Generalized Eulerian Polynomials with a Nonnegative Gamma Vector

Pasquale Petrullo and Domenico Senato Università degli Studi della Basilicata Dipartimento di Scienze Umane Via Nazario Sauro 85 85100 Potenza Italy

              pasquale.petrullo@unibas.it
               domenico.senato@unibas.it

Abstract

We define a family of generalized Eulerian polynomials depending on three parameters. We prove that these polynomials have a nonnegative gamma vector, and we provide a combinatorial description of the corresponding gamma coefficients. By assigning suitable integer values to the parameters, we obtain a new expansion of the nth Eulerian polynomial over the symmetric group 𝔖_{n−1}, a new description of the associated gamma vector, and an identity relating the derangements of 𝔖_{2n} to the alternating permutations of 𝔖_{2n+1}.

1 Introduction

The Eulerian polynomial A_{n}(x) counts the permutations of the symmetric group 𝔖_{n} by number of excedances. More precisely, we have

A_{n}(x) = ∑_{w∈𝔖_{𝔫}} x^{exc(w)},  (1)

where

exc(w) := |{i | 1 ≤ i ≤ n, w(i) > i}|.

As A_{n}(x) is a palindromic polynomial of degree n − 1, namely

A_{n}(x) = x^{n−1} A_{n}(1/x),

we may consider its expansion in terms of the so-called gamma-basis:

A_{n}(x) = ∑_{k=0}^{⌊ \frac{n−1}{2} ⌋} γ_{n,k}x^{k}(x + 1)^{n−1−2k}.  (2)

The array (γ_{n,k})_{k} is known as the gamma vector associated with A_{n}(x). The coefficients γ_{n,k} are known to be nonnegative integers, and that is why the polynomial A_{n}(x) is said gamma-nonnegative (or gamma-positive).

The idea behind gamma-nonnegativity is contained in the pioneering paper of Foata and Schützenberger [8]. An explicit interpretation of γ_{n,k} in terms of permutation enumeration can be given by means of “valley-hopping” [12], which is a technique based on the results of Foata and Strehl [9]. Gamma-nonnegativity implies unimodality, which in turn has been the subject of a considerable amount of research in the recent past [5,6,16]. The literature around this topics is still growing, and provides interesting variants, refinements, and generalizations of Eulerian polynomials. See, for example, the beautiful survey by Athanasiadis [3], and the book of Petersen [12].

More recently, Petrullo [13] provided a characterization and a combinatorial description of gamma-nonnegative Sheffer sequences, obtaining a direct connection with classical orthogonal polynomials. Moreover, Agapito et al. [1,2] introduced a new family of generalized Eulerian polynomials through the action of the Weyl algebra on formal power series. Then, as sometimes happens in the practice of mathematical investigation, in an attempt to prove that some of these polynomials are gamma-nonnegative, we came to define a new family of polynomials with nonnegative gamma vector.

In this paper we show that some of the most remarkable Eulerian-like polynomials can be obtained by assigning suitable integer values to the parameters q, t, y in the polynomials P_{n}(q, t; x, y) defined by

∑_{n≥0} P_{n}(q, t; x, y)\frac{z^{n}}{n!} = (\frac{x − y}{xe^{yz} − ye^{xz}} )^{q}e^{(x+y)qtz}.

As P_{n}(q, t; x, y) is symmetric and homogeneous of degree n in x, y, by the fundamental theorem of symmetric functions, there exist uniquely determined coefficients γ_{n,k}(q, t) such that

P_{n}(q, t; x, y) = ∑_{k=0}^{⌊ \frac{n}{2} ⌋} γ_{n,k}(q, t) x^{k}y^{k} (x + y)^{n−2k}.

