The r-bell Numbers. 1 Introduction

Transcription

1 Jounal of Intege Sequences, Vol. 4 (, Aticle.. The -Bell Numbes István Meő Depatment of Applied Mathematics and Pobability Theoy Faculty of Infomatics Univesity of Debecen P. O. Box H-4 Debecen Hungay meo.istvan@inf.unideb.hu Abstact The notion of -Stiling numbes implies the definition of genealied Bell (o - Bell numbes. The -Bell numbes have appeaed in seveal wos, but thee is no systematic teatise on this topic. In this pape we fill this gap. We discuss the most impotant combinatoial, algebaic and analytic popeties of these numbes, which genealie simila popeties of the Bell numbes. Most of these esults seem to be new. It tuns out that in a pape of Whitehead, these numbes appeaed in a vey diffeent context. In addition, we study the so-called -Bell polynomials. Intoduction The Bell numbe B n [] counts the patitions of a set with n elements. The Stiling numbe with paametes n and, denoted by { n }, enumeates the numbe of patitions of a set with n elements consisting disjoint, nonempty sets. We get immediately that B n can be given by the sum { } n B n. ( The numbes { n } ae also called as Stiling patition numbes. The n-th Bell polynomial is B n (x { } n x.

2 These numbes and polynomials have many inteesting popeties and appea in seveal combinatoial identities. A compehensive pape is []. A moe geneal notion can be intoduced. The -Stiling numbe of the second ind with paametes n enumeates the patitions of a set of n elements into nonempty, disjoint subsets such that the fist elements ae in distinct subsets. It is denoted by { } n. A systematic teatment on the -Stiling numbes is given in [4], and a diffeent appoach is descibed in [6, 7]. Accoding to (, it seems to be natual to define the numbes B n, { } n +. ( + (It is obvious that B n B n,, because { { n } n } by the definitions. The vey fist question is on the meaning of the -Bell numbes. By (, B n, is the numbe of the patitions of a set with n + element such that the fist elements ae in distinct subsets in each patition. The name of -Stiling numbes suggests the name fo the numbes B n, : we call them as -Bell numbes, and the name of the polynomials B n, (x { } n + x + will be -Bell polynomials (see also the title of []. Thus B n, B n, ( and B n, (x B n (x, the odinay Bell polynomial.. Some elementay facts about the -Bell polynomials Actually, the coefficients of B n, (x ae polynomials in, since { } n + + i ( { } n i n i. (3 i That is, ( ( { n i B n, (x } n i x. (4 i i { The equality (3 can be poven easily: n+ } enumeates the ( + -patitions of n + + elements such that the fist elements ae in distinct subsets. The numbe of such patitions can be enumeated in the following way. We sepaate,..., into singletons, and we ceate additional blocs to have + blocs. To fill the blocs, we choose i elements fom { +,...,n + } into them. This can happen ( { n i way. We can constuct i } diffeent -patitions fom these elements. The emaining n i elements fom { +,...,n + } go beside the fist elements. We may choose these blocs independently, so we have n i possibilities. Finally we sum on i.

3 A consequence is that the -Bell polynomials can be expessed by the Bell polynomials: B n, (x ( n B n (x. To see the validity of this identity, just change the ode of the summations in (4. As fa as we now, this pape is the fist one fully devoted to the -Bell numbes, although Calit [6, 7] defined these numbes and poved some identities fo them. His oiginal notation was B(n, such that B n, B(n,. Example and tables The following example illuminates again the meaning of the -Bell numbes. By definition, { } { } { } B, { 4 } counts the patitions of 4 element into subsets such that the fist element ae in distinct subsets:. { 4 3} {, 3, 4}, {} ; {}, {, 3, 4} ; {, 3}, {, 4} ; {, 4}, {, 3}. belongs to the patitions {}, {}, {3, 4} ; {, 3}, {}, {4} ; {, 4}, {}, {3} ; {}, {, 3}, {4} ; {}, {, 4}, {3}. Finally, { } 4 equals to the numbe of patitions of 4 elements into 4 subsets (and necessaily, 4 the fist two elements ae in distinct subsets: That is, B, { } 4 + {}, {}, {3}, {4}. { } { } is the numbe of patitions of the set {,, 3, 4} such that the fist two elements ae in distinct subsets. 3 Geneating functions We stat to deive the popeties of -Bell numbes and polynomials. Fist of all, the geneating functions ae detemined. 3

4 Figue : The fist few -Bell numbes n n n n 3 n 4 n 5 n Figue : The fist few -Bell polynomials B, (x B, (x x + B, (x x + ( + x + B 3, (x x 3 + (3 + 3x + ( x + 3 B 4, (x x 4 + (4 + 6x 3 + ( x + ( x + 4 4

