Symmetrized Poly-Bernoulli Numbers and Combinatorics

Toshiki Matsusaka Institute for Advanced Research Nagoya University Furo-cho, Chikusa-ku Nagoya 464-8601 Japan

         matsusaka.toshiki@math.nagoya-u.ac.jp

Abstract

Poly-Bernoulli numbers are one of the generalizations of the classical Bernoulli numbers. Since a negative indexed poly-Bernoulli number is an integer, it is an interesting problem to study this number from a combinatorial viewpoint. In this short article, we give a new combinatorial relation between symmetrized poly-Bernoulli numbers and Dumont-Foata polynomials.

1 Introduction

A poly-Bernoulli polynomial B_{m}^{(ℓ)}(x) of index ℓ ∈ ℤ is defined by the generating series

e^{−xt} \frac{Li_{ℓ}(1 − e^{−t})}{1 − e^{−t}} = ∑_{m=0}^{∞} B_{m}^{(ℓ)}(x)\frac{t^{m}}{m!},

where Li_{ℓ}(z) is the polylogarithm function given by

Li_{ℓ}(z) = ∑_{m=1}^{∞} \frac{z^{m}}{m^{ℓ}} (|z| < 1).

The polynomial B_{m}^{(1)}(x) coincides with the classical Bernoulli polynomial B_{m}(1 − x) = (−1)^{m}B_{m}(x) since Li_{1}(z) = − log(1 − z) holds. Following Kaneko [7], the special value B_{m}^{(ℓ)} := B_{m}^{(ℓ)}(0) is called a poly-Bernoulli number of index ℓ.

The aim of this study is giving a combinatorial perspective to the special values of B_{m}^{(ℓ)}(x) at integers k ∈ ℤ. We assume that the index ℓ ≤ 0. In this case the values B_{m}^{(ℓ)}(k) are always integers. One of the first such investigations was Brewbaker’s study [4]. He noticed the coincidence of two numbers, the poly-Bernoulli number B_{m}^{(ℓ)}(0) and the number of 01 lonesum matrices of size m × |ℓ|. Recently, Bényi and Hajnal [3] also established more combinatorial relations in this direction.

In this article, we take a step in another direction similar to Kaneko, Sakurai, and Tsumura [9]. To describe this more precisely, let G_{n} be the Genocchi number \underline{A110501} defined by G_{n} = 2(2^{n+2} − 1)|B_{n+2}|, where B_{m} = B_{m}^{(1)}(1) is the classical Bernoulli number. In [9, Theorem 4.2] the authors showed

∑_{ℓ=0}^{n}(−1)^{ℓ}B_{n−ℓ}^{(−ℓ−1)}(1) = (−1)^{n/2}G_{n}  (1)

for any n ≥ 0. As they mentioned, this equation is an analogue of the result of Arakawa and Kaneko [2],

∑_{ℓ=0}^{n}(−1)^{ℓ}B_{n−ℓ}^{(−ℓ)}(0) = \begin{cases}1, & if n = 0; \\ 0, & if n > 0. \end{cases}  (2)

In addition, Sakurai asked in her master’s thesis whether we can generalize these equations for any positive integers x = k, and give some combinatorial meaning to them. Our main result provides an answer to these two questions in terms of the Dumont-Foata polynomial as follows.

Theorem 1. Let G_{n}(x, y, z) be the n-th Dumont-Foata polynomial defined in (3), and ℬ_{m}^{(−ℓ)}(k) the symmetrized poly-Bernoulli number defined in (5). Then we have

∑_{ℓ=0}^{n}(−1)^{ℓ}ℬ_{n−ℓ}^{(−ℓ)}(k) = k! · (−1)^{n/2}G_{n}(1, 1, k)

for any non-negative integers n, k ≥ 0. In particular, both sides equal zero for odd n.

This theorem recovers the equations (1) and (2) since ℬ_{m}^{(−ℓ)}(0) = B_{m}^{(−ℓ)}(0), ℬ_{m}^{(−ℓ)}(1) = B_{m}^{(−ℓ−1)}(1), and G_{n}(1, 1, 1) = G_{n} hold as we see later. Remark 2. I feel that there are a lot of possibilities of establishing similar identities as this theorem. Recent work of Bényi and Hajnal [3, Section 6] pointed out that the sequence of diagonal sums

∑_{ℓ=0}^{n} B_{n−ℓ}^{(−ℓ−1)}(1) (n ≥ 0)

appears as \underline{A136127} in the On-Line Encyclopedia of Integer Sequences (OEIS) [11]. Furthermore, there are many types of generalizations of poly-Bernoulli numbers as referred in [5], and analogues such as poly-Euler numbers [10], and poly-cosecant numbers [8].