So, P_{n}(q, t; x) := P_{n}(q, t; x, 1) is a palindromic polynomial in x, with gamma coefficients γ_{n,k}(q, t). To achieve a combinatorial description of P_{n}(q, t; x), we define

C_{n}(x) := ∑_{w∈ℭ_{n}} x^{exc(w)},

where ℭ_{n} is the set of all n-cycles in 𝔖_{n}. Then, from C_{n}(x) = x A_{n−1}(x) we recover the generating function (Proposition 1)

∑_{n≥1} C_{n}(x)\frac{z^{n}}{n!} := z + log(\frac{x − 1}{xe^{z} − e^{xz}} ),

so that we may write

∑_{n≥0} P_{n}(q, t; x, y)\frac{z^{n}}{n!} = e^{qtz+q ∑_{n≥2} C_{n}(x) \frac{z^{n}}{n!}} = e^{qtz+q ∑_{n≥2} x A_{n−1}(x) \frac{z^{n}}{n!}} .

Given this, the enumerative properties of P_{n}(q, t; x) (Theorem 2), as well as its gammanonnegative expansion (Theorem 4), plainly follow from the classical results on Eulerian polynomials by means of the exponential formula. Moreover, by generalizing some results on derangement polynomials [3, 15], we obtain a combinatorial interpretation of γ_{n,k}(q, t). Finally, starting from the curious special case A_{n+1}(x) = P_{n}(2, 1/2; x, 1), we establish a new expansion of A_{n}(x) over the symmetric group 𝔖_{n−1}, a new description of the associated gamma vector, and a new identity relating the derangements of 𝔖_{2n} to the alternating permutations of 𝔖_{2n+1} (Example 5.4).

2 The generalized Eulerian polynomials

Let q, t, x, y, z denote indeterminates over the field ℚ of rational numbers, set

P(q, t; x, y; z) := (\frac{x − y}{xe^{yz} − ye^{xz}} )^{q}e^{(x+y)qtz},  (3)

and consider the polynomials P_{n}(q, t; x, y) that occur in the following formal power series expansion of P(q, t; x, y; z):

P(q, t; x, y; z) = ∑_{n≥0} P_{n}(q, t; x, y)\frac{z^{n}}{n!} .  (4)

Note that, if α is any further independent indeterminate, then we have

P(q, t; αx, αy; z) = P(q, t; x, y; αz) and P(q, t; x, y; z) = P(q, t; y, x; z).

The identities above say us that P_{n}(q, t; x, y) is a degree n homogeneous symmetric polynomial in x, y. In particular, this means that P_{n}(q, t; x) := P_{n}(q, t; x, 1) is palindromic of degree n in x, namely we have

x^{n} P_{n}(q, t; 1/x) = P_{n}(q, t; x).

Recall that the Eulerian polynomials (1) have the following exponential generating function:

A(x; z) := 1 + ∑_{n≥1} A_{n}(x)\frac{z^{n}}{n!} = \frac{1 − x}{e^{(x−1)z} − x} = \frac{x − 1}{xe^{z} − e^{xz}} e^{z}.  (5)

So, comparing with (3), we have

A_{n}(x) = P_{n}(1, 1/(x + 1); x).  (6)

Now, for all S ⊆ {1, 2, …, n}, we let ℭ(S) denote the set of all w ∈ 𝔖_{n} that cyclically permute the elements of S, and that fix every i ∉ S. Then, we define

C_{S}(x) := ∑_{w∈ℭ(S)} x^{exc(w)}.  (7)

As C_{S}(x) depends only on |S|, we will often write C_{n}(x) instead of C_{S}(x) whenever |S| = n. The exponential generating function of the polynomials C_{n}(x) is given below.

