Poly-Bernoulli Numbers and Eulerian Numbers

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1 Journal of Integer Sequences, Vol. 21 (2018, Article Poly-Bernoulli Nubers and Eulerian Nubers Beáta Bényi Faculty of Water Sciences National University of Public Service H-1441 Budapest, P.O. Box 60 Hungary beata.benyi@gail.co Péter Hanal Bolyai Institute University of Szeged H-6720 Szeged, Dugonics square 13 Hungary and Alfréd Rényi Institute of Matheatics Hungarian Acadey of Sciences 1245 Budapest, P.O. Box 1000 Hungary hanal@ath.u-szeged.hu Abstract In this note we prove, using cobinatorial arguents, soe new forulas connecting poly-bernoulli nubers with negative indices to Eulerian nubers. 1

2 1 Introduction Kaneko [10] introduced the poly-bernoulli nubers A during his investigations on ultiple zeta values. He defined these nubers by their generating function: B n (k x n n! = Li k(1 e x, (1 1 e x where n=0 Li k (z = is the classical polylogarithic function. As the nae indicates, poly-bernoulli nubers are generalizations of the Bernoulli nubers. For k = 1 B n (1 are the classical Bernoulli nubers with B 1 = 1. For negative k-indices poly-bernoulli nubers are integers (see the values for 2 sall n, k in Table 1 and have interesting cobinatorial properties. i=1 k 0 1 n z i i k Table 1: The poly-bernoulli nubers B ( k n Poly-Bernoulli nubers enuerate several cobinatorial obects arisen in different research areas, such as lonesu atrices, Γ-free atrices, acyclic orientations of coplete bipartite graphs, alternative tableaux with rectangular shape, perutations with restriction on the distance between positions and values, perutations with excedance set [k], etc. In [3, 4] the authors suarize the known interpretations, present connecting biections and give further references. In this note we are concerned only with poly-bernoulli nubers with negative indices. For convenience, we let B n,k denote the poly-bernoulli nubers B n ( k. Kaneko derived two forulas for the poly-bernoulli nubers with negative indices: a forula that we call the basic forula, and an inclusion-exclusion type forula. The basic forula is B n,k = in(n,k (! 2 { n }{ } k +1, (2 +1

3 where { n k} denotes the Stirling nuber of the second kind A that counts the nuber of partitions of an n-eleent set into k non-epty blocks [9]. The inclusion-exclusion type forula is { } n B n,k = ( 1 n+! (+1 k. (3 n=0 Kaneko s proofs were algebraic, based on anipulations of generating functions. The first cobinatorial investigation of poly-bernoulli nubers was done by Brewbaker [7]. He defined B n,k as the nuber of lonesu atrices of size n k. He proved cobinatorially both forulas; hence, he proved the equivalence of the algebraic definition and the cobinatorial one. Bayad and Haahata [2] introduced poly-bernoulli polynoials by the following generating function: B n (k (x tn n! = Li k(1 e t e xt. 1 e t n=0 For negative indices the polylogarithic function converges for z < 1 and equals to k k =0 z k Li k (z =, (4 (1 z k+1 where k is the Eulerian nuber [9] A given, for instance, by = i=0 ( 1 i ( k +1 i In [2] the authors used analytical ethods to show that for k 0 B (k n (x = k =0 k k ( i k. (5 ( k ( 1 (x+ k 1 n. (6 The evaluation of (6 at x = 0 leads to a new explicit forula of the poly-bernoulli nubers involving Eulerian nubers. Theore 1. [2] For all k > 0 and n > 0 we have B n,k = =0 k ( k ( 1 (k +1 n. (7 We see that the Eulerian nubers and the defining generating function of poly-bernoulli nubers for negative k are strongly related. In this note we prove this forula purely cobinatorially. Moreover, we show four further new forulas for poly-bernoulli nubers involving Eulerian nubers. 3

