arXiv:1401.1625v1 [math.CO] 8 Jan 2014
AN EXPLICIT FORMULA FOR COMPUTING BELL NUMBERS IN TERMS OF LAH AND STIRLING NUMBERS FENG QI Abstract. In the paper, the author finds an explicit formula for computing Bell numbers in terms of Lah numbers and Stirling numbers of the second kind.
In combinatorics, Bell numbers, usually denoted by Bn for n ∈ {0} ∪ N, count the number of ways a set with n elements can be partitioned into disjoint and non-empty subsets. These numbers have been studied by mathematicians since the 19th century, and their roots go back to medieval Japan, but they are named after Eric Temple Bell, who wrote about them in the 1930s. Every Bell number Bn may be generated by ∞ X x Bk k ee −1 = x (1) k! k=0
or, equivalently, by
e
e−x −1
=
∞ X
(−1)k Bk
k=0
xk . k!
(2)
In combinatorics, Stirling numbers arise in a variety of combinatorics problems. They are introduced in the eighteen century by James Stirling. There are two kinds of Stirling numbers: Stirling numbers of the first and second kinds. Every Stirling number of the second kind, usually denoted by S(n, k), is the number of ways of partitioning a set of n elements into k nonempty subsets, may be computed by k 1 X i k S(n, k) = (k − i)n , (3) (−1) i k! i=0 and may be generated by
∞ X xn (ex − 1)k = S(n, k) , k! n!
k ∈ {0} ∪ N.
(4)
n=k
In combinatorics, Lah numbers, discovered by Ivo Lah in 1955 and usually denoted by L(n, k), count the number of ways a set of n elements can be partitioned into k nonempty linearly ordered subsets and have an explicit formula n − 1 n! L(n, k) = . (5) k − 1 k!
Lah numbers L(n, k) may also be interpreted as coefficients expressing rising factorials (x)n in terms of falling factorials hxin , where ( x(x + 1)(x + 2) · · · (x + n − 1), n ≥ 1, (x)n = (6) 1, n=0
2010 Mathematics Subject Classification. Primary 11B73; Secondary 11B75, 33B10, 33C15. Key words and phrases. explicit formula; Bell number; Lah number; Stirling number of the second kind; Kummer confluent hypergeometric function. This paper was typeset using AMS-LATEX. 1
2
F. QI
and hxin =
(
x(x − 1)(x − 2) · · · (x − n + 1), n ≥ 1, 1, n = 0.
(7)
In [4, Theorem 2] and its formally published paper [7, Theorem 2.2], the following explicit formula for computing the n-th derivative of the exponential function e±1/t was inductively obtained: e±1/t
(n)
= (−1)n e±1/t
n X
(±1)k L(n, k)
k=1
1 tn+k
(8)
.
The formula (8) have been applied in [2, 3, 5, 6]. In combinatorics or number theory, it is common knowledge that Bell numbers Bn may be computed in terms of Stirling numbers of the second kind S(n, k) by n X
Bn =
(9)
S(n, k).
k=1
In this paper, we will find a new explicit formula for computing Bell numbers Bn in terms of Lah numbers L(n, k) and Stirling numbers of the second kind S(n, k). Theorem 1. For n ∈ N, Bell numbers Bn may be computed in terms of Lah numbers L(n, k) and Stirling numbers of the second kind S(n, k) by " k # n X X n−k Bn = (−1) L(k, ℓ) S(n, k). (10) k=1
ℓ=1
Proof. In combinatorics, Bell polynomials of the second kind, or say, partial Bell polynomials, denoted by Bn,k (x1 , x2 , . . . , xn−k+1 ) for n ≥ k ≥ 1, are defined by X
Bn,k (x1 , x2 , . . . , xn−k+1 ) =
1≤i≤n,ℓ i ∈N Pn iℓi =n Pi=1 n i=1 ℓi =k
n! Qn−k+1 i=1
ℓi !
n−k+1 Y i=1
xi ℓi . i!
