we have e
Φ(σ1)6=e
Φ(σ2). And |Sn(231)|=|Sn(321)|=Cn,
so e
Φis a bijection from Sn(231) to Sn(321). Finally, the equidistribution (2.44) follows
from Lemma 2.1.
Theorem 2.12. Let e
Ψbe the restriction of Ψon Sn(213). Then e
Ψis a bijection from
Sn(213) to Sn(321).

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 17
2-Mn
Sn+1(213) Sn+1(321)
e
ψF V e
ψY ZL
e
Ψ
2-M∗
n
Sn(231) Sn(321)
e
φF V e
φF Z
e
Φ
Figure 4. Two factorizations: e
ψF V =e
ψY ZL ◦e
Ψand e
φF V =e
φF Z ◦e
Φ
Proof. If σ∈Sn(213), then (2-13)iσ= 0 for i∈[n]. Thus, (2-31)1ˆσ= 0, and by Lemma 2.3,
(2-31)i+1 ˆσ= 1 if i+ 1 is a nondescent top and (2-31)i+1 ˆσ= 0 otherwise. By definition of Φ,
we construct two biwords, f
f0and g
g0, where f(resp. g) is the subword of descent bottoms
(resp. nondescent bottoms) in ˆσordered increasingly, and f0(resp. g0) is the permutation
of descent tops (resp. nondescent tops) in ˆσsuch that the letters (resp. except 1at the
end) in f0(resp. g0) are in increasing order.
Let τ= Φ(ˆσ). It is not hard to verify that nesti(τ) = 1 if i∈g0\ {1}and nesti(τ) = 0
otherwise. Thus, by (2.21), we have nest(e
Ψ(σ)) = 0. By Lemma 2.9,e
Ψ(σ)∈Sn(321).
For σ1, σ2∈Sn(213), since Ψis a bijection, we have e
Ψ(σ1)6=e
Ψ(σ2). And |Sn(213)|=
|Sn(321)|=Cn, so e
Ψis a bijection from Sn(213) to Sn(321).
Example 2.13. If σ=168972534, then ˆσ=27910836451, and reading
from left to right, we obtain the corresponding numbers (2-31)i: 1,1,1,0,0,1,0,1,0,0for
i= 2,7,...,1, andf
f0=1
5
3
6
4
8
8
10,g
g0=2
2
5
3
6
4
7
7
9
9
10
1.
Hence
w=f
f0
g
g0=1
5
3
6
4
8
8
10
2
2
5
3
6
4
7
7
9
9
10
1→10
1
2
2
5
3
6
4
1
5
3
6
7
7
4
8
9
9
8
10.
Thus τ= Φ(ˆσ)=10256137498, and e
Ψ(σ) = τ(2) . . . τ (10) = 2 5 6 1 3 7 4 9 8.
Combining Theorems 2.10,2.11,2.12 and Figure 1we obtain the diagrams in Figure 4.
3. Main results
For a finite set of permutations Ωand mstatistics stat1,...,statmon Ω, we define the
generating polynomial
P(stat1,...,statm)(Ω; t1, . . . , tm) := X
σ∈Ω
tstat1σ
1. . . tstatmσ
m.(3.1)

18 B. HAN, J. MAO, AND J. ZENG
We define the polynomial
An(p, q, t) := X
σ∈Sn
pnest σqcros σtexc σ.(3.2)
The following is a generalization of Stembridge’s identity (1.4).
Theorem 3.1. For n≥1, we have
An(p, q, t) = 1 + xt
1 + xn−1
P(nest,cros,cpk∗,exc)Sn;p, q, (1 + x)2t
(x+t)(1 + xt),x+t
1 + xt,(3.3)
equivalently,
P(nest,cros,cpk∗,exc)(Sn;p, q, x, t) = 1 + u
1 + uv n−1
An(p, q, v),(3.4)
where u=1+t2−2xt−(1−t)√(1+t)2−4xt
2(1−x)tand v=(1+t)2−2xt−(1+t)√(1+t)2−4xt
2xt .
By Corollary 2.2, we obtain the following linear generalization of Stembridge’s identity.
Corollary 3.1. For n≥1, we have
An(p, q, t) = 1 + xt
1 + xn−1
P(2-13,31-2,pk−1,des)Sn;p, q, (1 + x)2t
(x+t)(1 + xt),x+t
1 + xt,(3.5)
equivalently,
P(2-13,31-2,pk−1,des)(Sn;p, q, x, t) = 1 + u
1 + uv n−1
An(p, q, v),(3.6)
where u=1+t2−2xt−(1−t)√(1+t)2−4xt
2(1−x)tand v=(1+t)2−2xt−(1+t)√(1+t)2−4xt
2xt .
Remark 3.2. When x= 1 or p=q= 1 we recover two special cases of (3.3) due to
Brändén [3, Eq (5.1)] and Zhuang [29, Theorem 4.2], respectively.
With Lemma 2.8 and (2.6), letting p= 0 (resp. q= 0) in Corollary 3.1, we obtain the
following corollary.
Corollary 3.2. For all positive integers nand each triple statistic
(τ, stat1,stat2)∈{(213,31-2,val),(312,2-13,val)},
we have
P(stat1,des)(Sn(τ); q, t)
=1 + xt
1 + xn−1
P(stat1,stat2,des)Sn(τ); q, (1 + x)2t
(x+t)(1 + xt),x+t
1 + xt,(3.7)
equivalently,
P(stat1,stat2,des)(Sn(τ); q, x, t) = 1 + u
1 + uv n−1
P(stat1,des)(Sn(τ); q, v),(3.8)