Proposition 1. We have

C(x; z) := ∑_{n≥1} C_{n}(x)\frac{z^{n}}{n!} = z + ln(\frac{x − 1}{xe^{z} − e^{xz}} ).  (8)

Proof. Recall that every permutation w ∈ 𝔖_{n} induces a partition π = {S_{1}, S_{2}, …, S_{k}} of the set {1, 2, …, n} as follows: set a, b ∈ S_{i} if and only if b = σ^{j}(a) for a suitable j. We write c_{1}, c_{2}, …, c_{k} to denote the disjoint cycles (1-cycles included) of w, and we assume that c_{i} ∈ ℭ(S_{i}) for 1 ≤ i ≤ k. Because exc(w) = exc(c_{1}) + exc(c_{2}) + ⋯ + exc(c_{k}), we may write

A_{n}(x) = ∑_{w∈𝔖_{n}} x^{exc(w)} = ∑_{π∈Π_{n}} ∏_{S∈π} C_{S}(x),

where Π_{n} is the set of all partitions of {1, 2, …, n}. Now, the exponential formula ensures us that

A(x; z) = e^{C(x;z)},

and (8) comes directly from (5).

Identity (8) is our starting point to develop a combinatorial description of the polynomial P_{n}(q, t; x, y). To this aim, for every w ∈ 𝔖_{n}, we let fix(w) denote the number of fixed points of w. Moreover, we let cyc(w) denote the number of disjoint cycles of w.

Theorem 2. We have

P_{n}(q, t; x, y) = ∑_{w∈𝔖_{n}} q^{cyc(w)}t^{fix(w)}x^{exc(w)}y^{exc(w^{−1})}(x + y)^{fix(w)}.  (9)

Proof. From (3), (4), and (8), it follows that

e^{q[C(x;z)−(1−t)z]} = (\frac{x − 1}{xe^{z} − e^{xz}} )^{q}e^{qtz} = 1 + ∑_{n≥1} P_{n}(q, \frac{t}{x + 1}; x, 1)\frac{z^{n}}{n!} .

On the other hand, again by the exponential formula, we recover

P_{n}(q, t/(x + 1); x) = ∑_{π∈Π_{n}} q^{|π|}t^{sing(π)} ∏_{S∈π} C_{S}(x),  (10)

where |π| is the number of blocks of π, and sing(π) is the number of singletons of π. Now, as we have |π| = cyc(w) and sing(π) = fix(w) for every w that induces π, by virtue of (7) we get

∑_{π∈Π_{n}} q^{|π|}t^{sing(π)} ∏_{S∈π} C_{S}(x) = ∑_{w∈𝔖_{n}} q^{cyc(w)}t^{fix(w)}x^{exc(w)}.  (11)

Finally, since P_{n}(q, t; x, y) is homogeneous of degree n in x, y, from (10) and (11) we conclude

P_{n}(q, t; x, y) = y^{n}P_{n}(q, t; x/y) =

= ∑_{w∈𝔖_{n}} q^{cyc(w)}t^{fix(w)}x^{exc(w)}y^{n−exc(w)−fix(w)}(x + y)^{fix(w)}.

At this point, (9) is obtained by observing that exc(w) + fix(w) + exc(w^{−1}) = n.

Remark 3. A slightly different version of the generating function (3) has been studied by Zeng [17] in connection with continued fractions.

3 Gamma-nonnegativity

As we have already pointed out, P_{n}(q, t; x, y) is symmetric and homogeneous polynomial of degree n in x, y. So, by the fundamental theorem of symmetric functions, there exist uniquely determined coefficients γ_{n,k}(q, t) such that

P_{n}(q, t; x, y) = ∑_{k=0}^{⌊ \frac{n}{2} ⌋} γ_{n,k}(q, t) x^{k}y^{k}(x + y)^{n−2k}.  (12)

We are going to prove that P_{n}(q, t; x) is gamma-nonnegative as a polynomial in x. Namely, we will show that each γ_{n,k}(q, t) occurring in (12) is itself a polynomial in q, t with nonnegative integer coefficients. To this goal, we first recall that A_{n}(x) is gamma-nonnegative, which means that the coefficients γ_{n,k} in (2) are nonnegative integers. In addition, we observe that from (5) and (8) it follows that

