On the other hand, the sum [B.sub.m] = [[summation].sup.m.sub.k=0] [s.sub.m,k] of numbers of the mth row of S is called a Bell number, so {[B.sub.m] | m [greater than or equal to] 0} = {1, 1, 2, 5, 15, ...} counts the number of partitions of an m elements set.
So the 6th row of P begins with 52 followed by 67, 87, 114, and 151 and then reach 203, the next Bell number of 52.
(i) Each ith row begins with the Bell number [p.sup.(k).sub.i,1] = [B.sup.(0).sub.i-1].
Like the Bell numbers which are the sum of each row of S, we will take sum of each row of the k-Stirling matrix [S.sup.(k)] and call this the k-Bell number and denote it by [B.sup.(k)] = {[B.sup.(k).sub.i] | i [greater than or equal to] 0}.
For an information table of modest size with 10 attributes with 20 values each, the number of possible instances is [5.sup.*][10.sup.137] (the computations involve the
Bell number which exhibits a superexponential growth).
Among their topics are basic tools, Stirling and
Bell numbers, normal ordering in the Weyl algebra, a generalization of the Weyl algebra, and the q-deformed generalized Weyl algebra.
which consequently yields the Bell numbers [B.sub.n] when x = 1.
The following beautiful integral representation of the Bell numbers [B.sub.n] was first obtained by Cesaro [12]:
This expression was generalized by Mezo [11] using a kind of generalization of the classical Bell numbers called r-Bell numbers [B.sub.n,r].
Aigner [19] established the Hankel transform of the classical Bell numbers. A similar identity was obtained by Mezo [11] for the Hankel transform of the r-Bell numbers.