4 2 Main results Inourproofsaspecialclassofperutationsplaysthekeyrole. Wecallthisperutationclass Callan perutations because Callan introduced this class as a cobinatorial interpretation of the poly-bernoulli nubers [8]. We use the well-known notation [N] for {1,2,...,N}. Definition 2. Callan perutation of [n+k] is a perutation such that each substring whose support belongs to N = {1,2,...,n} or K = {n+1,n+2,...,n+k} is increasing. Let C k n denote the set of Callan perutations of [n+k]. We call the eleents in N the left-value eleents and the eleents in K the right-value eleents. For instance, for n = 2 and k = 2, the Callan perutations are (writing the left-value eleents in red, right-value eleents in blue as follows: 1234, 1324, 1423, 1342, 2314, 2413, 2341, 3124, 3142, 3241, 3412, 4123, 4132, It is easy to see that Callan perutations are enuerated by the poly-bernoulli nubers, but for the sake of copleteness, we recall a sketch of the proof. Theore 3. [8] in(n,k { }{ } n+1 k +1 (! 2 = B n,k Proof. (Sketch We extend our universe with 0, a special left-value eleent, and n+k +1, a special right-value eleent. Define N = N {0} and K = K {n+k +1}. Let π C k n. Let π = 0π(n+k +1. Divide π into axial blocks of consecutive eleents in such a way that each block is a subset of N (left blocks or a subset of K (right blocks. The partition starts with a left block (the block of 0 and ends with a right block (the block of (n+k +1. So the left and right blocks alternate, and their nuber is the sae, say +1. Describing a Callan perutation is equivalent to specifying, a partition Π N of N into +1 classes (one class is the class of 0, the other classes are called ordinary classes, a partition Π K of K into +1 classes ( of the not containing (n+k +1, these are the ordinary classes, and two orderings of the ordinary classes. After specifying the coponents, we need to erge the two ordered set of classes (starting with the nonordinary class of N and ending with the nonordinary class of K, and list the eleents of classes in increasing order. The classes of our partitions will for the blocks of the Callan perutations. We will refer to the blocks coing fro ordinary classes as ordinary blocks. This proves the clai. The ain results of this note are the next five forulas for the nuber of Callan perutations and hence, for the poly-bernoulli nubers. We present eleentary cobinatorial proofs of the theores in the next section. Theore 8 is equivalent to Theore 1; we recall the theore in the cobinatorial setting. 4

5 Theore 4. For all k > 0 and n > 0 the identity holds. in(n,k n i=0 =0 n i k ( n+1 i +1 i Theore 5. For all k > 0 and n > 0 the identity holds. =0 k k+2 ( k +2 Theore 6. For all k > 0 and n > 0 the identity holds. =0 Theore 7. For all k > 0 and n > 0 the identity holds. =0 ( k +1 = B n,k (8 +1 { } n (+ 1! + 1 = B n,k (9 1 ( 1 ( 1 (k +1 n = B n,k (10 +1 ( +1 (+k! Theore 8. [2] For all k > 0 and n > 0 the identity holds. =0 k ( k ( 1 { } n +k = B n,k (11 (k +1 n = B n,k. (12 3 Proofs of the ain results Eulerian nubers play the crucial role in these forulas. Though Eulerian nubers are well-known, we think it could be helpful for readers who are not so failiar with this topic to recall soe basic cobinatorial properties. 5

6 3.1 Eulerian nubers First, we need soe definitions and notation. Let π = π 1 π 2 π n be a perutation of [n]. We call i [n 1] a descent (resp., ascent of π if π i > π i+1 (resp., π i < π i+1. Let D(π (resp., A(π denote the set of descents (resp., the set of ascents of the perutation π. For instance, π = has 3 descents and D(π = {2,5,7}, while it has 5 ascents and A(π = {1,3,4,6,8}. Eulerian nubers k counts the perutations of [k] with 1 descents. A perutation π S n with 1 descents is the union of increasing subsequences of consecutive entries, so called ascending runs. So, in other words k is the nuber of perutations of [k] with ascending runs. In our exaple, π is the union of 4 ascending runs: 36, 148, 79, and 25. There are several identities involving Eulerian nubers, see, for instance, [6, 9]. We will use a strong connection between the surections/ordered partitions and Eulerian nubers: { } k r! = r r =0 k ( k r. (13 Proof. We take all the partitions of [k] into r classes. Order the classes, and list the eleents in increasing order. This way we obtain perutations of [k]. Counting by ultiplicity we get r! { k r} perutations. All of these have at ost r ascending runs. Take a perutation with ( r ascending runs. How any ties did we list it in the previous paragraph? We split the ascending runs by choosing r places out of the k ascents to obtain all the initial r blocks. The ultiplicity is ( k r. This proves our clai. Inverting (13 iediately gives = { k ( 1 r r! r r=1 }( k r r In the previous section we entioned the close relation between Eulerian nubers and the polylogarithic function Li k (x. Here we recall one possible derivation of the identity (4 following [6]. =0 x = = =0 i=0 x = l=0 =0 i=0 ( 1 i ( k +1 i = (1 x k+1 Li k (x. i=0. ( k +1 ( 1 i ( i k x = i ( k +1 l k x i+l = ( 1 i x i l k x l i Substituting in (5 for k, changing the order of the suation, using the transforation l = i, and finally applying the binoial theore, we get the result. 6 l=0