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 19
where u=1+t2−2xt−(1−t)√(1+t)2−4xt
2(1−x)tand v=(1+t)2−2xt−(1+t)√(1+t)2−4xt
2xt .
Remark 3.3. When x= 1, (3.7) reduces to [14, Eqs. (1.5) and (1.6)]. When (τ, stat1,stat2) =
(213,31-2,val)and q= 1, (3.7) reduces to [29, Corollary 5.3].
From (2.37) and (3.3) we derive the following result, which is an extension of Shin and
Zeng [22, Theorem 1].
Corollary 3.3. For n≥1,
X
σ∈Sn
qinv σ−exc σtexc σ
=1 + xt
1 + xn−1
P(2-13,31-2,pk−1,des)Sn;q2, q, (1 + x)2t
(x+t)(1 + xt),x+t
1 + xt.
(3.9)
Define the cycle-refinement of the Eulerian polynomial An(t)by
A(cyc∗−fix∗,exc)
n(q, t) := X
σ∈Sn
q(cyc∗−fix∗)σtexc σ,
we obtain a cyclic analogue of Zhuang’s formula [29, Theorem 4.2].
Theorem 3.4. For n≥1, we have
A(cyc∗−fix∗,exc)
n(q, t)
=1 + xt
1 + xn−1
P(cyc∗−fix∗,cpk∗,exc)Sn;q, (1 + x)2t
(x+t)(1 + xt),x+t
1 + xt,(3.10)
equivalently,
P(cyc∗−fix∗,cpk∗,exc)(Sn;q, x, t) = 1 + u
1 + uv n−1
An(q, v),(3.11)
where u=1+t2−2xt−(1−t)√(1+t)2−4xt
2(1−x)tand v=(1+t)2−2xt−(1+t)√(1+t)2−4xt
2xt .
Let p=q= 1 in (3.3) or q= 1 in (3.10), we get the following corollary.
Corollary 3.4. For n≥1, we have
An(t) = 1 + xt
1 + xn−1
P(cpk∗,exc)Sn;(1 + x)2t
(x+t)(1 + xt),x+t
1 + xt,(3.12)
equivalently,
P(cpk∗,exc)(Sn;x, t) = 1 + u
1 + uv n−1
An(v),(3.13)
where u=1+t2−2xt−(1−t)√(1+t)2−4xt
2(1−x)tand v=(1+t)2−2xt−(1+t)√(1+t)2−4xt
2xt .

20 B. HAN, J. MAO, AND J. ZENG
Recall that a permutation σ∈Snis a derangement if it has no fixed points, i.e., σ(i)6=i
for all i∈[n]. Let
D(stat1,stat2)
n(q, t) := X
σ∈Dn
qstat1σtstat2σ,
where Dnis the set of derangements in Sn.
Taking (p, q, tq, r)=(q, 1, t, 0) (resp. (p, q, tq, r)=(q2, q, tq, 0)) in Theorem 3.6 and by
(2.37), we obtain the following corollary.
Corollary 3.5. For all positive integers nand for each statistic stat ∈ {nest,inv},
D(stat,exc)
n(q, t) = 1 + xt
1 + xn
P(stat,cpk,exc)Dn;q, (1 + x)2t
(x+t)(1 + xt),x+t
1 + xt,(3.14)
equivalently,
P(stat,cpk,exc)(Dn;q, x, t) = 1 + u
1 + uv n
D(stat,exc)
n(q, v),(3.15)
where u=1+t2−2xt−(1−t)√(1+t)2−4xt
2(1−x)tand v=(1+t)2−2xt−(1+t)√(1+t)2−4xt
2xt .
By (2.37) and Lemma 2.9, the r= 0 case of (3.22) yields the following result in parallel
with Corollary 3.5, which generalizes Lin’s identity [16, Theorem 1.4].
Corollary 3.6. For n≥1,
P(inv,exc)(Dn(321); q, t)(3.16)
=1 + xt
1 + xn
P(inv,cpk,exc)Dn(321); q, (1 + x)2t
(x+t)(1 + xt),x+t
1 + xt,
equivalently,
P(inv,cpk,exc)(Dn(321); q, x, t) = 1 + u
1 + uv n
P(inv,exc)(Dn(321); q, v),(3.17)
where u=1+t2−2xt−(1−t)√(1+t)2−4xt
2(1−x)tand v=(1+t)2−2xt−(1+t)√(1+t)2−4xt
2xt .
Moreover, we have the the following formula.
Theorem 3.5. For all positive integers n,
D(cyc,exc)
n(q, t) = 1 + xt
1 + xn
P(cyc,cpk,exc)Dn;q, (1 + x)2t
(x+t)(1 + xt),x+t
1 + xt,(3.18)
equivalently,
P(cyc,cpk,exc)(Dn;q, x, t) = 1 + u
1 + uv n
D(cyc,exc)
n(q, v),(3.19)
where u=1+t2−2xt−(1−t)√(1+t)2−4xt
2(1−x)tand v=(1+t)2−2xt−(1+t)√(1+t)2−4xt
2xt .

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 21
Theorem 3.6. For n≥1,
P(nest,cros,exc,fix)(Sn;p, q, tq, r)
=1 + xt
1 + xn
P(nest,cros,cpk,exc,fix)Sn;p, q, (1 + x)2t
(x+t)(1 + xt),q(x+t)
1 + xt ,(1 + x)r
1 + xt ,(3.20)
equivalently,
P(nest,cros,cpk,exc,fix)(Sn;p, q, x, qt, r)
=1 + u
1 + uv n
P(nest,cros,exc,fix)Sn;p, q, qv, (1 + uv)r
1 + u,(3.21)
where u=1+t2−2xt−(1−t)√(1+t)2−4xt
2(1−x)tand v=(1+t)2−2xt−(1+t)√(1+t)2−4xt
2xt .
Remark 3.7. Cooper et al. [7, Theorem 11] have recently proved the p=q= 1 case of
(3.20) by applying Sun and Wang’s CMFS action [26], see (4.1).
Applying Lemma 2.9 and Theorem 3.6 with p= 0, we obtain the following result.
Corollary 3.7. For n≥1,
P(cros,exc,fix)(Sn(321); q, tq, r)
=1 + xt
1 + xn
P(cros,cpk,exc,fix)Sn(321); q, (1 + x)2t
(x+t)(1 + xt),q(x+t)
1 + xt ,(1 + x)r
1 + xt ,(3.22)
equivalently,
P(cros,cpk,exc,fix)(Sn(321); q, x, qt, r)
=1 + u
1 + uv n
P(cros,exc,fix)Sn(321); q, qv, (1 + uv)r
1 + u,(3.23)
where u=1+t2−2xt−(1−t)√(1+t)2−4xt
2(1−x)tand v=(1+t)2−2xt−(1+t)√(1+t)2−4xt
2xt .
Consider the generalized q-Narayana polynomials Nn(t, q, r)defined by
Nn(t, q, r) := X
σ∈Sn(321)
texc σqinv σrfix σ.(3.24)
In particular, we have
Nn(t/q, q, 1) = X
σ∈Sn(321)
texc σqinv σ−exc σ,(3.25)
Nn(t, q, t) = X
σ∈Sn(321)
twex σqinv σ.(3.26)
Fu et al. [14] gave more interpretations of Nn(t/q, q, 1) and Nn(t, q, t)in terms of n-
permutation patterns. We further prove the following interpretations by using the (n−1)-
permutation patterns.