5 Theoem 3.. The exponential geneating function fo the -Bell polynomials is n B n, (x n n! ex(e +. Poof. Bode [4] gave the double geneating function of -Stiling numbes ( { n + }x n + n! + ex(e. n The inne sum is exactly ou polynomial B n, (x. We note that this identity is emaed in [6, eq. (3.9] We ema that the non-polynomial vesion was poven by Calit [6, eq. (3.8]. In ode to detemine the odinay geneating function we need some othe notions. The falling factoial of a given eal numbe x is denoted and defined by x n x(x (x (x n +, (n,,... (5 and (x, while the ising factoial (a..a. Pochhamme symbol is (x n x n x(x + (x + (x + n (n,,... (6 with (x. It is obvious that ( n n!. Fitting ou notations to the theoy of hypegeometic functions defined below, we apply the notation (x n instead of x n. The next tansfomation fomula holds x n ( n ( x n. (7 The hypegeometic function (o hypegeometic seies is defined by ( a, a,..., a p pf q b, b,..., b q t (a (a (a p t (b (b (b q!. The odinay geneating function of B n, (x can be given by this function. Theoem 3.. The -Bell polynomials have the geneating function B n, (x n e x F x. n Poof. It is nown [4] that fo the Stiling numbes n { } n n m This can be ewitten as { } n + m + nm ( + m ( ( ( + ( m n (m. m ( ( ( + ( (m +. 5

6 We tansfom the denominato using the falling factoial: Hence ( ( ( + ( (m + ( ( ( ( (m + m++ ( ( ( ( m+ (. m { } + m + Equality (7 and definitions (5-(6 give that Consequently, ( ( m++. ( m++ ( ( m++ m++ ( + + ( ( m++ + nm { } n + m + n Since ( ( + we get that nm { } n + n m + ( ( + (,. m ( ( m++ + m ( m ( + We multiply both sides by x m and tae summation ove the non-negative integes: B n, (x n n m ( x m ( + m m ( F.. + x. Finally, we apply Kumme s fomula [, p. 55] ( ( a e x F b a b x F b x with b + and a. 6

7 4 Basic ecuences In an ealie pape of the autho [8], the polynomials B n, (x wee intoduced because of a vey diffeent eason. These functions wee used to study the unimodality of -Stiling numbes and some popeties of them wee poven in that pape. We epeat those esults without poof. Theoem 4.. We have the following ecusive identities: ( d B n, (x x dx B n,(x + B n, (x + B n, (x, e x x B n, (x x d dx (ex x B n, (x. Moeove, all eos of B n, (x ae eal and negative. Staightfowad coollaies ae that fo a fixed the constant tem of the n-th polynomial is n : B n, ( n, and that the deivative of an -Bell polynomial is detemined by the elation The identity d dx B n,(x B n+,(x B n,(x B n, (x. x x { } n + + { } n + ( + was poven in [4, p. 45] and implies the ecuence elation { } n + + B n, (x xb n,+ (x + B n, (x. Theoem 4.. The next polynomial identity is valid: n ( n B n, (x B n, (x + x B, (x. Poof. We give a combinatoial poof fo the non-polynomial vesion (x. Fist we eaange the sum on the ight hand side: n ( n n ( n n B, B n, n Hence we need to pove that n ( n B n, B n, + B n,. 7 ( n B n,.

8 If we constuct patitions on n + elements and the fist elements ae in distinct blocs, then we have two possibilities: the last element, n +, belongs to a bloc containing one of the fist elements. Such patition can be constucted such that we constuct a patition of {,,...,n + } and then put the last element into the bloc containing o... o. We see that thee ae B n, possibilities. the last element belongs to a bloc not containing,,... and. Now we may choose othe elements fom { +,...,n + } into the bloc of n. Thee ae ( n ways to do this. Then the emaining n elements build up a patition (such that,..., ae in diffeent blocs. This can be done B n, ways. Last, we tae summation ove all the possible values of. Closing this section, we cite Calit s identities [6, eq. (3.-3.3]: Hee [ ] n m B n+m, B n,+m m j m j { } m + B n,+j, (8 j + ] ( m j [ m + j + B n+j,. is an -Stiling numbe of the fist ind (see [4, 6, 7]. 5 Dobinsi s fomula The Bell numbes ae involved in Dobinsi s nice fomula [9, 3, 4, 9]: B n e n!. Ou goal is to genealie this identity to ou case. Theoem 5. (Dobinsi s fomula. The -Bell polynomials satisfy the identity B n, (x ( + n x. e x! Consequently, the -Bell numbes ae given by B n, e ( + n.! Poof. The -Stiling numbes fo a fixed n (and have the hoiontal geneating function [4] { } n + (x + n x, + 8