7 3.2 Cobinatorial proofs of the theores We turn our attention now to the proofs of our theores. For the sake of convenience, thanks to our color coding (left-value eleents are red, and right-value eleents are blue, werewritethesetofright-valueeleentsask = {1,2,...,k}, and K = K {k +1}. Wecan do this without essentially changing Callan perutations, since we ust need the distinction between the eleents N and K and an order in N and K. If we separately consider the left-value eleents and right-value eleents in the perutation π, the eleents of N for a perutation of [n], and the eleents of K for a perutation of [k]. We let π r denote the perutation restricted to the right-value eleents, and we let π l denote the perutation restricted to the left-value eleents. Further, let π r and π l denote the extended versions of the perutations. For instance, for π = , we have π r = , while π l = Proof. We consider the last entries of the blocks in the restricted perutations π l (resp., π r. Soe of the blocks end with a descent and soe of the blocks do not. (The special eleents 0 and k +1 are neither descents nor ascents of the perutations. Let i be the nuber of ascending runs in π l and be the nuber of ascending runs in π r. Further, let be the nuber of ordinary blocks of both types. The i 1 descents of π l deterine i last eleents of blocks; hence, we are issing (i 1 blocks with an ascent as last eleent. Siilarly, the 1 descents of π r deterine 1 blocks and there are ( 1 additional blocks with an ascent as last eleent. Given a pair ( π l, π r with D( π l = i 1 and D( π r = 1, we can construct a Callan perutation, according to the above arguents. In our running exaple, π l = , we need to choose 3 2 = 1 fro 9 2 possibilities. In general, we need to choose (i 1 (as last eleents of blocks fro n (i 1 possibilities. And analogously for π r, we need to choose ( 1 fro k ( 1 possibilities. Hence, for a given pair ( π l, π r with D( π l = i 1 and D( π r = 1 we have ( ( n+1 i k i +1 differentcorrespondingcallanperutations. Sincethenuberofpairs( π l, π r with D( π l = i 1 and D( π r = 1 is n i k The identity (8 is proven. Note that (8 is actually a rewriting of the basic cobinatorial forula (2 in ters of Eulerian nubers using the relation (13 between the nuber of ordered partitions and Eulerian nubers. NowweenuerateCallanperutationsaccordingtothenuberofdescentsinπ r. Given a perutation π r with 1 descents, we deterine the nuber of ways to erge π r with left-value eleents to obtain a valid Callan perutation. Let D = {d 1,d 2,...,d 1 } be the set of descents of π r. In our running exaple, = 3 and D = {3,5}. We code the positions of the left-values coparing to the right-values by a word w. We let w i be the nuber of 7