22 B. HAN, J. MAO, AND J. ZENG
Table 1. Five choices of (τ, stat1,stat2,stat3)
#τstat1stat2stat3
1321 exc inv fix
2231 des des + 31-2fmax
3132 asc asc + 2-13 amax
4312 des des + 2-31 amin
5213 asc asc + 13-2fmin
Theorem 3.8. For n≥1, the following identities hold
Nn(t/q, q, 1) = X
σ∈Sn−1(τ)
tstat1σqstat2σ(1 + t)stat3σ,(3.27)
Nn(t, q, t) = tnX
σ∈Sn−1(τ)
(q/t)stat1σqstat2σ(1 + q/t)stat3σ,(3.28)
where five choices for the quadruples (τ , stat1,stat2,stat3)are listed in Table 1.
For 0≤k≤n, define the sets
e
Sn,k(321) = {σ∈Sn(321) : exc σ=k, cda σ= 0},(3.29a)
e
Sn,k(213) = {σ∈Sn(213) : asc σ=k, rda σ= 0},(3.29b)
e
Sn,k(312) = {σ∈Sn(312) : des σ=k, ldd σ= 0},(3.29c)
e
Sn,k(132) = {σ∈Sn(132) : asc σ=k, rda σ= 0},(3.29d)
e
Sn,k(231) = {σ∈Sn(231) : des σ=k, ldd σ= 0},(3.29e)
and e
Sn(τ) = ∪n
k=0 e
Sn,k(τ)for τ∈S3.
Theorem 3.9. For n≥1, the following q-analogue of (1.10)holds
Nn(t/q, q, 1) =
bn−1
2c
X
k=0
γn−1,k(q)tk(1 + t)n−1−2k,(3.30a)
where
γn−1,k(q) = X
π∈
e
Sn−1,k(321)
qinv π(3.30b)
=X
π∈
e
Sn−1,k(231)
q(31-2) π+des π=X
π∈
e
Sn−1,k(312)
q(2-31) π+des π(3.30c)
=X
π∈
e
Sn−1,k(132)
q(2-13) π+asc π=X
π∈
e
Sn−1,k(213)
q(13-2) π+asc π.(3.30d)

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 23
Theorem 3.10. For n≥1, the following q-analogue of (1.10)holds
Nn(t, q, t) =
bn+1
2c
X
k=1 eγn−1,k−1(q)tk(1 + t/q)n+1−2k,(3.31)
where
eγn−1,k−1(q) = X
π∈
e
Sn−1,k(321)
qn−1+inv π−exc π(3.32)
=X
π∈
e
Sn−1,k−1(231)
qn−1+(31-2) π=X
π∈
e
Sn−1,k−1(312)
qn−1+(2-31) π(3.33)
=X
π∈
e
Sn−1,k−1(132)
qn−1+(2-13) π=X
π∈
e
Sn−1,k−1(213)
qn−1+(13-2) π.(3.34)
Remark 3.11. Other interpretations for γn−1,k(q)and eγn−1,k−1(q)are given in [14,16,17].
Let Bnbe the set of permutations σof {±1,...,±n}with σ(−i) = −σ(i)for every
i∈[n]. From Steingrímsson [23, Definition 3], we define the excedance of σ∈ Bnby
i <fσ(i)for i∈[n], in the friends order <fof {±1,...,±n}:
1<f−1<f2<f−2<f··· <fn <f−n,
and denote the number of excedances of σby excB(σ). Following Brenti [4] we say that
i∈ {0,1, . . . , n −1}is a B-descent of σif σ(i)> σ(i+ 1) in the natural order <of
{±1,...,±n}:
−n < ··· <−2<−1<1<2<··· < n,
where σ(0) = 0. Denote the number of B-descents of σby desB(σ). Brenti [4, Theorem 3.4]
considered the Eulerian polynomials of type B
Bn(y, t) := X
σ∈Bn
yneg σtdesBσ(3.35)
and proved the following exponential g.f.
X
n≥0
Bn(y, t)zn
n!=(1 −t)ez(1−t)
1−tez(1−t)(1+y)
=ey(t−1)zS((1 + y)z;t),(3.36)
where S(z;t) := (1−t)ez(1−t)
1−tez(1−t)is the exponential g.f. of type A Eulerian polynomials An(t).
Our main results for the polynomials Bn(y, t)are the following two theorems.
Theorem 3.12. We have
Bn(y, t) = X
σ∈Bn
yneg σtexcBσ.(3.37)

24 B. HAN, J. MAO, AND J. ZENG
Theorem 3.13. For n≥1,
Bn(y, t) = (1 + yt)nP(cpk,exc)Sn;(1 + y)2t
(y+t)(1 + yt),y+t
1 + yt ,(3.38a)
equivalently,
P(cpk,exc)(Sn;y, t) = 1
(1 + uv)nBn(u, v),(3.38b)
where u=1+t2−2yt−(1−t)√(1+t)2−4yt
2(1−y)tand v=(1+t)2−2yt−(1+t)√(1+t)2−4yt
2yt .
4. Proofs using group actions
In this section, using group actions we shall prove Theorem 3.5 and Theorem 3.6, re-
spectively, in the following two subsections.
4.1. Proof of Theorem 3.5.Let σ∈Snwith convention 0–∞. For any x∈[n], the
x-factorization of σreads σ=w1w2xw3w4,where w2(resp. w3) is the maximal contiguous
subword immediately to the left (resp. right) of xwhose letters are all smaller than x.
Following Foata and Strehl [11] we define the action ϕxby
ϕx(σ) = w1w3xw2w4.
Note that if xis a double ascent (resp. double descent), then w3=∅(resp. w2=∅),
and if xis a valley then w2=w3=∅. For instance, if x= 5 and σ= 26471583 ∈S7,
then w1= 2647, w2= 1, w3=∅and w4= 83. Thus ϕ5(σ) = 26475183. Clearly, ϕxis an
involution acting on Snand it is not hard to see that ϕxand ϕycommute for all x, y ∈[n].
Brändén [3] modified the map ϕxto be
ϕ0
x(σ) := (ϕx(σ)if xis not a peak of σ,
σif xis a peak of σ.
It is clear that ϕ0
xis involution and commutes with ϕ0
yfor x6=y. For any subset S⊆[n]
with S={x1, . . . , xr}we then define the map ϕ0
S:Sn→Snby
ϕ0
S(σ) = Y
x∈S
ϕ0
x(σ)
where Qx∈Sϕ0
x=ϕ0
x1◦ ··· ◦ ϕ0
xr. Hence the group Zn
2acts on Snvia the functions ϕ0
S,
S⊆[n]. This action is called the Modified Foata–Strehl action (MFS-action for short).
Recall that a permutation σ∈Sncan be factorized into distinct cycles, say C1, C2,··· , Ck,
where each cycle Ccan be written as a sequence C= (a, σ(a), . . . , σr−1((a)) with σr(a) = a
for some a, r ∈[n]. We say that stan(σ) := C1C2···Ckis the standard cycle representation
of σif
•the largest element of each cycle is at the first position,
•the cycles are arranged in increasing order according to their largest elements.

