Generalized Bell polynomials and the combinatorics of Poisson central moments

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1 Generaized Be poynomias and the combinatorics of Poisson centra moments Nicoas Privaut Division of Mathematica Sciences Schoo of Physica and Mathematica Sciences Nanyang Technoogica University SPMS-MAS-05-43, 21 Nanyang Lin Singapore February 25, 2011 Abstract We introduce a famiy of poynomias that generaizes the Be poynomias, in connection with the combinatorics of the centra moments of the Poisson distribution. We show that these poynomias are dua of the Charier poynomias by the Stiring transform, and we study the resuting combinatoria identities for the number of partitions of a set into subsets of size at east 2. Key words: Be poynomias, Poisson distribution, centra moments, Stiring numbers. Mathematics Subject Cassification: 11B73, 60E07. 1 Introduction The moments of the Poisson distribution are we-nown to be connected to the combinatorics of the Stiring and Be numbers. In particuar the Be poynomias B n (λ) satisfy the reation B n (λ) E λ [Z n ], n IN, (1.1) where Z is a Poisson random variabe with parameter λ > 0, and B n (1) S(n, c) (1.2) nprivaut@ntu.edu.sg c0 1

2 is the Be number of order n, i.e. the number of partitions of a set of n eements. In this paper we study the centra moments of the Poisson distribution, and we show that they can be expressed using the number of partitions of a set into subsets of size at east 2, in connection with an extension of the Be poynomias. Consider the above mentioned Be (or Touchard) poynomias B n (λ) defined by the exponentia generating function e λ(et 1) n0 n! B n(λ), (1.3) λ, t IR, cf. e.g of [4], and given by the Stiring transform B n (λ) λ c S(n, c), (1.4) c0 where S(n, c) 1 c! c c ( 1) c n (1.5) denotes the Stiring number of the second ind, i.e. the number of ways to partition a set of n objects into c non-empty subsets, cf. 1.8 of [7], Proposition 3.1 of [3] or 3.1 of [6], and Reation (1.2) above. In this note we define a two-parameter generaization of the Be poynomias, which is dua to the Charier poynomias by the Stiring transform. We study the ins of these poynomias with the combinatorics of Poisson centra moments, cf. Lemma 3.1, and as a byproduct we obtain the binomia identity S 2 (m, n) ( m ( 1) ) S(m, n ), (1.6) where S 2 (n, a) denotes the number of partitions of a set of size n into a subsets of size at east 2, cf. Coroary 3.2 beow, which is the binomia dua of the reation S(m, n) m S 2 (m, n ), 2

3 cf. Proposition 3.3 beow. We proceed as foows. Section 2 contains the definition of our extension of the Be poynomias. In Section 3 we study the properties of the poynomias using the Poisson centra moments, and we derive Reation (1.6) as a coroary. Finay in Section 4 we state the connection between these poynomias and the Charier poynomias via the Stiring transform. 2 An extension of the Be poynomias We et (B n (x, λ)) n IN denote the famiy of poynomias defined by the exponentia generating function e ty λ(et t 1) n0 n! B n(y, λ), λ, y, t IR. (2.1) Ceary from (1.3) and (2.1), the definition of B n (x, λ) generaizes that of the Be poynomias B n (λ), in that When λ > 0, Reation (2.1) can be written as which yieds the reation B n (λ) B n (λ, λ), λ IR. (2.2) e ty IE λ [e t(z λ) ] n0 n! B n(y, λ), y, t IR, B n (y, λ) E λ [(Z + y λ) n ], λ, y IR, n IN, (2.3) which is anaog to (1.1), and shows the foowing proposition. Proposition 2.1 For a n IN we have B n (y, λ) (y λ) n λ i S(, i), y, λ IR, n IN. (2.4) 3

4 Proof. Indeed, by (2.3) we have B n (y, λ) E λ [(Z + y λ) n ], (y λ) n E λ [Z ] (y λ) n B (λ) (y λ) n λ i S(, i), y, λ IR. 3 Combinatorics of the Poisson centra moments As noted in (1.1) above, the connection between Poisson moments and poynomias is we understood, however the Poisson centra moments seem to have received ess attention. In the seque we wi need the foowing emma, which expresses the centra moments of a Poisson random variabe using the number S 2 (n, b) of partitions of a set of size n into b subsets with no singetons. Lemma 3.1 Let Z be a Poisson random variabe with intensity λ > 0. We have Proof. B n (0, λ) E λ [(Z λ) n ] λ a S 2 (n, a), n IN. (3.1) a0 We start by showing the recurrence reation n 1 [ E λ [(Z λ) n+1 ] λ E ] λ (Z λ) i, n IN, (3.2) i for Z a Poisson random variabe with intensity λ. We have E λ [(Z λ) n+1 ] e λ λ ( λ)n+1! 4