8 right-values that are to the left of the left-value i. In our exaple, w 1 = 5, since there are 5 right-value eleents preceding the left-value 1, w 2 = 0, because there are no right-value eleents preceding the left-value 2, etc. Hence, w = Note that the blocks of the left-value eleents can be recognized fro the word: The positions i, for which the values w i are the sae, are the eleents of the sae block. We call a word valid with respect to π r if the augentation of π r according to the word w leads to a valid Callan perutation. Observation 9. A word w is valid with respect to a perutation π r if and only if it contains every value d i of the descent set of π r. Proof. In a Callan perutation the substrings restricted to K or N are increasing subsequences. Given π r with descent set D, each d i D has to be the last eleent of a block in the ordered partition of the set K, the set of right-value eleents. Hence, each right-value with position d i in π r has to be followed by a left-value eleent in the Callan perutation. In our word w at the position of this left-value eleent there is a d i. For the converse, assue that our word w contains at least one d i, for any d i D(π r. There is at least one left-value eleent with d i in w at its position. This iplies that if we cobine π r and π l then in π r the position of the descent will be interrupted by a left-value eleent. The cobined perutation will be a Callan perutation. Corollary 10. The nuber of valid words with respect to π r depends only on the nuber of descents in π r. We let w 1 denote a word that is valid to a π r with 1 descents and W(π r denote the set of words w 1. The nuber of Callan perutations of size n + k is the nuber of pairs (π r,w 1, where π r is a perutation of [k] with 1 descents and w 1 W(π r. We denote W(π r by w( 1. Hence, =1 w( 1. The next two proofs are based on two different ways of deterining w( 1, i.e., enuerating those w 1 s that are valid to a π r with 1 descents. Proof. Fix π r and take a w 1 W(π r. Then w 1 corresponds to an ordered partition of [n] into at least 1 blocks. Let 1+ be the nuber of the blocks. First, we take an ordered partition of {1,2,...,n } into + 1 non-epty blocks in (+ 1! { n + 1} ways. Then we refine the partition of π r, defined by the descents. For the refineent we need to choose additional places for the blocks. These places can be before the first eleent of π r, or at an ascent. We have ( k+2 choices. This proves (9. Proof. Now we calculate w( 1 using the inclusion-exclusion principle. The total nuber of words of length n with entries {0,1,...,k} (w i = 0 if the left-value i is in the first block of the Callan perutation is (k + 1 n. We have to reduce this nuber with the nuber 8

9 of invalid words with respect to π r, with the words that do not contain at least one of the d i D. Let A s be the set of words that do not contain the value s. The quantity s D A s is to be deterined. Clearly, A s = (k n and this nuber does not depend on the choice of s; hence, we have s D A s = k n ( 1. Then A s A t = (k n and s,t D A s A t = (k 1 n( 1 2. Analogously, l=1 A sl = (k + 1 n( 1. The inclusion-exclusion principle gives and this iplies (10. 1 ( 1 w( 1 = ( 1 (k +1 n, Proof. Finally, the identities (11 and (12 follow by the syetry of Eulerian nubers. If we reverse a perutation of [k] with 1 descents we obtain a perutation with k ( 1 1 descents. According to our previous arguents a pair (π r,w k, where π r is a perutation with k descents and w k is a valid word with respect to π r deterines a Callan perutation. Hence, k w(k = k +1 =1 We have two forulas for w(k. w(k = This iplies (11 and ( ( +1 =1 (+k! { w(k. } n, +k k ( k w(k = ( 1 (k +1 n. 4 Acknowledgents The authors thank the referee for carefully reading the anuscript and providing helpful suggestions. References [1] T. Arakawa, T. Ibukiyaa, and M. Kaneko, Bernoulli Nubers and Zeta Functions, Springer Monographs in Matheatics, Springer,

10 [2] A. Bayad and Y. Haahata, Polylogariths and poly-bernoulli polynoials, Kyushu J. Math. 65 (2011, [3] B. Bényi and P. Hanal, Cobinatorics of poly-bernoulli nubers, Studia Sci. Math. Hungarica 52 (2015, [4] B. Bényi and P. Hanal, Cobinatorial properties of poly-bernoulli relatives, Integers 17 (2017, A31. Available at [5] B. Bényi and G. V. Nagy, Biective enuerations of Γ-free 0-1 atrices, Adv. Appl. Math. 96 (2018, [6] M. Bóna, Cobinatorics of Perutations, Discrete Matheatics and its Applications, Chapan Hall/CRC, [7] C. R. Brewbaker, A cobinatorial interpretation of the poly-bernoulli nubers and two Ferat analogues, Integers 8 (2008, A02. Available at [8] D. Callan, Coent on sequence A099594, in [11]. [9] L. Graha, D. Knuth, and O. Patashnik, Concrete Matheatics, Addison-Wesley, [10] M. Kaneko, Poly-Bernoulli nubers, J. Théor. Nobres Bordeaux 9 (1997, [11] N. J. A. Sloane, The on-line encyclopedia of integer sequences, Matheatics Subect Classification: Priary 05A05; Secondary: 05A15, 05A19, 11B83. Keywords: cobinatorial identity, Eulerian nuber, poly-bernoulli nuber. (Concerned with sequences A008277, A008282, and A Received April ; revised versions received April ; July Published in Journal of Integer Sequences, July Return to Journal of Integer Sequences hoe page. 10