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 25
We define ι(σ)to be the permutation obtained from stan(σ)by erasing the parentheses
of cycles. For example, for σ= 26471583 ∈D8, then stan(σ) = (6512)(8347) and ι(σ) =
65128347.
In this section, we consider the statistics of ι(σ)with the convention 0–∞.
Lemma 4.1. For σ∈Dn, we have
cval σ=lval ι(σ) = lpk T(σ) = cpk σ, lda ι(σ) = exc σ−cpk σ,
ldd ι(σ) =n−cpk σ−exc σ, lda ι(σ) + ldd ι(σ) = n−2cpk σ.
Proof. The first two identities are easily seen by the definitions of σand ι(σ). For the third
identity,
ldd ι(σ) =n−(lpk ι(σ) + lval ι(σ) + lda ι(σ))
=n−(cpk σ+cval σ+exc σ−cval σ)
=n−cpk σ−exc σ.
With the second and third identities, the fourth identity can be derived directly. 
For σ∈Dn, define the map τc
x:Dn7→ Dnby
τc
x(σ) := ι−1(ϕ0
x(ι(σ))).
It is easy to see that τc
xis an involution and commutes with τc
yfor x, y ∈[n]. Let S⊆[n],
we define τc
S:Dn→Dnby
τc
S(σ) = Y
x∈S
τc
x(σ).(4.1)
Sun and Wang [26] defined the group action of Zn
2on Dnvia the involutions τc
Sover all
S⊆[n]; this group action is called the cyclic modified Foata–Strehl action, abbreviated
CMFS-action, see Figure 5for an illustration. For any permutation σ∈Sn, let Orb(σ) =
{g(σ) : g∈Zn
2}be the orbit of σunder the CMFS-action.
0
6
5
1
2
8
3
4
7
∞
Figure 5. CMFS-actions on (6512)(8347)

26 B. HAN, J. MAO, AND J. ZENG
Remark 4.2. The CMFS-action divides the set Dninto disjoint orbits. Moreover, for
σ∈Dn,xis a double drop (resp. double excedance) of σif and only if xis a double
excedance (resp.double drop) of τc
x(σ). A double drop (resp. double excedance) xof σ
remains a double drop (resp. double excedance) of τc
y(σ)for any y6=x. Hence, there is
a unique permutation in each orbit which has no double excedance. Let ˇσbe this unique
element in Orb(σ), and for any other σ0∈Orb(σ), it can be obtained from ˇσby repeatedly
applying τc
xfor some double drop xof ˇσ. Each time this happens, exc increases by 1and
cdd decreases by 1. Thus by Lemma 4.1, we have
X
σ∈Orb σ
texc σ=texc ˇσ(1 + t)cdd ˇσ=tcpk ˇσ(1 + t)n−2cpk ˇσ.(4.2)
We obtain gamma expansion of derangement polynomials immediately by summing over
all the orbits that form Dn.
We can give a more general version of Theorem 3.5. For any subset Π⊆Snlet
A(exc,cyc)(Π; w, t) := X
σ∈Π
wcyc σtexc σ.
The set Πis invariant under the CMFS-action if τc
S(σ)∈Πfor any σ∈Πand any S⊆[n].
Theorem 4.3. If Π⊆Dnis invariant under the CMFS-action, then
A(cyc,exc)(Π; w, t) = 1 + xt
1 + xn
P(cyc,cpk,exc)Π; w, (1 + x)2t
(x+t)(1 + xt),x+t
1 + xt,(4.3)
equivalently,
P(cyc,cpk,exc)(Π; x, t, w) = 1 + u
1 + uv n+1
A(cyc,exc)(Π; w, v),
where u=1+t2−2xt−(1−t)√(1+t)2−4xt
2(1−x)tand v=(1+t)2−2xt−(1+t)√(1+t)2−4xt
2xt .
First we prove the following identity.
Lemma 4.4. Let σ∈Dn. We have
(1 + x)cda σ+cdd σX
σ0∈Orb(σ)
texc σ0=X
σ0∈Orb(σ)
(1 + xt)cdd σ0(x+t)cda σ0tcval σ0.(4.4)
Proof. Let j=cda σ+cdd σ. By (4.2) the left-hand side of (4.4) is equal to
(1 + x)jtcval ˇσ(1 + t)j=tcval ˇσ(1 + xt +x+t)j.
Let J(σ)be the set of indices of double excedances and double drops of σ, i.e.,
J(σ) := {i∈[n] : σ(i)is a double excedance or double drop}.
Clearly |J(σ)|=j. By (4.1) CMFS-action establishes a bijection from the set of subsets
of J(σ)to Orb(σ)such that if S⊂J(σ)then |S|=cdd σ0with σ0=τc
S(σ). Hence the

