Symmetrized poly-Bernoulli numbers and combinatorics
Toshiki Matsusaka
math.NT math.CO




 


Faculty of Mathematics, Kyushu University, Motooka 744, Nishi-ku Fukuoka 819-0395, Japan,
toshikimatsusaka@gmail.com

[2010]Primary 11B68, Secondary 05A15.


This work was supported by JSPS KAKENHI Grant Numbers 18J20590.

Symmetrized poly-Bernoulli numbers and combinatorics
    Toshiki Matsusaka
    December 30, 2023
====================================================




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








§ INTRODUCTION




A poly-Bernoulli polynomial B_m^(ℓ) (x) of index ℓ∈ is defined by the generating series

    e^-xt_ℓ (1 - e^-t)/1 - e^-t = ∑_m=0^∞ B_m^(ℓ) (x) t^m/m!,

where _ℓ (z) is the polylogarithm function given by

    _ℓ (z) = ∑_m=1^∞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 _1(z) = -log(1-z) holds (see <cit.>). Following Kaneko <cit.>, the special value B_m^(ℓ) := B_m^(ℓ)(0) is particularly called a poly-Bernoulli number of index ℓ. 


The aim of this study is to give a combinatorial perspective to the special values of B_m^(ℓ)(x) at integers k ∈. We now assume that the index ℓ≤ 0, so that the values B_m^(ℓ)(k) are always integers. One of the first such investigations is the coincidence of two numbers, the poly-Bernoulli number B_m^(ℓ)(0) and the number of 01 lonesum matrices of size m × |ℓ| (see <cit.>). A recent work by Bényi-Hajnal <cit.> established more combinatorial relations in this direction.
 

In this article, we take a step in another direction along with Kaneko-Sakurai-Tsumura <cit.>. To describe this more precisely, let G_n be the Genocchi number defined by G_n = 2(2^n+2 - 1)|B_n+2|, where B_m = B_m^(1)(1) is the classical Bernoulli number. Then they showed in <cit.> that 

    ∑_ℓ =0^n (-1)^ℓ B_n-ℓ^(-ℓ -1) (1) = (-1)^n/2 G_n

for any n ≥ 0. As they mentioned, this equation is an analogue of Arakawa-Kaneko's result <cit.>

    ∑_ℓ =0^n (-1)^ℓ B_n-ℓ^(-ℓ) (0) = {[          1  if  n = 0;          0 if  n > 0. ].

In addition, Sakurai asked in her master 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.


	Let G_n(x,y,z) be the n-th Dumont-Foata polynomial defined in (<ref>), and _m^(ℓ)(k) the symmetrized poly-Bernoulli number defined in (<ref>). Then we have
	
    ∑_ℓ=0^n (-1)^ℓ_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 (<ref>) and (<ref>) since _m^(-ℓ) (0) = B_m^(-ℓ) (0), _m^(-ℓ) (1) = B_m^(-ℓ-1)(1), and G_n(1,1,1) = G_n hold as we see later. 


Recent work of Bényi-Hajnal <cit.> pointed out that the sequence of diagonal sums

    ∑_ℓ = 0^n B_n-ℓ^(-ℓ -1) (1)    (n ≥ 0)

appears in OEIS <cit.> A136127. It might be an interesting problem to extend our theorem in this direction.






§ DEFINITIONS








 §.§ Dumont-Foata polynomials




We review the work of Dumont-Foata <cit.> here. For a positive even integer n ∈ 2, 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 (surjective) pistol of size n, and corresponds to the following diagram. Here we draw one example for n = 6 given by p(1) = 2, p(2) = 4, p(3) = 6, p(4) = 4, p(5) = 6, p(6) = 6.



Let _n be the set of all pistols of size n. For each pistol 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 ∈_n if p(y) < p(x) for any 0< y < x. Then we denote by b(p) 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.



Next, the point x ∈{1, 2, …, n} is called a fixed point of p ∈_n if p(x) = x. Finally, the point x ∈{1, 2, …, n-1} is a maximal point of p ∈_n if p(x) = n. Then we denote by f(p) and m(p) the numbers of fixed points and maximal points of p ∈_n, respectively. For the above example, we have f(p) = 2 and m(p) = 2. Under these notations, Dumont-Foata established the following interesting theorem.

<cit.>
	Let n ∈ 2_>0 be a positive even integer. The polynomial defined by
	
    G_n(x,y,z) := ∑_p ∈_n x^b(p) y^f(p) z^m(p)

	is a symmetric polynomial in three variables, and gives the Genocchi number G_n as
	
    G_n(1,1,1) = G_n.



In addition, we put G_0(x,y,z) = 1 and G_n(x,y,z) = 0 for a positive odd integer n ∈. The polynomial G_n(x,y,z) is called the n-th Dumont-Foata polynomial. Furthermore, they showed that the polynomial for n > 0 has the form G_n(x,y,z) = xyz F_n(z,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 satisfies

    G_n+2(z) = z(z+1) G_n(z+1) - z^2 G_n(z)

with G_0(z) = 1, G_1(z) = 0. 


For instance, there exist three pistols of size 4. Each pistol has (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 y z^2 + x y^2 z^2 = xyz(xy + yz + zx).

In fact, G_4(1,1,1) = 3 coincides the 4-th Genocchi number given by G_4 = 2(2^6 -1) |B_6| = 126 × 1/42 = 3.