2 Definitions

2.1 Dumont-Foata polynomials

We review the work of Dumont and Foata [6] here. For a positive even integer n ∈ 2ℤ, we consider the surjective map p : {1, 2, …, n} → {2, 4, …, n} with p(x) ≥ x for each x ∈ {1, 2, …, n}. This map is called a (surjective) pistol of size n, and corresponds to the following diagram. Here we draw an example for n = 6.

Figure 1: p(1) = 2, p(2) = p(4) = 4, p(3) = p(5) = p(6) = 6

Let P_{n} be the set of all pistols of size n. For each pistol p ∈ P_{n}, we define three quantities called bulging, fixed, and maximal points. First, the number x ∈ {1, 2, …, n} is a bulging point of p ∈ P_{n} if p(y) < p(x) for any 0 < y < x. We let b(p) denote the number of bluging points of p. In the diagram, b(p) corresponds to the number of steps of the minimal stair covering all check marks. For the above example p, the points x = 1, 2, 3 are bulging points, so that b(p) = 3.

Figure 2: minimal stair (left), not minimal stair (right)

Next, the point x ∈ {1, 2, …, n} is called a fixed point of p ∈ P_{n} if p(x) = x. Finally, the point x ∈ {1, 2, …, n − 1} is a maximal point of p ∈ P_{n} if p(x) = n. We let f(p) and m(p) denote the numbers of fixed points and maximal points of p ∈ P_{n}, respectively. For the above example, we have f(p) = 2 and m(p) = 2. Dumont and Foata [6, Théorème 1a, 2] established the following interesting theorem.

Theorem 3. [6] Let n ∈ 2ℤ_{>0} be a positive even integer. The polynomial defined by

G_{n}(x, y, z) := ∑_{p∈P_{n}} x^{b(p)}y^{f(p)}z^{m(p)}  (3)

is a symmetric polynomial in three variables, and satisfies G_{n}(1, 1, 1) = G_{n}.

In addition, G_{0}(x, y, z) = 1 and G_{n}(x, y, z) = 0 for a positive odd integer n ∈ ℤ. The polynomial G_{n}(x, y, z) is called the n-th Dumont-Foata polynomial. Furthermore, Dumont and Foata showed that the polynomial for n > 0 has the form G_{n}(x, y, z) = xyzF_{n}(x, y, z), and the polynomial F_{n}(x, y, z) satisfies the recurrence relation

F_{n}(x, y, z) = (x + z)(y + z)F_{n−2}(x, y, z + 1) − z^{2}F_{n−2}(x, y, z)

with initial values F_{1}(x, y, z) = 0 and F_{2}(x, y, z) = 1. This implies that the polynomial G_{n}(z) := G_{n}(1, 1, z) called the Gandhi polynomial \underline{A036970} satisfies

G_{n+2}(z) = z(z + 1)G_{n}(z + 1) − z^{2}G_{n}(z)  (4)

with G_{0}(z) = 1, G_{1}(z) = 0.

For instance, there exist three pistols of size 4. The pistols have (b(p), f(p), m(p)) = (2, 2, 1), (2, 1, 2) and (1, 2, 2), so that the Dumont-Foata polynomial is given by

G_{4}(x, y, z) = x^{2}y^{2}z + x^{2}yz^{2} + xy^{2}z^{2} = xyz(xy + yz + zx).

Indeed, G_{4}(1, 1, 1) = 3 coincides the 4-th Genocchi number given by G_{4} = 2(2^{6} − 1)|B_{6}| = 3.

Figure 3: All elements of P_{4}

2.2 Symmetrized poly-Bernoulli numbers

Table 1. includes the first few values of poly-Bernoulli numbers {B_{m}^{(−ℓ)}(0)} and {B_{m}^{(−ℓ)}(1)} with m, ℓ ≥ 0.

\begin{matrix}\begin{matrix}ℓ\m & 0 & 1 & 2 & 3 & 4 \\ 0 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 2 & 4 & 8 & 16 \\ 2 & 1 & 4 & 14 & 46 & 146 \\ 3 & 1 & 8 & 46 & 230 & 1066 \\ 4 & 1 & 16 & 146 & 1066 & 6902 \end{matrix} & \begin{matrix}ℓ\m & 0 & 1 & 2 & 3 & 4 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 \\ 2 & 1 & 3 & 7 & 15 & 31 \\ 3 & 1 & 7 & 31 & 115 & 391 \\ 4 & 1 & 15 & 115 & 675 & 3451 \end{matrix} \\ Table 1: B_{m}^{(−ℓ)}(0) \underline{A099594}, B_{m}^{(−ℓ)}(1) \underline{A136126} \end{matrix}