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 27
right-hand side of (4.4) is equal to
tcvalˇσX
S⊂J(σ)
(1 + xt)|S|(x+t)j−|S|.
Eq. (4.4) follows then from PS⊂[j](1 + xt)|S|(x+t)j−|S|= (1 + xt +x+t)j.
Proof of Theorem 4.3.With Lemma 4.1 and Eq. (4.4), we have
X
σ0∈Orb(σ)
texc σ0(1 + x)n−2cpk σ0=X
σ0∈Orb(σ)
(1 + xt)n−exc σ0−cpk σ0(x+t)exc σ0−cpk σ0tcpk σ0,
which is equivalent to
X
σ0∈Orb(σ)
texc σ0=X
σ0∈Orb(σ)
(1 + xt)n−exc σ0−cpk σ0(x+t)exc σ0−cpk σ0tcpk σ0
(1 + x)n−2cpk σ0.
Then, summing over all the orbits leads to
X
σ∈Π
texc σ=X
σ∈Π
(1 + xt)n−exc σ−cpk σ(x+t)exc σ−cpk σtcpk σ
(1 + x)n−2cpk σ.
For σ0∈Orb(σ), first we have cyc(σ0) = cyc(σ). From the definition of o(σ), we have
cyc(σ)is equal to the number of left-to-right maximum of o(σ). It is easy to see that the
number of left-to-right maximum is invariant under MFS-action. Thus the number cyc(σ0)
is invariant for any σ0∈Orb(σ). Therefore,
X
σ∈Π
texc σwcyc σ=X
σ∈Π
(1 + xt)n−exc σ−cpk σ(x+t)exc σ−cpk σtcpk σ
(1 + x)n−2cpk σwcyc σ
=1 + xt
1 + xn
P(cpk,exc,cyc)Π; (1 + x)2t
(x+t)(1 + xt),x+t
1 + xt, w.
The proof is completed. 
Remark 4.5. Recently, using the joint distribution of the cyclic valley number and ex-
cedance number statistics Cooper, Jones and Zhuang [7] have generalized the formula of
Stembridge by applying Sun and Wang’s CMFS-action. In particular they also obtained
the w= 1 case of Theorem 4.3.
4.2. Proof of Theorem 3.6.For our purpose we define a 3-Motzkin path of length nas
a word s:= s1. . . snon the alphabet {U,D,Ly,Lb,Lr}such that |s1. . . sn|U= [s1. . . sn|D
and the height of the ith step is nonnegative, i.e.,
hi(s) := |s1. . . si|U−[s1. . . si|D≥0 (i= 1, . . . , n),(4.5)
where |s1. . . si|Uis the number of letters Uin the word s1. . . si. Let
α(s) := {i∈[n] : si=α}for α∈ {U,D,Ly,Lb,Lr}.
Avariant restricted Laguerre history of length nis a pair (s,p), where sis a 3-Motzkin
path s1. . . snand p= (p1, . . . , pn)with 0≤pi≤hi−1(s)if si=U,0≤pi≤hi−1(s)−1

28 B. HAN, J. MAO, AND J. ZENG
if si=D,Lb,Lrand pi=hi−1if si=Lywith h0(s) = 0. Let LH0
nbe the set of variant
restricted Laguerre histories of length n.
We use a variant of Foata-Zeilberger’s bijection φF Z :Sn→ LH0
n(cf. (2.40)). Given
σ∈Sn, we construct the variant restricted Laguerre history φ0
F Z (σ) := (s,p)∈ LH0
nas
follows. For i= 1, . . . , n, let
si=












Uif i∈Cval σ,
Dif i∈Cpk σ,
Lrif i∈Cdd σ,
Lbif i∈Cda σ,
Lyif i∈Fix σ,
(4.6)
with pi=nestiσ.
Lemma 4.6. If φ0
F Z (σ) = (s,p)∈ LH0
nwith σ∈Sn, then
Fix σ=Ly(s),(4.7a)
Exc σ=Lb(s)∪U(s),(4.7b)
nest σ=
n
X
i=1
pi,(4.7c)
exc σ+cros σ+nest σ=
n
X
i=1
hi−1(s),(4.7d)
where Exc σdenotes the set of excedances of σ.
Proof. From the construction of φ0
F Z , it is easy to see (4.7a)-(4.7c). Define
exciσ=1if σ(i)> i,
0if σ(i)≤i.
By inductions on i∈[n]we verify that
exciσ+nestiσ+crosiσ=










hi−1(s)+1,if si=U,
hi−1(s)if si=Lr,
hi−1(s)−1if si=D,
hi−1(s)if si=Lb,
hi−1(s)if si=Ly.
(4.8)
This implies (4.7d) immediately. 
We define a Zn
2-action on LH0
n, which is similar to Yan-Zhou-Lin’s group action on
LHnin [27] and a generalization of Lin’s group action on 2-M∗
nin [16]. Let i∈[n]and
(s,p)∈ LH0
n. Define the group action θias follows,
θi((s,p)) = (s,p)if i∈Ly(s),
(s0,p)otherwise,

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 29
where s0is the 3-Motzkin path obtained from sby changing sias Lb↔Lr. For any subset
S⊆[n]define the mapping θ0
S:LH0
n→ LH0
nby
θ0
S((s,p)) = Y
i∈S
θi((s,p)).(4.9)
Hence the group Zn
2acts on LH0
nvia the function θ0
S. Note that the three sequences Ly(s),
pand (h0(s),...,(hn−1(s)) are invariant under the group action. This action divides the set
LH0
ninto disjoint orbits and each orbit has a unique restricted Laguerre history whose level
steps are 6=Lyor Lb. For any fixed (s,p)∈ LH0
nlet Orb((s,p)) := {θ0
S((s,p)) |S⊆[n]}.
For 0≤j≤nwe define
Sn,j ={σ∈Sn:fixσ=j},(4.10a)
Rn,j ={(s,p)∈ LH0
n:|s|Ly=j}.(4.10b)
where |s|ameans the number of letters ain the word s.
Lemma 4.7. Let (s,p)∈ Rn,j. We have
(1 + x)|s|Lb+|s|LrX
(s0,p)∈Orb(s,p)
t|s0|Lb
=X
(s0,p)∈Orb(s,p)
(1 + xt)|s0|Lr(x+t)|s0|Lb.(4.11)
Proof. Let L(s) = {i∈[n] : si=Lbor si=Lr}with cardinality `=|s|Lr+|s|Lb. By (4.9),
the group action establishes a bijection from the set of subsets of L(s)to Orb(s,p)such
that if S⊂L(s)then S=Lr(s0)with (s0,p) = θ0
S((s,p)). Eq. (4.11) is equivalent to
(1 + x)`X
S⊂L(s)
t|S|=X
S⊂L(s)
(1 + xt)|S|(x+t)`−|S|,
namely, (1 + x)`·(1 + t)`= (1 + xt +x+t)`.
Proof of Theorem 3.6.For (s,p)∈ Rn,j , since 2|s|U+|s|Lr+|s|Lb+j=n, by (4.11), we
have
(1 + x)n−j−2|s0|UX
(s0,p)∈Orb(s,p)
t|s0|Lb(4.12)
=X
(s0,p)∈Orb(s,p)
(1 + xt)n−j−2|s0|U−|s0|Lb(x+t)|s0|Lb,
that is,
X
(s0,p)∈Orb(s,p)
t|s0|Lb=X
(s0,p)∈Orb(s,p)
(1 + xt)n−j−2|s0|U−|s0|Lb(x+t)|s0|Lb
(1 + x)n−j−2|s0|U.
Summing over all the orbits leads to

