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Extensions of the Combinatorics of Poly-Bernoulli Numbers 3

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Extensions of the Combinatorics of Poly-Bernoulli Numbers 3 EXTENSIONS OF THE COMBINATORICS OF POLY-BERNOULLI NUMBERS BEATA´ BENYI´ AND TOSHIKI MATSUSAKA Abstract. In this paper we extend the combinatorial theory of poly-Bernoulli numbers. We prove combinatorially some identities involving poly-Bernoulli polynomials. We in- troduce a combinatorial model for poly-Euler numbers and provide combinatorial proofs of some identities. We redefine r-Eulerian polynomials and verify Stephan’s conjecture concerning poly-Bernoulli numbers and the central binomial series. 1. Introduction Poly-Bernoulli numbers were introduced by Kaneko [14] using polylogarithm function. His work motivated several studies in the same manner, such as further generalizations of these numbers and analogue definitions based on the polylogarithm function. These numbers and their relatives have also beautiful combinatorial properties and arise in dif- ferent research fields, such as discrete tomographie, mathematics of origami, biology, etc. [13, 25, 27, 34]. In this work we show some possible directions of studies motivated by the combinatorial treatment of these numbers. First, we introduce two models for interpreting the poly- Bernoulli polynomials and in order to show their usefulness we provide combinatorial proofs for some known identities. In Section 3 we consider poly-Euler numbers and extend one of the combinatorial interpretations of poly-Bernoulli numbers that helps in the combinatorial analysis of poly-Euler numbers of the first and second kind studied in [20, 32]. Section 4 is devoted to r-Eulerian polynomials, which we define a new way and use for verifying a conjecture concerning a relation between the central binomial series and poly-Bernoulli numbers observed by Stephan and stated by Kaneko [15]. arXiv:2106.05585v1 [math.CO] 10 Jun 2021 2. Poly-Bernoulli polynomials Poly-Bernoulli polynomials were defined in different ways in the literature [3, 11, 21]. We consider in this paper the definition of Bayad and Hamahata [3]. Date: June 11, 2021. 2010 Mathematics Subject Classification. 05A05, 05A19, 11B68. Key words and phrases. Poly-Bernoulli polynomials, poly-Euler numbers. The second author was supported by JSPS KAKENHI Grant Number 20K14292. 1 2 BEATA´ BENYI´ AND TOSHIKI MATSUSAKA (k) ∞ Definition 2.1. For every integer k the polynomials (Bn (x))n=0 are called poly-Bernoulli polynomials and are defined by ∞ tn Li (1 e−t) B(k)(x) = k − ext, n n! 1 e−t n=0 X − ∞ zi where Lik(z)= i=1 ik is the k-th polylogarithm function. (k) (k) The numbers PBn := Bn (0) are the poly-Bernoulli numbers, introduced by Kaneko [14] and studied since then extensively [1, 4, 5, 15, 16, 29]. On the other hand, ( 1)nB(1)( x)= B (x) − n − n are the classical Bernoulli polynomials. We describe two ways to interpret these polynomials based on the trivial interpretation of poly-Bernoulli numbers, Callan sequences and on two generalizations of Stirling numbers of the second kind. Actually, our models are straightforward combinations of known results and basic techniques in combinatorics. In the first model, x is assumed to be a positive integer, while in the second x marks a weight, hence, it is a formal variable, being arbitrary. These two different approaches are general in combinatorics. One gives a very concrete picture, the other allows more generality, greater freedom of abstraction. Both approaches have their own advantages. Let us recall now the definition of Callan sequences that are enumerated by the poly- Bernoulli numbers with negative k indices. Let N = 1,..., n (referred to as red elements) and K = 1,..., k (referred { }∪{∗} ∗ { }∪{∗} to as blue elements). Let R1,...,Rm, R be a partition of the set N into m + 1 non-empty ∗ blocks (0 m n) and B1,...,Bm, B a partition of the set K into m + 1 non-empty blocks. The≤ blocks≤ containing and are denoted by B∗ and R∗, respectively. We call B∗ and R∗ extra blocks, while the∗ other∗ blocks ordinary blocks. We call a pair of a blue and a red block, (Bi, Ri) for an i a Callan pair. A Callan sequence of size n k (or an (n, k)-Callan sequence) is a linear arrangement of Callan pairs augmented by× the extra pair (B , R )(B , R ) (B , R ) (B∗, R∗). 