We can see the symmetric property of these numbers at a glance. On the other hand, for k ≥ 2 it seems unlikely that such a simple symmetric property can be given. In order to reproduce the symmetric properties for any k ≥ 2, Kaneko-Sakurai-Tsumura [9] considered combinations of B_{m}^{(−ℓ)}(k). To make this precise, let m, ℓ, k ≥ 0 be non-negative integers. We now define the symmetrized poly-Bernoulli number ℬ_{m}^{(−ℓ)}(k) by

ℬ_{m}^{(−ℓ)}(k) = ∑_{j=0}^{k} [\begin{matrix}k \\ j \end{matrix}] B_{m}^{(−ℓ−j)}(k),  (5)

where [\begin{matrix}k \\ j \end{matrix}] is the unsigned Stirling number of the first kind \underline{A130534} defined in [1, Definition

2.5]. This number satisfies the symmetry property

ℬ_{m}^{(−ℓ)}(k) = ℬ_{ℓ}^{(−m)}(k)

for any m, ℓ, k ≥ 0. Note that

ℬ_{m}^{(−ℓ)}(0) = B_{m}^{(−ℓ)}(0), ℬ_{m}^{(−ℓ)}(1) = B_{m}^{(−ℓ−1)}(1).

\begin{matrix}ℓ\m & 0 & 1 & 2 & 3 & 4 \\ 0 & 1 & −1 & 1 & −1 & 1 \\ 1 & 1 & 0 & 0 & 0 & 0 \\ 2 & 1 & 2 & 2 & 2 & 2 \\ 3 & 1 & 6 & 18 & 42 & 90 \\ 4 & 1 & 14 & 86 & 374 & 1382 \end{matrix}  \begin{matrix}ℓ\m & 0 & 1 & 2 & 3 & 4 \\ 0 & 2 & 2 & 2 & 2 & 2 \\ 1 & 2 & 8 & 20 & 44 & 92 \\ 2 & 2 & 20 & 104 & 416 & 1472 \\ 3 & 2 & 44 & 416 & 2744 & 15032 \\ 4 & 2 & 92 & 1472 & 15032 & 120632 \end{matrix}

Table 2: B_{m}^{(−ℓ)}(2) and ℬ_{m}^{(−ℓ)}(2)

Moreover, the authors showed the following explicit formula for ℬ_{m}^{(−ℓ)}(k).

ℬ_{m}^{(−ℓ)}(k) = ∑_{j=0}^{min(m,ℓ)} j!(k + j)! \begin{cases}m + 1 \\ j + 1 \end{cases}\} \begin{cases}ℓ + 1 \\ j + 1 \end{cases}\} ,  (6)

where \begin{cases}k \\ j \end{cases}\} is the Stirling number of the second kind \underline{A008277} defined in [1, Definition 2.2].

We prove our main theorem using this formula in the next section.

3 Proof

To prove Theorem 1, it suffices to show that the function

G̃_{n}(k) := \frac{(−1)^{n/2}}{k!} ∑_{ℓ=0}^{n}(−1)^{ℓ}ℬ_{n−ℓ}^{(−ℓ)}(k)

satisfies the recurrence relation (4) for any integer k ≥ 0. First, we can easily see that G̃_{0}(k) = 1 and G̃_{1}(k) = 0, which are the initial cases. Moreover, for any odd integer n, G̃_{n}(k) = 0 follows from the symmetric property of ℬ_{m}^{(−ℓ)}(k). For an even integer n ≥ 2, by the formula (6) we have

(−1)^{n/2}k!(k(k + 1)G̃_{n}(k + 1) − k^{2}G̃_{n}(k) − G̃_{n+2}(k))

= k ∑_{j=0}^{n/2} j!(k + j + 1)! ∑_{ℓ=j}^{n−j}(−1)^{ℓ} \begin{cases}n − ℓ + 1 \\ j + 1 \end{cases}\} \begin{cases}ℓ + 1 \\ j + 1 \end{cases}\}  (7)

− k^{2} ∑_{j=0}^{n/2} j!(k + j)! ∑_{ℓ=j}^{n−j}(−1)^{ℓ} \begin{cases}n − ℓ + 1 \\ j + 1 \end{cases}\} \begin{cases}ℓ + 1 \\ j + 1 \end{cases}\}  (8)