30 B. HAN, J. MAO, AND J. ZENG
X
(s,p)∈Rn,j
t|s|Lb=X
(s,p)∈Rn,j
(1 + xt)n−j−2|s|U−|s|Lb(x+t)|s|Lb
(1 + x)n−j−2|s|U.
Let
|p|=
n
X
i=1
piand h(s) =
n
X
i=1
hi−1(s).
Since U(s),pand (h0(s), . . . , hn−1(s)) are invariant under the group action,
X
(s,p)∈Rn,j p|p|qh(s)−|p|t|s|Lb+|s|U
=X
(s,p)∈Rn,j p|p|qh(s)−|p|(1 + xt)n−j−2|s|U−|s|Lb(x+t)|s|Lbt|s|U
(1 + x)n−j−2|s|U.(4.13)
By Lemma 4.6, as the bijection φ0
F Z maps Sn,j to Rn,j with corresponding statistics, we
can rewrite (4.13) as
X
σ∈Sn,j pnest σqcros σ+exc σtexc σ(4.14)
=X
σ∈Sn,j pnest σqcros σ+exc σ(1 + xt)n−j−exc σ−cpk σ(x+t)exc σ−cpk σtcpk σ
(1 + x)n−j−2cpk σ
=1 + xt
1 + xn−j
P(nest,cros,cpk,exc)Sn,j;p, q, (1 + x)2t
(x+t)(1 + xt),q(x+t)
1 + xt .
Multiplying (4.14) by rjand summing over jyields (3.20). By using the substitution
u=(1+x2)t
(x+t)(1+xt)and v=x+t
1+xt as in (3.20), we obtain (3.21) immediately. 
Remark 4.8. We show that Eq. (4.13) implies also two other known results in the litter-
ature. When x= 1 Eq.(4.13) reduces to
X
(s,p)∈Rn,j p|p|qh(s)−|p|t|s|Lb+|s|U=X
(s,p)∈Rn,j p|p|qh(s)−|p|(1 + t)n−j−2|s|Ut|s|U
2n−j−2|s|U.(4.15)
Let
On,k,j ={(s,p)∈ Rn,j :|s|Lb= 0 and |s|U=k}.
By the group action on Rn,j, we see that there are 2n−j−2|s|Uelements in each orbit, and
then
2n−2k−j|On,k,j |=|{(s,p)∈ Rn,j :|s|U=k}|.

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 31
Hence (4.15) is equivalent to
X
(s,p)∈Rn,j p|p|qh(s)−|p|t|s|Lr+|s|U=
n
X
k=0 X
(s,p)∈On,j,kp|p|qh(s)−|p|(1 + t)n−j−2ktk.(4.16)
Thanks to the bijection φ0
F Z and (4.7a)–(4.7d) we obtain Theorem 8 in [21],
X
π∈Sn
(tq)exc πpnest πqcros πrfix π
=
n
X
j=0
rj
b(n−j)/2c
X
k=0 X
σ∈Sn,k,j
pnest σqcros σ+exc σtk(1 + t)n−j−2k,(4.17)
where
Sn,k,j ={σ∈Sn,cpk σ=k, fix σ=j, cda σ= 0}.
By Lemma 2.9 and (2.37), letting p= 0 in (4.17) yields Theorem 2.4 in [16],
X
σ∈Sn(321)
texc σqinv σrfix σ
=
n
X
j=0
rj
b(n−j)/2c
X
k=0 
X
σ∈Sn,k,j (321)
qinv σ
tk(1 + t)n−j−2k,(4.18)
where
Sn,k,j (321) := {σ∈Sn(321) : fix σ=j, exc σ=k, cda σ= 0}.
Note that when r= 1 + t, Eq. (4.18) reduces to Eq. (3.30a) with the γ-coefficients in
(3.30b)–(3.30d).
5. Proofs via continued fractions
For convenience, we use the following compact notation for the J-type continued fraction
J[z;bn, λn] = 1
1−b0z−λ1z2
1−b1z−λ2z2
1−b2z−λ3z2
1− · ··
.(5.1)
We shall use the notation [n]p,q := (pn−qn)/(p−q)for n∈N.

32 B. HAN, J. MAO, AND J. ZENG
5.1. Some combinatorial continued fractions. We first recall a standard contraction
formula for continued fractions, see [20, Eq. (44)].
Lemma 5.1 (Contraction formula).The following contraction formulae hold
1
1−α1z
1−α2z
1−α3z
1−α4z
1− · ··
=1
1−α1z−α1α2z2
1−(α2+α3)z−α3α4z2
1− · ··
= 1 + α1z
1−(α1+α2)z−α2α3z2
1−(α3+α4)z−α4α5z2
1− · ··
.
The following four combinatorial continued fraction expansions are proved by Shin and
Zeng [21]. Let
An(p, q, t, u, v, w) := X
σ∈Sn
pnest σqcros σtexc σucdd∗σvcda∗+fix∗σwcpk∗σ(5.2a)
=X
σ∈Sn
p(2-13) σq(31-2) σtdes σuda σvdd σwpk σ−1,(5.2b)
where the equality of the two enumerative polynomials follows from Lemma (2.2).
Lemma 5.2. [21, Eq. (28)] We have
X
n≥0
An+1(p, q, t, u, v, w)zn=J[z;bn, λn](5.3a)
with
bn= (u+tv)[n+ 1]p,q , λn= [n]p,q [n+ 1]p,q .(5.3b)
Let
Bn(p, q, t, u, v, w, y) := X
σ∈Sn
pnest σqcros σtexc σucdd σvcda σwcval σyfix σ.(5.4)
Lemma 5.3. [21, Eq. (34)] We have
1 +
∞
X
n=1
Bn(p, q, t, u, v, w, y)zn=J[z;bn, λn],(5.5a)
with
bn=ypn+ (qu +tv)[n]p,q , λn=tw[n]2
p,q.(5.5b)