5 e λ λ ( 1)! ( λ)n λe λ 1 λe λ λe λ λ n 1! n 1 λe λ i n 1 λ i λ! (( + 1 λ)n ( λ) n ) ( λ) i i λ ( λ)i! ) E λ [(Z λ) i ]. λ ( λ)n! Next, we show that the identity n 1 E λ [(Z λ) n ] λ a a +1 1 a1 0 1 a+1 n 1 (3.3) hods for a n 1, where a b means a < b 1. Note that the degree of (3.3) in λ is the argest integer d such that 2d n, hence it equas n/2 or (n 1)/2 according to the parity of n. Ceary, the identity (3.3) is vaid when n 1 and when n 2. Assuming that it hods up to the ran n 2, from (3.2) we have n 1 [ E λ [(Z λ) n+1 ] λ E ] λ (Z λ) n 1 [ λ + λ E ] λ (Z λ) 1 n 1 1 λ + λ λ b 1 1 b1 n 1 n λ + λ λ + λ n 1 b 1 b b2 ) b b2 5 λ b b+1 1 λ b b b b b

6 n 1 λ + λ λ + λ λ + b1 b 1 b2 b λ b 1 b b 1 b2 b2 λ b λ b λ b b b+1 n b b+1 n b+1 n b+1 n ( +1 1 ), and it remains to note that 0 1 b+1 n +1 1 S 2 (n, b) (3.4) 1 equas the number S 2 (n, b) of partitions of a set of size n into b subsets of size at east 2. Indeed, any contiguous such partition is determined by a sequence of b 1 integers 2,..., b with 2b n and 0 2 b n so that subse o i has size i+1 i 2, i 1,..., b, with b+1 n, and the number of noecessariy contiguous partitions of that size can be computed inductivey on i 1,..., b as 1 n 1 b )( b 1 b 1 b 1 ) For this, at each step we pic an eement which acts as a boundary point in the subset n o i, and we do not count it in the possibe arrangements of the remaining i+1 1 i eements among i+1 1 paces. Lemma 3.1 and (3.4) can aso be recovered by use of the cumuants (κ n ) n 1 of Z λ, defined from the cumuant generating function og E λ [e t(z λ) ] λ(e t 1) i.e. κ 1 0 and κ n λ, n 2, which shows that E λ [(Z λ) n ] a1 n1 1 κ n n!, B 1,...,B a κ B1 κ Ba, 6

7 where the sum runs over the partitions B 1,..., B a of {1,..., n} with cardina B i by the Faà di Bruno formua, cf. 2.4 of [5]. Since κ 1 0 the sum runs over the partitions with cardina B i at east equa to 2, which recovers E λ [(Z λ) n ] λ a S 2 (n, a), (3.5) a1 and provides another proof of (3.4). In addition, (3.2) can be seen as a consequence of a genera recurrence reation between moments and cumuants, cf. Reation (5) of [8]. In particuar when λ 1, (3.1) shows that the centra moment B n (0, 1) E 1 [(Z 1) n ] S 2 (n, a) (3.6) is the number of partitions of a set of size n into subsets of size at east 2, as a counterpart to (1.2). a0 By (2.3) we have B n (y, λ) y n E λ [(Z λ) ] y IR, λ > 0, n IN, hence Lemma 3.1 shows that we have B n (y, λ) y n y n B (0, λ), λ c S 2 (, c), λ, y IR, n IN. (3.7) c0 As a consequence of Reations (2.4) and (3.7) we obtain the foowing binomia identity. Coroary 3.2 We have Proof. S 2 (n, c) c ( 1) S(n, c ), 0 c n. (3.8) By Reation (2.4) we have B n (y, λ) n )(y λ) λ i S(n, i) 7

8 ) ( n )y ( λ) λ i S(n, i) ) n )y ( λ) λ i S(n, i) n ) n )y ( λ) λ i S(n, i) n ) b y ( λ) n b λ i S(b, i) b )( b )y n ( λ) b λ i S( b, i) b ) y n ( λ) b λ c b S( b, c b) b n b0 b0 b0 ) y n c0 and we concude by Reation (3.7). λ c cb c ( 1) b S( b, c b), b b0 y, λ IR, As a consequence of (3.7) and (3.8) we have the identity c B n (0, λ) E λ [(Z λ) n ] λ c ( 1) a S(n a, c a), a for the centra moments of a Poisson random variabe Z with intensity λ > 0. c0 a0 The foowing proposition, which is the inversion formua of (3.8) has a natura interpretation by recaing that S 2 (m, b) is the number of partitions of a set of m eements made of b sets of cardina greater or equa to 2, as wi be seen in Proposition 3.4 beow. Proposition 3.3 We have the combinatoria identity b S(n, b) S 2 (n, b ), b, n IN. (3.9) Proof. By Reation (3.7) we have B n (λ) B n (λ, λ) 8