+ ∑_{j=0}^{n/2+1} j!(k + j)! ∑_{ℓ=j}^{n+2−j}(−1)^{ℓ} \begin{cases}n − ℓ + 3 \\ j + 1 \end{cases}\} \begin{cases}ℓ + 1 \\ j + 1 \end{cases}\} .  (9)

Since \begin{cases}k \\ 1 \end{cases}\} = 1 holds for any k ≥ 1, we can split the third line (9) according as j = 0 or not,

which equals

k! + ∑_{j=0}^{n/2}(j + 1)!(k + j + 1)! ∑_{ℓ=j+1}^{n+1−j}(−1)^{ℓ} \begin{cases}n + 3 − ℓ \\ j + 2 \end{cases}\} \begin{cases}ℓ + 1 \\ j + 2 \end{cases}\} .

Let

a_{n,j} := ∑_{ℓ=j}^{n−j}(−1)^{ℓ} \begin{cases}n − ℓ + 1 \\ j + 1 \end{cases}\} \begin{cases}ℓ + 1 \\ j + 1 \end{cases}\} .  (10)

Then the total of (7), (8), and (9) equals

k! + ∑_{j=0}^{n/2} j!(k + j)!(k(j + 1)a_{n,j} + (j + 1)(k + j + 1)a_{n+2,j+1}).  (11)

Once the sum is 0, the proof completes. By using the generating function given in [1, Proposition 2.6 (8)], we have

\frac{t^{j}}{(1 − t)(1 − 2t) ⋯ (1 − (j + 1)t)} = ∑_{ℓ≥j} \begin{cases}ℓ + 1 \\ j + 1 \end{cases}\} t^{ℓ} = ∑_{ℓ≤n−j} \begin{cases}n − ℓ + 1 \\ j + 1 \end{cases}\} t^{n−ℓ}

for any non-negative integers n, j ∈ ℤ_{≥0}. Multiplying these two expressions, we obtain

\frac{s^{j}t^{j}}{(1 − s)(1 − t) ⋯ (1 − (j + 1)s)(1 − (j + 1)t)} = ∑_{ℓ≥j} ∑_{k≤n−j} \begin{cases}ℓ + 1 \\ j + 1 \end{cases}\} \begin{cases}n − k + 1 \\ j + 1 \end{cases}\} s^{ℓ}t^{n−k}.

By specializing at s = −x, t = x,

\frac{(−1)^{j}x^{2j}}{(1 − x^{2}) ⋯ (1 − (j + 1)^{2}x^{2})} = ∑_{ℓ≥j} ∑_{k≤n−j}(−1)^{ℓ} \begin{cases}ℓ + 1 \\ j + 1 \end{cases}\} \begin{cases}n − k + 1 \\ j + 1 \end{cases}\} x^{n+ℓ−k}. (12)

Thus, we see that the number a_{n,j} defined in (10) appears as the n-th coefficient of (12). By the expression of the left-hand side of (12), we easily see that a_{n,j} = 0 when n is an odd integer or 2j > n. Further, we get the initial values a_{2j,j} = (−1)^{j}, a_{n,0} = 1 for even n, and the recurrence relation

a_{n+2,j} = (j + 1)^{2}a_{n,j} − a_{n,j−1}.

Applying this to the equation (11), we get

k! + ∑_{j=0}^{n/2}j!(k + j)!(k(j + 1)a_{n,j} + (j + 1)(k + j + 1)((j + 2)^{2}a_{n,j+1} − a_{n,j}))

= k! + ∑_{j=0}^{n/2}(j + 2)(j + 2)!(k + j + 1)!a_{n,j+1} − ∑_{j=0}^{n/2}(j + 1)(j + 1)!(k + j)!a_{n,j}.

Since a_{n,n/2+1} = 0 and a_{n,0} = 1, this equals 0, which concludes the proof of Theorem 1.

4 Acknowledgments

The author sincerely thanks the referees for helpful comments and suggestions. This work was supported by JSPS KAKENHI Grant Numbers 18J20590 and 20K14292.

References

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[11] N. J. A. Sloane et al., The on-line encyclopedia of integer sequences, 2020. Available at

    https://oeis.org.

2010 Mathematics Subject Classification: Primary 11B68; Secondary 05A15. Keywords: poly-Bernoulli number, Genocchi number, Dumont-Foata polynomial.

(Concerned with sequences \underline{A008277}, \underline{A036970}, \underline{A099594}, \underline{A110501}, \underline{A130534}, \underline{A136126}, and \underline{A136127}.)

Received April 2 2020; revised version received July 15 2020. Published in Journal of Integer Sequences, October 13 2020.

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