1 + \frac{1}{x} \frac{d}{dz} C(x, z) = \frac{1}{x} + \frac{x − 1}{xe^{z} − e^{xz}} e^{z} = \frac{1}{x} + A(x, z),

whence

C_{n}(x) = x A_{n−1}(x) = ∑_{k=0}^{⌊ \frac{n−2}{2} ⌋} γ_{n−1,k} x^{k+1} (x + 1)^{n−2(k+1)}, for n ≥ 2.  (13)

Moreover, note that every product A_{n_{1}}(x)A_{n_{2}}(x) ⋯ A_{n_{i}}(x) of Eulerian polynomials is a gamma-nonnegative polynomial of degree n_{1} + n_{2} + ⋯ + n_{i} − i. More precisely, we have

A_{n_{1}}(x)A_{n_{2}}(x) ⋯ A_{n_{i}}(x) = ∑_{k=0}^{⌊ \frac{n−i}{2} ⌋} γ_{n_{1},n_{2},...,n_{i};k} x^{k} (x + 1)^{n−i−2k},

where n := n_{1} + n_{2} + ⋯ + n_{i} and

γ_{n_{1},n_{2},...,n_{i};k} = ∑_{j_{1}+j_{2}+···+j_{i}=k} γ_{n_{1},j_{1}}γ_{n_{2},j_{2}} ⋯ γ_{n_{i},j_{i}}.  (14)

Theorem 4. Each polynomial γ_{n,k}(q, t) in (12) has nonnegative integer coefficients.

Proof. From (10) and (13) it follows that

P_{n}(q, t; x) = ∑_{π∈Π_{n}} q^{|π|}t^{sing(π)}x^{|π|−sing(π)}(x + 1)^{sing(π)} ∏_{S∈π} A_{|S|−1}(x).

As ∏_{S∈π} A_{|S|−1}(x) is a gamma-nonnegative polynomial of degree

n_{π} := n − 2|π| + sing(π),

there exist nonnegative integers γ_{π,i} satisfying

∏_{S∈π} A_{|S|−1}(x) = ∑_{i=0}^{⌊ \frac{nπ}{2} ⌋} γ_{π,i} x^{i} (x + 1)^{n_{π}−2i}.

Hence, we have

P_{n}(q, t; x) = ∑_{π∈Π_{n}} q^{|π|}t^{sing(π)} ∑_{i=0}^{⌊ \frac{nπ}{2} ⌋} γ_{π,i} x^{i+|π|−sing(π)} (x + 1)^{n−2(i+|π|−sing(π))},

which ensures us that γ_{n,k}(q, t) is a polynomial in q, t with nonnegative integer coefficients.

Since gamma-nonnegativity implies unimodality, the following result directly follows from Theorem 4.

Corollary 5. If q, t are nonnegative integers, then P_{n}(q, t; x) is a unimodal polynomial in x.

Remark 6. Thanks to the “transformation fondamentale” [8], the Eulerian polynomial A_{n}(x) can be defined in the equivalent way

A_{n}(x) = ∑_{w∈𝔖_{n}} x^{asc(w)},

where asc(w) := |{i |w_{i} < w_{i+1}, 1 ≤ i ≤ n − 1}| is the number of ascents in w = w_{1}w_{2} ⋯ w_{n}. Now, let ℭ_{n} := ℭ({1, 2, …, n}). For all w = (a_{1}, a_{2}, …, a_{n−1}, n) ∈ ℭ_{n}, set w′ := a_{1}a_{2} ⋯ a_{n−1} ∈ 𝔖_{n−1}. As w ↦ w′ is bijective, we have

C_{n}(x) = ∑_{w∈ℭ_{n}} x^{exc(w)} = ∑_{w′∈𝔖_{n−1}} x^{1+asc(w)} = xA_{n−1}(x),

and a bijective proof of (13) is obtained.