9 and we concude from (1.4). b0 λ n λ b b λ b S 2 (, b) b0 S 2 (n, b ), Reation (3.9) is in fact a particuar case for a 0 of the identity proved in the next proposition, since S( c, 0) 1 {c}. Proposition 3.4 For a a, b, n IN we have a + b b S(n, a + b) S( c, a)s 2 (n, b c). a c Proof. c0 c The partitions of {1,..., n} made of a+b subsets are abeed using a possibes vaues of {0, 1,..., n} and c {0, 1,..., }, as foows. For every {0, 1,..., n} and c {0, 1,..., } we decompose {1,..., n} into a subset ( 1,..., ) of {1,..., n} with ) possibiities, c singetons within ( 1,..., ), i.e. ( c) possibiities, a remaining subset of ( 1,..., ) of size c, which is partitioned into a IN (non-empty) subsets, i.e. S( c, a) possibiities, and a remaining set {1,..., n} \ ( 1,..., ) of size n which is partitioned into b c subsets of size at east 2, i.e. S 2 (n, b c) possibiities. In this process the b subsets mentioned above were counted ( with) their combinations a + b within a + b sets, which expains the binomia coefficient on the right-hand a side. 4 Stiring transform In this section we consider the Charier poynomias C n (x, λ) of degree n IN, with exponentia generating function e λt (1 + t) x n0 n! C n(x, λ), x, t, λ IR, 9

10 and C n (x, λ) cf. 3.3 of [7], where x s(, ) 1! ( λ) n s(, ), x, λ IR, (4.1) ( 1) i ( i) i is the Stiring number of the first ind, cf. page 824 of [1], i.e. ( 1) s(, ) is the number of permutations of eements which contain exacty permutation cyces, n IN. In the next proposition we show that the Charier poynomias C n (x, λ) are dua to the generaized Be poynomias B n (x λ, λ) defined in (2.1) under the Stiring transform. Proposition 4.1 We have the reations C n (y, λ) s(n, )B (y λ, λ) and B n (y, λ) S(n, )C (y + λ, λ), y, λ IR, n IN. Proof. For the first reation, for a fixed y, λ IR we et with A(t) e λt (1 + t) y+λ n0 A(e t 1) e t(y+λ) λ(et 1) n! C n(y + λ, λ), t IR, n0 n! B n(y, λ), t IR, and we concude from Lemma 4.2 beow. The second part can be proved by inversion using Stiring numbers of the first ind, as S(n, )C (y + λ, λ) S(n, )s(, )B (y, λ) B (y, λ) B n (y, λ), S(n, )s(, ) 10

11 from the inversion formua S(n, )s(, ) 1 {n}, n, IN, (4.2) for Stiring numbers, cf. e.g. page 825 of [1]. Next we reca the foowing emma, cf. e.g. Reation (3) page 2 of [2], which has been used in Proposition 4.1 to show that the poynomias B n (y, λ) are connected to the Charier poynomias. Lemma 4.2 Assume that the function A(t) has the series expansion A(t) t! a, t IR. Then we have with A(e t 1) b n a S(n, ), t! b, t IR, Finay we note that from (2.4) we have the reation B n (y, y + λ) which paraes (4.1). References n IN. (y + λ) ( λ) n S(, ), y, λ IR, n IN, [1] M. Abramowitz and I.A. Stegun. Handboo of mathematica functions with formuas, graphs, and mathematica tabes, voume 55. Dover Pubications, New Yor, th Edition. [2] M. Bernstein and N. J. A. Soane. Some canonica sequences of integers. Linear Agebra App., 226/228:57 72, [3] K.N. Boyadzhiev. Exponentia poynomias, Stiring numbers, and evauation of some gamma integras. Abstr. App. Ana., pages Art. ID , 18, [4] C.A. Charaambides. Enumerative combinatorics. CRC Press Series on Discrete Mathematics and its Appications. Chapman & Ha/CRC, Boca Raton, FL, [5] E. Luacs. Characteristic functions. Hafner Pubishing Co., New Yor, Second edition, revised and enarged. 11

12 [6] E. Di Nardo and D. Senato. Umbra nature of the Poisson random variabes. In Agebraic combinatorics and computer science, pages Springer Itaia, Mian, [7] S. Roman. The umbra cacuus, voume 111 of Pure and Appied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Pubishers], New Yor, [8] P.J. Smith. A recursive formuation of the od probem of obtaining moments from cumuants and vice versa. Amer. Statist., 49(2): ,