1 1 2 2 ··· m m ∪ (−k) (−k) It is known that the poly-Bernoulli number Bn = Bn (0) counts Callan sequences of size n k. × 2.1. Extended Callan sequences. This generalization is motivated by the so called r- Stirling numbers studied first by Broder [7]. The r-Stirling numbers of the second kind, n count partitions of [n + r]= 1, 2,...,n,n +1,...,n + r into k + r non-empty blocks k r { } such that the elements 1, 2,...,r are in distinct blocks. The generating function is given by { } ∞ n tn (et 1)k = ert − . k n! k! n=0 r X EXTENSIONS OF THE COMBINATORICS OF POLY-BERNOULLI NUMBERS 3 The r-Stirling numbers of the second kind and related r-generalizations are well studied, see for instance the book of Mez˝o[30] and the references therein. We extend the Callan sequences with some blocks of red elements that may be empty. Definition 2.2. An r-extended Callan sequence of size n k is a Callan sequence of size n k extended by r labeled possible non-empty blocks of× red elements. These blocks are called× special blocks. One could think on this extension as adding to the set N r marking elements, and saying that we partition the red elements into m + r + 1 blocks, where the r marking elements, n +1, n +2,..., n + r and are in distinct blocks, and the blocks that do not include any{ of these elements are} used{∗ for} creating the ordinary Callan pairs. Example 2.3. Let n = 9, k = 6, r = 3. The first special block is 5, 10 , the second is 11 , the third is 2, 7, 12 , and then an 3-extended Callan sequence{ is } { } { } (5, 6, 8)(3, 6, 4, 9)(1, 2, 4, 3)( , 1, ) 5, 10 11 2, 7, 12. ∗ ∗ | | | From the definition Theorem 2.4 is straightforward. Theorem 2.4. Let (n, k; r) denote the set of r-extended Callan sequences of size n k. Then we have EC × B(−k)(r)= (n, k; r) . n |EC | Proof. An r-extended Callan sequence of size n k has the form of × (B , R ) (B , R ) (B∗, R∗) (R , n +1) (R , n + r) 1 1 ··· m m ∪ ∪ 1 ∪···∪ r with 0 m n. Note that the blocks R1,..., Rr could be empty. Choose j red elements ≤ ≤ n j and arrange them in the special blocks in j r ways. The remaining elements form Callan (−k) sequences in Bn−j (0) ways as shown in [5, Theorem 5]. Thus, we have n n (2.1) (n, k; r) = B(−k)(0)rj, |EC | j n−j j=0 X (−k) which coincides with the formula for Bn (r) given in [17, p.204]. The combinatorial proofs of the following theorems easily follow. Theorem 2.5. [18] For n,k,r 0, ≥ min(n,k) k +1 n (2.2) B(−k)(r)= (m!)2 , n m +1 m m=0 r+1 X Proof. We partition N n +1,..., n + r into m+r+1 blocks such that , n +1,..., n + r are in distinct blocks and∪{ K into m + 1 blocks.} The blocks with and ∗will be the extra pair, the m blue blocks with the m red blocks that do not contain∗ any∗ special elements, will create the ordinary pairs. The blocks containing the special elements are the special blocks in the r-extended Callan sequence. 4 BEATA´ BENYI´ AND TOSHIKI MATSUSAKA Theorem 2.6. [18] For n,k,r 0, ≥ k k (2.3) B(−k)(r)= ( 1)k+mm!(m + r + 1)n. n m − m=0 X Proof. We enumerate r-extended Callan sequences having at most m ordinary blocks. Par- tition first the set 1, 2,..., k into m blocks and permute these blocks. Add after the sequence of these blocks{ the element} and the red elements n +1, n +2,..., n + r in this order. There are m + r + 1 objects∗ in this arrangements.{ Now insert the elements} 1, 2,..., n after any of these objects. The number of constructing so a sequence is { k } n m m!(m + r +1) . Using the inclusion-exclusion principle we obtain Equation (2.3). We do not include any further proofs explicitly here, but encourage the interested reader to find combinatorial proofs for other identities involving poly-Bernoulli polynomials. Instead, we recall a function defined by Bayad–Hamahata [3], k k C(−k)(x, y)= B(−j)(x)yk−j. n j n j=0 X After a short thought it should be clear that the function symmetrize in some sense the (−k) roles of n and k. For r and s positive integers, Cn (r, s) enumerates (r, s)-extended Callan sequences that we define as Callan sequences extended not only by red special blocks, but also by blue special blocks. Theorem 2.7 is immediate. Theorem 2.7. For n,k,r,s non-negative integers, we have min(n,k) n k C(−k)(r, s)= (m!)2 . n m m m=0 r+1 s+1 X The duality Theorem 1.7, the inversion formula Theorem 1.8, and the closed formula Theorem 1.9 in [3] follow from this combinatorial interpretation, we left the details to the reader. 2.2. Abundant Callan sequences. In this interpretation we rely on Carlitz’s weighted Stirling numbers [9, 10]. We redefine combinatorially these polynomials the following way.
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