4 A combinatorial description of gamma coefficients

Given w ∈ 𝔖_{n} and i ∈ {1, 2, …, n}, we say that i is a double excedance for w if and only if we have w^{−1}(i) < i < w(i). Let dexc(w) denote the number of double excedances of w. Moreover, let 𝔇_{n} ⊂ 𝔖_{n} denote the set of derangements (permutations with no fixed points) in 𝔖_{n}, and consider the expansion of the derangement polynomial d_{n}(x) in terms of the gamma basis [3]:

d_{n}(x) := ∑_{w∈𝔇_{n}} x^{exc(w)} = ∑_{k=0}^{⌊ \frac{n}{2} ⌋} ξ_{n,k}x^{k}(x + 1)^{n−2k}.

By means of “valley-hopping”, Athanasiadis and Savvidou [4] obtained the following neat combinatorial interpretation of ξ_{n,k}:

ξ_{n,k} = |{w | w ∈ 𝔇_{n}, exc(w) = k, dexc(w) = 0}|.  (15)

This last result plainly extends to the q-derangement polynomial d_{n}(q; x), for which we have [3,15]

d_{n}(q; x) := ∑_{w∈𝔇_{n}} q^{cyc(w)}x^{exc(w)} = ∑_{k=0}^{⌊ \frac{n}{2} ⌋} ξ_{n,k}(q)x^{k}(x + 1)^{n−2k},

where

ξ_{n,k}(q) = ∑_{{w∈𝔇n \atop exc(w)=k, dexc(w)=0}} q^{cyc(w)} = ∑_{i=1}^{n} ξ_{n,k,i}q^{i},

and

ξ_{n,k,i} = |{w | w ∈ 𝔇_{n}, exc(w) = k, dexc(w) = 0, cyc(w) = i}|.  (16)

With this established, a combinatorial description of γ_{n,k}(q) can be easily carried out.

Theorem 7. We have

γ_{n,k}(q, t) = ∑_{{w∈𝔖n \atop exc(w)=k, dexc(w)=0}} q^{cyc(w)}t^{fix(w)} = ∑_{i,j=0}^{n} γ_{n,k,i,j}q^{i}t^{j},  (17)

where

γ_{n,k,i,j} = |{w | w ∈ 𝔖_{n}, exc(w) = k, dexc(w) = 0, cyc(w) = i, fix(w) = j}|.  (18)

Proof. From (4) we obtain

P_{n}(q, t; x) = ∑_{j=0}^{n} \binom{n}{j}t^{j}q^{j}(x + 1)^{j}d_{n−j}(q; x),

then

γ_{n,k}(q, t) = ∑_{j=0}^{n} \binom{n}{j}t^{j}q^{j}ξ_{n−j,k}(q).  (19)

On the other hand, it is easily seen that

γ_{n,k,i,j} = \binom{n}{j}ξ_{n−j,k,i−j}.

Indeed, for every positive integer i, and for every F ⊆ {1, 2, …, n} such that |F | = j, there are exactly ξ_{n−j,k,i−j} permutations w ∈ 𝔖_{n} with i cycles, k excedances, no double excedances, and with the set F of fixed points. Hence, we may write

∑_{i,j=0}^{n} γ_{n,k,i,j}q^{i}t^{j} = ∑_{j=0}^{n} \binom{n}{j}q^{j}t^{j}ξ_{n−j,k}(q).  (20)

Finally, comparing (19) and (20), we obtain (17).

Corollary 8. We have P_{2n+1}(q, t; −1) = 0 and

(−1)^{n}P_{2n}(q, t; −1) = ∑_{w∈𝔇_{2n}} q^{cyc(w)}(−1)^{n−exc(w)}  (21)
= ∑_{{w∈𝔇_{2n} \atop exc(w)=n, dexc(w)=0}} q^{cyc(w)}.  (22)

Proof. Setting x = −1 and y = 1 in (9), we get (21). Analogously, setting x = −1 and y = 1 in (12), we obtain P_{2n+1}(q, t; −1) = 0 and

P_{2n}(q, t; −1) = (−1)^{n}γ_{2n,n}(q, t).