(11)
See [1, p. 134, Theorem A]. The famous Fa` a di Bruno formula may be described in terms of Bell polynomials of the second kind Bn,k (x1 , x2 , . . . , xn−k+1 ) by n X dn f ◦ h(t) = f (k) (h(t))Bn,k h′ (t), h′′ (t), . . . , h(n−k+1) (t) . n dt
(12)
k=1
See [1, p. 139, Theorem C]. Taking f (u) = e1/u and h(x) = ex in (12) and making use of (8) give dn ee d xn
−x
=
n X dk e1/u
d uk
k=1
=
n X
n−k+1
}| { z Bn,k (ex , ex , . . . , ex )
k 1/u
(−1) e
k=1
=e
e−x
ℓ=1
n X
k=1
Further by virtue of
k X
k
(−1)
n−k+1
1
}| { z L(k, ℓ) k+ℓ Bn,k (ex , ex , . . . , ex ) u
k X ℓ=1
n−k+1
1
z }| { L(k, ℓ) (k+ℓ)x Bn,k (ex , ex , . . . , ex ). e
Bn,k abx1 , ab2 x2 , . . . , abn−k+1 xn−k+1 = ak bn Bn,k (x1 , xn , . . . , xn−k+1 )
(13)
AN EXPLICIT FORMULA FOR COMPUTING BELL NUMBERS
and
3
n−k+1
z }| { Bn,k 1, 1, . . . , 1 = S(n, k) listed in [1, p. 135], where a and b are complex numbers, we obtain dn ee d xn
−x
=e
e−x
= ee
−x
n X
k X
(14)
n−k+1
1
z }| { (−1) L(k, ℓ) (k+ℓ)x e Bn,k (1, 1, . . . , 1) e k=1 ℓ=1 n X
k
(−1)k
k=1
k X
L(k, ℓ)
ℓ=1
kx
1 S(n, k). eℓx
Comparing this with the n-th derivative of the generating function (2) −x ∞ X dn ee −1 xk−n k = (−1) B k d xn (k − n)!
(15)
k=n
yields
e
∞ X
n
k
k=1
ℓ=1
X −x X xk−n 1 (−1) Bk = ee (−1)k L(k, ℓ) ℓx S(n, k). (k − n)! e k
k=n
Letting x → 0 in the above equation reveals (−1)n eBn = e
n X
(−1)k
k=1
k X
L(k, ℓ)S(n, k)
ℓ=1
which may be rearranged as (10). The proof of Theorem 1 is complete.
References [1] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, Revised and Enlarged Edition, D. Reidel Publishing Co., Dordrecht and Boston, 1974. 2, 3 [2] F. Qi, A recurrence formula for the first kind Stirling numbers, available online at http: //arxiv.org/abs/1310.5920. 2 [3] F. Qi, Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions, available online at http://arxiv.org/abs/1302. 6731. 2 [4] F. Qi, Properties of three functions relating to the exponential function and the existence of partitions of unity, Available online at http://arxiv.org/abs/1202.0766. 2 [5] F. Qi and C. Berg, Complete monotonicity of a difference between the exponential and trigamma functions and properties related to a modified Bessel function, Mediterr. J. Math. 10 (2013), no. 4, 1685–1696; Available online at http://dx.doi.org/10.1007/ s00009-013-0272-2. 2 [6] F. Qi and S.-H. Wang, Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions, available online at http://arxiv.org/abs/1210.2012. 2 [7] X.-J. Zhang, F. Qi, and W.-H. Li, Properties of three functions relating to the exponential function and the existence of partitions of unity, Int. J. Open Probl. Comput. Sci. Math. 5 (2012), no. 3, 122–127. 2 College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia Autonomous Region, 028043, China; Department of Mathematics, College of Science, Tianjin Polytechnic University, Tianjin City, 300387, China; Institute of Mathematics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China E-mail address:
[email protected],
[email protected],
[email protected] URL: http://qifeng618.wordpress.com