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 33
Let
Cn(q, t, u, v, w) := X
σ∈Sn
qcyc∗σ−fix∗σtwex∗σucda∗+fix∗σvcdd∗σwcval∗σ.(5.6)
Lemma 5.4. [21, Eq. (50)] We have
∞
X
n≥0
Cn+1(q, t, u, v, w)zn=J[z;bn, λn](5.7a)
with
bn= (n+ 1)(tu +v), λn=n(q+n)tw. (5.7b)
Let
Dn(q, t, u, v, w) := X
σ∈Dn
qcyc σtexc σucda σvcdd σwcval σ.(5.8)
Lemma 5.5. [21, Eq. (41)] We have
1 +
∞
X
n=1
Dn(q, t, u, v, w)zn=J[z;bn, λn](5.9a)
with
bn=n(tu +v), λn=n(q+n−1)tw. (5.9b)
5.2. Proof of Theorems 3.1,3.4–3.6,3.8–3.10.In the previous section Theorems 3.1
and 3.7 are proved using group actions. Here we shall give an alternative proof for Theo-
rems 3.1 and 3.7 using continued fractions.
Proof of Theorem 3.1.In view of (5.3a), we have
An(p, q, t, 1,1, x) = X
σ∈Sn
pnest σqcros σtexc σxcpk∗σ.
It follows that
∞
X
n=0
An+1(p, q, t, 1,1, x)zn=J[z;bn, λn](5.10a)
with
bn= [n+ 1]p,q(t+ 1), λn= [n]p,q[n+ 1]p,q tx. (5.10b)
When x= 1 we have the J-fraction for P∞
n=0 An+1(p, q, t)zn. The g.f. of the right side of
Eq. (3.3) is
X
n≥0
P(nest,cros,cpk∗,exc)Sn+1;p, q, (1 + x)2t
(x+t)(1 + xt),x+t
1 + xt(1 + xt)
1 + xn
zn.

34 B. HAN, J. MAO, AND J. ZENG
Substituting t, x and z, respectively, by
x+t
1 + xt,(1 + x)2t
(x+t)(1 + xt)and (1 + xt)z
1 + x
in (5.10a), we obtain the J-fraction of P∞
n=0 An+1(p, q, t)zn.
Proof of Theorem 3.4.By Eq. (5.6), the g.f. of the left side of Eq. (3.10) is
∞
X
n≥0
A(cyc∗−fix∗,exc)
n+1 (q, t)zn=J[z;bn, λn](5.11a)
with
bn= (n+ 1)(t+ 1), λn=n(q+n).(5.11b)
By definition, the g.f. of the right side of Eq. (3.10) is
X
n≥0
P(cyc∗−fix∗,cval∗,exc)Sn+1;q, (1 + x)2t
(x+t)(1 + xt),x+t
1 + xt(1 + xt)
1 + xn
zn.
Letting u=v= 1 and transforming x+t
1+xt ,(1+x)2t
(x+t)(1+xt)and z(1+xt)
(1+x)to t,xand zin (5.7a),
respectively, we obtain (5.11a) immediately. 
Proof of Theorem 3.5.Letting u=v=w= 1 in Eq. (5.9a), we see that the g.f. of the left
side of Eq. (3.18) is
1 +
∞
X
n=1
D(cyc,exc)
n(q, t)zn=J[z;bn, λn](5.12a)
with
bn=n(t+ 1), λn=n(q+n−1)t. (5.12b)
On the other hand, the g.f. of the right side of Eq. (3.18) is
X
n≥0
P(cyc,cpk,exc)Dn;q, (1 + w)2t
(w+t)(1 + wt),w+t
1 + wt (1 + wt)
1 + wn
zn.
Letting u=v= 1 in (5.9a) and substituting t,wand zwith
w+t
1 + wt ,(1 + w)2t
(w+t)(1 + wt)and (1 + wt)z
(1 + w),
respectively, we obtain the J-fraction in (5.12a) immediately. 
Proof of Theorem 3.6.We prove that both sides of Eq. (3.20) have the same g.f. by
comparing their continued fraction expansions. By (5.4), we have
Bn(p, q, qt, 1,1,1, r) = P(nest,cros,exc,fix)(Sn;p, q, tq, r).

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 35
It follows from Eq. (5.5a) that
X
n≥0
P(nest,cros,exc,fix)(Sn;p, q, tq, r)zn=J[z;bn, λn](5.13a)
with
bn=rpn+q(1 + t)[n]p,q , λn=tq[n]2
p,q.(5.13b)
On the other hand, by definition and invoking the equality cpk =cval (cf. (2.3)), the g.f.
of the right-hand side of Eq. (3.20) is
X
n≥01 + xt
1 + xn
P(nest,icr,cpk,exc,fix)Sn;p, q, (1 + x)2t
(x+t)(1 + xt),q(x+t)
1 + xt ,(1 + x)r
1 + xt zn.(5.14)
In Eq. (5.5a) letting u=v= 1 and making the substitution
t←q(x+t)
1 + xt , w ←(1 + x)2t
(x+t)(1 + xt), y ←(1 + x)r
1 + xt , z ←(1 + xt)z
(1 + x),
we see that the g.f. (5.14) has the same J-fraction expansion as (5.13a). 
Proof of Theorem 3.8.Recall [5, Theorem 7.2] that
∞
X
n=0
Nn(t, q, r)zn=1
1−rz −tqz2
1−(1 + t)qz −tq3z2
1−(1 + t)q2z−tq5z2
...
.(5.15)
Writing P∞
n=1 Nn−1(t, q, r)zn=zP∞
n=0 Nn(t, q, r)znwe have
1 + X
n≥1
Nn−1(t, q, 1 + t)zn= 1 + z
1−(1 + t)z−tqz2
1−q(t+ 1)z−tq3z2
1− · ··
,(5.16)
which is Pn≥0Nn(t/q, q, 1)znby applying Lemma 5.1. Thus
Nn(t/q, q, 1) = Nn−1(t, q, 1 + t).(5.17)
By (2.5) and and Lemma 2.9, we see that π∈Sn(321) if and only if π−1∈Sn(321). As
wex π−1=n−drop π−1=n−excπand inv π−1=inv πwe have
Nn(t, q, t) = X
π∈Sn(321)
twex π−1qinv π−1
=tnX
π∈Sn(321)
(1/t)exc πqinv π=tnNn(1/t, q, 1).