9 whence, fo an abitay intege m, (m + n m! m { } n + + (m!. In the next step we multiply both sides by x m and sum fom m to. Then (m + n x m m! m e x ( m m { } n + + x m { } n + x e x B n, (x. + (m! We can detemine some inteesting sums with the aid of -Bell numbes. Fo example, we now fom the second paagaph that B,, so e ( +!. 6 An integal epesentation In 885, Cesào [8] found a emaable integal epesentation of the Bell numbes (see also [3, 5]: B n n! π πe Im e eeiθ sin(nθdθ. It is not had to deduce the -Bell vesion. Theoem 6.. The -Bell numbes have the integal epesentation B n, n! πe Im π e eeiθ e eiθ sin(nθdθ. Poof. In [6] we find that { } n +! + ( ( j (j + n. (9 j j In the next step we use the next equality [5]: Im π e jeiθ sin(nθdθ π j n n!. ( 9

10 Unifying equations (9 and (, we get that whence { } π n + n! +! j [ π (! Im ( j j π Im and the esult follows. j ( e eiθ! ( ( j j ( j] e eiθ e eiθ sin(nθdθ, { } n + n! π + π Im π Im e (j+eiθ sin(nθdθ ( e eiθ! e eiθ sin(nθdθ e eiθ sin(nθdθ, The imaginay pat of the above integal can be calculated with a bit of effot: B n, n! πe π e ecosθ cos sin θ+ cos θ [cos(e cos θ sin sin θ sin( sin θ + sin(e cos θ sin sin θ cos( sin θ ] sin(nθdθ. Without the -Bell numbes in bacgound, the evaluation of this integal seems to be impossible... Citing the geneal vesion of Dobinsi s fomula we find the compelling identity ( + n! n! π π Im e eeiθ e eiθ sin(nθdθ. 7 Hanel tansfomation and log-convexity Since e t n Cauchy s poduct immediately implies the next B n, (x tn n! ex(et +(+t, Theoem 7.. The -Bell polynomials satisfy the elations B n,+ (x B n, (x ( n ( n B, (x, ( n B,+ (x.

11 An inteesting coollay is connected to the Hanel tansfom. The H Hanel matix [6] of an intege sequence (a n is a a a a 3 a a a 3 a 4 H a a 3 a 4 a 5, while the Hanel matix of ode n, denoted by h n, is the uppe-left submatix of H of sie n n. The Hanel tansfom of the sequence (a n is again a sequence fomed by the deteminants of the matices h n. A notable esult of Aigne and Lenad [, 7] is that the Hanel tansfom of the Bell numbes is (!,!!,!!3!,..., that is, fo any fixed n, B B B B n B B B 3 B n+ n i!.... i B n B n+ B n+ B n We can detemine the Hanel tansfom of -Bell numbes easily. To each this aim, we ecall the next notion. If (a n is a sequence, then its binomial tansfom (b n is defined by the elation ( n b n ( n a, while the invese tansfom is a n ( n b. See the pape [] on these tansfomations, fo instance. A useful theoem of Layman [6] states that any intege sequence has the same Hanel tansfom as its binomial tansfom. Then Theoem 7. yields the next Coollay 7.. The -Bell numbes have the Hanel tansfom B, B, B, B n, B, B, B 3, B n+,.... B n, B n+, B n+, B n, Pofesso J. Cigle [] calculated moe geneal identities with espect to Hanel deteminants involving not only -Bell numbes but polynomials. We cite his unpublished esults hee. Let d(n, det(b i+j+, (x n i,j. Cigle s esults ae the following: d(n, x (n n!, n i! i