 §.§ Symmetrized poly-Bernoulli numbers




The following are the tables of poly-Bernoulli numbers {B_m^(-ℓ)(0)} and {B_m^(-ℓ)(1)} with m, ℓ≥ 0.



We can find the symmetric properties 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 <cit.> 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 _m^(-ℓ) (k) by

    _m^(-ℓ) (k) = ∑_j=0^k k 
     j B_m^(-ℓ - j) (k),

where k 
 j is the Stirling number of the first kind defined in <cit.>. Then this number satisfies _m^(-ℓ) (k) = _ℓ^(-m) (k) for any m,ℓ, k ≥ 0, and

    _m^(-ℓ) (0) = B_m^(-ℓ) (0),   _m^(-ℓ) (1) = B_m^(-ℓ-1) (1).




Moreover they showed the following explicit formula for _m^(-ℓ) (k).

    _m^(-ℓ) (k) = ∑_j=0^min(m, ℓ) j! (k+j)! {m + 1 
     j+1}{ℓ + 1 
     j+1},

where {[ k; j ]} is the Stirling number of the second kind defined in <cit.>. By using this formula, we prove our main theorem in the next section.





§ PROOF




To prove Theorem <ref>, it suffices to show that the function

    G̃_n(k) := (-1)^n/2/k!∑_ℓ=0^n (-1)^ℓ_n-ℓ^(-ℓ) (k)

satisfies the recurrence relation (<ref>) for any integer k ≥ 0. First we can easily see that G̃_0(k) = 1 and G̃_1(k) = 0, which are initial cases. Moreover for any odd integer n,  G̃_n(k) = 0 follows from the symmetric property of _m^(-ℓ)(k). For an even integer n ≥ 2, by the formula (<ref>) 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{n-ℓ + 1 
     j+1}{ℓ + 1 
     j+1}
            -k^2 ∑_j = 0^n/2 j! (k+j)! ∑_ℓ = j^n-j (-1)^ℓ{n-ℓ + 1 
     j+1}{ℓ + 1 
     j+1}
            + ∑_j=0^n/2+1 j! (k+j)! ∑_ℓ = j^n+2-j (-1)^ℓ{n-ℓ +3 
     j+1}{ℓ + 1 
     j+1}.

Since {[ k; 1 ]} = 1 holds for any k ≥ 1, the third line (<ref>) is divided 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{n +3-ℓ
     j+2}{ℓ+1 
     j +2}.

Let

    a_n,j := ∑_ℓ = j^n-j (-1)^ℓ{n-ℓ + 1 
     j+1}{ℓ + 1 
     j+1},

then the total of (<ref>), (<ref>), and (<ref>) 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].

Once the sum is 0, the proof completes. By using the generating function given in <cit.>, we have

    t^j/(1-t)(1-2t) ⋯ (1-(j+1)t) = ∑_ℓ≥ j{ℓ+1 
     j+1} t^ℓ = ∑_ℓ≤ n - j{n-ℓ +1 
     j+1} t^n-ℓ

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

    s^j t^j/(1-s)(1-t) ⋯ (1-(j+1)s)(1-(j+1)t) = ∑_ℓ≥ j∑_k ≤ n - j{ℓ +1 
     j +1}{n-k+1 
     j+1} s^ℓ t^n-k.

By specializing at s = -x, t = x,

    (-1)^j x^2j/(1-x^2) ⋯ (1-(j+1)^2 x^2) = ∑_ℓ≥ j∑_k ≤ n -j (-1)^ℓ{ℓ +1 
     j +1}{n-k+1 
     j+1} x^n+ℓ -k.

Thus we see that the number a_n,j defined in (<ref>) appears as the n-th coefficient of (<ref>). By the expression of the left-hand side of (<ref>), 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 term in (<ref>), 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  <ref>.



99
	AIK14 T. Arakawa, T. Ibukiyama, M. Kaneko,
		Bernoulli numbers and zeta functions, With an appendix by Don Zagier, Springer Monographs in Mathematics. Springer, Tokyo, 2014. xii+274 pp.
	AK99 T. Arakawa, M. Kaneko,
		On Poly-Bernoulli numbers, Comment. Math. Univ. Sanct. Pauli, vol. 48-2, (1999), 159–167.
	BH17 B. Bényi, P. Hajnal,
		Combinatorial properties of poly-Bernoulli relatives, Integers 17, (2017), A31, 26 pp.
	Bre08 C. Brewbaker,
		A combinatorial interpretation of the poly-Bernoulli numbers and two Fermat analogues, Integers 8 (2008), A02, 9pp.
	DF76 D. Dumont, D. Foata,
		Une propriété de symétrie des nombres de Genocchi, Bulletin de la Société Mathématique de France, Tome 104, (1976), 433–451.
	Kan97 M. Kaneko,
		Poly-Bernoulli numbers, J. Théor. Nombres Bordeaux, 9, (1997), 199–206.
	KST18 M. Kaneko, F. Sakurai, H. Tsumura,
		On a duality formula for certain sums of values of poly-Bernoulli polynomials and its application, J. de Théorie des Nombres de Bordeaux, 30-1, (2018), 203–218.
	OEIS N. J. A. Sloane, 
		The on-line encyclopedia of integer sequences, http://oeis.orghttp://oeis.org.