36 B. HAN, J. MAO, AND J. ZENG
It follows from (5.17) that
Nn(t, q, t) = tnNn−1(q/t, q, 1 + q/t).(5.18)
In view of (3.24) identities (5.17) and (5.18) provide the first interpretation in Table 1.
Other interpretations in Table 1 can be derived from the equidistribution results in (2.7a)
and (2.11a)–(2.11d)

Proof of Theorem 3.9.By Lemma 2.9, (2.37) and (5.4), we have
Bn(0, q, tq, 1,0,1,1) = X
σ∈
e
Sn(321)
qinv σtexc σ.(5.19)
It follows from Lemma 5.3 that
∞
X
n=0 X
σ∈
e
Sn(321)
qinv σtexc σzn=1
1−z−tqz2
1−qz −tq3z2
1−q2z−tq5z2
...
.(5.20)
Now, the g.f. of the right-hand side of Eq. (3.30a) is
GF : = 1 + zX
n≥1
bn−1
2c
X
k=0 
X
σ∈
e
Sn−1,k(321)
qinv σ
tk(1 + t)n−1−2kzn−1
=1 + zX
n≥0X
σ∈
e
Sn(321)
qinv σt
(1 + t)2exc σ
((1 + t)z)n.
By (5.20) and Lemma 5.1 we see that
GF = 1 + z
1−(1 + t)z−tqz2
1−(1 + t)qz −tq3z2
1−(1 + t)q2z−tq5z2
...
=1
1−z−tz2
1−(q+t)z−tq2z2
1−(q+t)qz −tq4z2
···
,
which is equal to P
n≥0
Nn(t/q, q, 1)znby (5.15). Other interpretations can be obtained by
the equidistribution results of (2.7a) and (2.11a)–(2.11d). 

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 37
Proof of Theorem 3.10.The g.f. of the right side of (3.31) can be written as
1 + zX
n≥1
bn+1
2c
X
k=1 
X
σ∈
e
Sn−1,k−1(321)
qn−1+inv σ−exc σ
tk(1 + t/q)n+1−2kzn−1
=1 + zt X
n≥0
X
σ∈
e
Sn(321)
qinv σ−exc σtq2
(q+t)2exc σ
((q+t)z)n.(5.21)
By using the second claim of Lemma 2.9, Eq.(2.37) and (5.4), we have
Bn(0, q, t, 1,0,1,1) = X
σ∈
e
Sn(321)
qinv σ−exc σtexc σ,
Lemma 5.3 implies that
∞
X
n=0 X
σ∈
e
Sn(321)
qinv σ−exc σtexc σzn=1
1−z−tz2
1−qz −tq2z2
1−q2z−tq4z2
...
.
Making the substitution z7→ (q+t)zand t7→ tq2/(q+t)2in the above equation and
applying the contarction formulae, we obtain
1 + zt
1−(q+t)z−tq2z2
1−(q+t)qz −tq4z2
1−(q+t)q2z−tq6z2
...
(5.22)
=1
1−tz −tqz2
1−(1 + t)qz −tq3z2
1−(1 + t)q2z−tq5z2
···
,(5.23)
which is equal to P
n≥0
Nn(t, q, t)znby (5.15). Other interpretations can be obtained by the
equidistribution results of (2.7a) and (2.11a)–(2.11d). 
5.3. Proof of Theorems 3.12 and 3.13.Recall the color order <cof {±1,...,±n}:
−1<c−2<c··· <c−n <c1<c2<c··· <cn,

38 B. HAN, J. MAO, AND J. ZENG
and define the following statistics:
fix σ= #{i∈[n] : i=σ(i)},
excAσ= #{i∈[n] : i <cσ(i)},
wexAσ= #{i∈[n] : i≤cσ(i)}=excAσ+fixσ,
wexCσ= #{i∈[n] : i≤ |σ(i)|and σ(i)<0},
neg σ= #{i∈[n] : σi<0}.
Let
Fn(q, t, w, r, y) = X
σ∈Bn
qcros σtwexAσwwexCσrfix σyneg σ.
The following result is the r= 2 case of [22, Lemma 16].
Lemma 5.6. We have X
n≥0
Fn(q, t, w, r, y)zn=J[z;bn, λn],(5.24)
with
λn= (t+wyqn−1)(1 + yqn)[n]2
q,
bn= (1 + yqn)[n]q+t(r+q[n]q) + wyqn[n+ 1]q.
We need the following lemma, see [28, Lemma 12] and [15, p. 307].
Lemma 5.7. If two sequences {µn}nand {νn}nsatisfy the equation
X
n≥0
µn
zn
n!=eαz X
n≥0
νn
zn
n!,
then X
n≥0
νnzn=J[z;bn, λn] =⇒X
n≥0
µnzn=J[z;bn+α, λn].
Proof of Theorem 3.12.Since exc = (wexA−fix) + wexC, see [22, Eq. (4.5)], we have
Fn(1, t, t, 1/t, y) = X
σ∈Bn
texcBσyneg σ
and formula (5.24) becomes
X
n≥0 X
σ∈Bn
texcBσyneg σ!zn=J[z;bn, λn],(5.25)
where bn= (n+ 1)(1 + yt) + n(t+y)and λn=n2(1 + y)2t. By Lemma 5.7 we derive
from (1.6b) and (3.36) that Pn≥0Bn(y, t)znhas the same continued fraction expansion in
(5.25). 

EULERIAN POLYNOMIALS AND EXCEDANCE STATISTICS 39
Proof of Theorem 3.13.By (5.4), we have
P(cpk,exc)(Sn;w, t) = Bn(1,1, t, 1,1, w, 1).
Specializing (p, q, t, u, v, w, y)in Lemma 5.3 yields
∞
X
n=0
P(cpk,exc)(Sn;w, t)zn=J[z;bn, λn],
where bn= 1 + n(1 + t)and λn=n2tw. It follows that the series
X
n≥0
P(cpk,exc)Sn;(1 + y)2t
(y+t)(1 + yt),y+t
1 + yt ((1 + yt)z)n
has the same continued fraction expansion for Pn≥0Bn(y, t)znin (5.25). 
Acknowledgement
The first two named authors were supported by the China Scholarship Council. The
second author’s work was done during his visit at Institut Camille Jordan, Université
Claude Bernard Lyon 1 in 2018-2019. The first author was supported by the Israel Science
Foundation (grant no. 1970/18) during the revision of this work.
An extended abstract of this paper will appear in the Proceedings of the 32nd Conference
on Formal Power Series and Algebraic Combinatorics (FPSAC’20).
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(Bin Han) Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut
Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France
Email address:han@math.univ-lyon1.fr, han.combin@hotmail.com.
(Bin Han) Current address: Department of Mathematics, Bar-Ilan University, Ramat-Gan
52900, Israel.
Email address:han@math.biu.ac.il, han.combin@hotmail.com.
(Jianxi Mao) School of Mathematic Sciences, Dalian University of Technology, Dalian
116024, P. R. China
Email address:maojianxi@hotmail.com
(Jiang Zeng) Univ Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut
Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France
Email address:zeng@math.univ-lyon1.fr