12 and d(n, x (n n! ( n x ( n. 8 Some occuences of the -Bell numbes Supisingly, the -Bell numbes tuned up in a table of Whitehead s pape []. In his table, the (n,i-enty is denoted by b n,i and it is the sum of the coefficients of the polynomial x i (x n i with espect to the so-called complete gaph base. A moe detailed desciption on this gaph theoetical notion can be found in the pape [] and the efeences theein. Ou -Bell numbes ae exactly the enties of that table, moe exactly, Fom this obsevation we get staightaway the next identity. Theoem 8.. We have fo all n that B n, b n+,n (n. ( B n, B n, + B n,+. Poof. Accoding to [], the enties b n,i satisfy the ecuence (n ib n,i + b n+,i b n+,i+. Then ( implies the statement. On the othe hand, this theoem is a special case of (8 but it is wothwhile to give a diffeent viewpoint. We note that the ow sum in the table of Whitehead can be expessed by the -Bell numbes, too. b n,i B i,n i. i Identification ( gives also that the -Bell numbes have meaning in the theoy of chomatic polynomials. Anothe occuence is the following. The -Bell numbes come fom a poblem on the maximum of -Stiling numbes (see [8]. The autho poved thee that all eos of the polynomial B n, (x ae eal. This implies that i { } { } { } n n n, + which is an impotant elation fo example in the theoy of combinatoial sequences. In addition, the maximiing index of -Stiling numbes of the second ind can be expessed appoximately by the -Bell numbes [8]. Namely, ( K Bn+, ( + <, B n,

13 whee K is the paamete, fo which { } n + K { } n + fo all, +,...,n +. We ema that (beside the papes cited above, thee ae othe aticles in which the -Bell numbes (at least implicitly appea. C. B. Cocino [] deals with the asymptotic popeties of these numbes. The pape of Hsu and Shiue [5] concens the Stiling-type pais. In that aticle a genealied Dobinsi fomula is pesented. 9 Acnowledgement I than Pofesso Cigle fo his suggestions and esults on Hanel deteminants of -Bell polynomials. I also appeciate that Jonathan Vos Post uploaded the table of -Bell numbes (see A3498 in []. Moeove, I would lie to than the efeee fo his/he useful suggestions and impovements. Refeences [] M. Abamowit and I. A. Stegun, eds., Handboo of Mathematical Functions with Fomulas, Gaphs, and Mathematical Tables (9th pinting, Dove, 97. [] M. Aigne, A chaacteiation of the Bell numbes, Discete Math. 5 (999, 7. [3] H. W. Bece and D. H. Bowne, Poblem E46 and solution, Ame. Math. Monthly 48 (94, [4] A. Z. Bode, The -Stiling numbes, Discete Math. 49 (984, [5] D. Callan, Cesao s integal fomula fo the Bell numbes (coected. [6] L. Calit, Weighted Stiling numbes of the fist and second ind I, Fibonacci Quat. 8 (98, [7] L. Calit, Weighted Stiling numbes of the fist and second ind II, Fibonacci Quat. 8 (98, [8] M. E. Cesào, Su une équation aux difféences melées, Nouv. Ann. Math. 4 (885, [9] S. Chowla and M. B. Nathanson, Mellin s fomula and some combinatoial identities, Monat. Math. 8 (976,

14 [] J. Cigle, Pesonal communication. [] L. Comtet, Advanced Combinatoics, D. Reidel, 974. [] C. B. Cocino, An asymptotic fomula fo the -Bell numbes, Matimyás Mat. 4 (, 9 8. [3] G. Dobińsi, Summiung de Reihe n m /n! fü m,, 3, 4, 5,..., Ach. fü Mat. und Physi 6 (877, [4] R. L. Gaham, D. E. Knuth, and O. Patashni, Concete Mathematics, Addison-Wesley, 994. [5] L. C. Hsu and P. J-S. Shiue, A unified appoach to genealied Stiling numbes, Adv. Appl. Math. (998, [6] J. W. Layman, The Hanel tansfom and some of its popeties, J. Intege Seq. Vol. 4 (, Aticle..5. [7] M. Gadne, Factal Music, Hypecads, and Moe...: Mathematical Receations fom Scientific Ameican Magaine, W. H. Feeman, 99, pp [8] I. Meő, On the maximum of -Stiling numbes, Adv. Appl. Math. 4 (8, [9] J. Pitman, Some pobabilistic aspects of set patitions, Ame. Math. Monthly 4 (997, 9. [] J. Riodan, Invese elations and combinatoial identities, Ame. Math. Monthly 7 (964, [] N. J. A. Sloane, The On-Line Encyclopedia of Intege Sequences. Published electonically at [] E. G. Whitehead, Stiling numbe identities fom chomatic polynomials, J. Combin. Theoy Se. A 4 (978, Mathematics Subject Classification: Pimay 5A8; Seconday 5A5. Keywods: Bell numbes, -Bell numbes, Stiling numbes, -Stiling numbes, Hanel deteminants, esticted patitions. (Concened with sequences A, A5493, A5494, A45379, and A3498. Received Novembe ; evised vesion eceived Decembe 9. Published in Jounal of Intege Sequences, Decembe 9. Retun to Jounal of Intege Sequences home page. 4