Besides, from (17) we have

P_{2n}(q, t; −1) = (−1)^{n} ∑_{{w∈𝔖_{2n} \atop exc(w)=n, dexc(w)=0}} q^{cyc(w)}t^{fix(w)}.  (23)

Now, given w ∈ 𝔖_{2n} such that exc(w) = n and dexc(w) = 0, we have that i is an excedance for w if and only if i < w(i) > w^{2}(i). So, w^{2}(i) is an excedance for w^{−1}, and exc(w^{−1}) = n. Finally, as exc(w) + exc(w^{−1}) + fix(w) = 2n, we deduce that fix(w) = 0, and we may replace 𝔖_{2n} with 𝔇_{2n} in (23) to obtain (22).

Note that the permutations w ∈ 𝔖_{2n} satisfying exc(w) = n and dexc(w) = 0 bijectively correspond to the alternating permutations via the “transformation fondamentale” [8]. For instance, starting from the standard representation of w = (5, 1, 4, 2)(8, 3, 7, 6) ∈ 𝔖_{8} (with exc(w) = 4 and dexc(w) = 0), via the “transformation fondamentale”we obtain ŵ = 51428376, which in fact is an alternating permutation. So, (22) can be seen as a q-analogue of the classical identity A_{2n+1}(−1) = (−1)^{n}E_{2n+1} relating Eulerian polynomials to the Euler numbers E_{n}.

5 Some special cases and new results on Eulerian polynomials

The polynomial P_{n}(q, t; x, y) reduces to well-known gamma-nonnegative polynomials once that the parameters q, t, y are suitably specialized. For each of these special cases, (9) provides the corresponding interpretation in terms of enumeration of permutations, while (17) and (18) lead to the combinatorial description of the associated gamma vector.

5.1 Powers of the binomial

From (4) we have

lim_{q→0} P_{n}(q, \frac{t}{q}; x, y) = t^{n}(x + y)^{n}.

In this case, the identity permutation e is the unique w ∈ 𝔖_{n} providing a nonzero contribution in (9). In particular, we have cyc(e) = fix(e) = n and dexc(e) = exc(e) = 0.

5.2 Derangement polynomials and their q-analogues

This case was essentially discussed in the previous section. We recall that we have

d_{n}(q; x) = P_{n}(q, 0; x, 1).  (24)

By virtue of (4), the corresponding generating function is

∑_{n≥0} d_{n}(q; x)\frac{z^{n}}{n!} = (\frac{x − 1}{xe^{z} − e^{xz}} )^{q}.  (25)

The combinatorial expansion obtained from (9) is

d_{n}(q; x) = ∑_{w∈𝔇_{n}} q^{cyc(w)}x^{exc(w)}.

Of course, for the gamma vector we recover the known expression [3,4,15]

γ_{n,k}(q, 0) = ξ_{n,k}(q).

The derangement polynomial arises as d_{n}(x) = d_{n}(1; x), for which we have γ_{n,k}(1, 0) = ξ_{n,k}.

5.3 Binomial Eulerian polynomials

Set

Ã_{n}(x) := P_{n}(1, 1; x) = ∑_{w∈𝔖_{n}} x^{exc(w)}(x + 1)^{fix(w)}.  (26)

From (3), (4), and (5) we recover

∑_{n≥0} Ã_{n}(x)\frac{z^{n}}{n!} = \frac{x − 1}{xe^{z} − e^{xz}} e^{(x+1)z} = A(x; z) e^{xz} = e^{z} + x(A(x; z) − 1) e^{z},

then

Ã_{n}(x) = ∑_{k=0}^{n} \binom{n}{k}x^{n−k}A_{k}(x) = 1 + x ∑_{k=1}^{n} \binom{n}{k}A_{k}(x),  (27)

where A_{0}(x) := 1. This says that Ã_{n}(x) is a binomial Eulerian polynomials [3]. For these polynomials, (17) and (18) lead to the known interpretation of the gamma vector γ̃_{n,k}:

γ̃_{n,k} = γ_{n,k}(1, 1) = |{w ∈ 𝔖_{n} | exc(w) = k, dexc(w) = 0}|.

5.4 Eulerian polynomials

Classical Eulerian polynomials arise by means of a less obvious specialization. In fact, (4) and (5) give

\frac{d}{dz} A(x, z) = (\frac{x − 1}{xe^{z} − e^{xz}} )^{2}e^{(x+1)z} = P(2, 1/2; x, 1; z),

from which it follows that

A_{n+1}(x) = P_{n}(2, 1/2; x, 1), for n ≥ 1.

Now, (9) leads to the following alternative combinatorial interpretation of the Eulerian polynomial A_{n}(x):

A_{n}(x) = ∑_{w∈𝔖_{n−1}} 2^{cyc(w)−fix(w)}x^{exc(w)}(x + 1)^{fix(w)}.  (28)

Moreover, from (17) and (18) we obtain a new description of the gamma vector of A_{n}(x):

γ_{n,k} = γ_{n,k}(2, 1/2) = ∑_{{w∈𝔖_{n−1} \atop exc(w)=k, dexc(w)=0}} 2^{cyc(w)−fix(w)}.  (29)

Finally, from (28) we get the following identity relating the derangements of 𝔖_{2n} to the alternating permutations of 𝔖_{2n+1}:

∑_{w∈𝔇_{2n}} 2^{cyc(w)}(−1)^{exc(w)} = (−1)^{n}E_{2n+1}.  (30)

An analogous relation involving the derangements and the alternating permutations of 𝔖_{2n} was stated by Roselle [14]:

∑_{w∈𝔇_{2n}}(−1)^{exc(w)} = (−1)^{n}E_{2n}.

It would be interesting to achieve neat bijective proofs of (28), (29), and (30).

5.5 A q,t-analogue of Eulerian polynomials

A very natural generalization of the Eulerian polynomial A_{n}(x) is given by

Q_{n}(q, t; x) := ∑_{w∈𝔖_{n}} q^{cyc(w)}t^{fix(w)}x^{exc(w)} = P_{n}(q, t/(x + 1); x).  (31)

This polynomial reduces to A_{n}(x) when q = t = 1. Also, Q_{n}(q, t; x) gives the q-derangement polynomial d_{n}(q; x) when t = 0, and the derangement polynomial d_{n}(x) when q = 1 and t = 0. Combinatorial properties of Q_{n}(q, t; x) have been studied by G. Kasavrelof and J. Zeng [10], and by S.-M. Ma [11]. Moreover, by setting t = 1 in (31) we obtain the polynomial A_{n}(x, q) = x Q_{n}(q, 1; x) introduced by Brenti [7]. It is easily seen that Q_{n}(q, t; x) is not palindromic in x, hence there is no gamma vector associated with Q_{n}(q, t; x). Nevertheless, by comparing (31) with (3) and (4) we recover

∑_{n≥0} P_{n}(q, t; x)\frac{z^{n}}{n!} = e^{xqtz} ∑_{n≥0} Q_{n}(q, t; x)\frac{z^{n}}{n!} .

So, we obtain

P_{n}(q, t; x) = ∑_{k=0}^{n} \binom{n}{k}(qtx)^{n−k}Q_{k}(q, t; x),  (32)

which is a generalization of the first identity in (27). In fact, the first part of (27) comes from (32) by setting q = t = 1.

References

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2020 Mathematics Subject Classification: Primary 05A15; Secondary 05A10, 05A19, 11B65, 11B83.

Keywords: Eulerian polynomial, excedance, derangement, alternating permutation.

(Concerned with sequences \underline{A000166}, \underline{A001250}, and \underline{A008292}.)

Received August 5 2023; revised version received August 6 2023; November 18 2023; November 20 2023. Published in Journal of Integer Sequences, January 13 2024.

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