The Stirling number
[s.sub.m,n] of the second kind is the number of ways to partition an m elements set into n nonempty subsets for any m, n [greater than or equal to] 0.
Let S(n, k) be the Stirling number
of the second kind, which counts the number of partitions of a set with n elements in k disjoint nonempty subsets.
For 1 [less than or equal to] k [less than or equal to] n, recall that the Stirling number
(of the second kind) Stir(n, k) counts the number of partitions of [n] with exactly k blocks.
Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the Stirling number
of the first kind (unsigned version).
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Let [S.sub.n,m] denote the set of permutations of [S.sub.n] with m cycles, [s.sub.n,m], the unsigned Stirling number
of the first kind, denote the cardinality of [S.sub.n,m], and [mu] = m/n.
In particular, we determine the stirling number
of the second kind [s.sub.2](n, 4), when n [greater than or equal to] 4 is given.
(see [15, 16, 18-23]), where [S.sub.1](n, k) is the Stirling numbers
of the first kind and [S.sub.2](n, k) is the Stirling numbers
of the second kind as follows:
Combinatorial Identities for Stirling Numbers
: The Unpublished Notes of H.W.
The numbers [W.sub.m](n,k) can be shown to be a kind of generalization of the famous Stirling numbers
of the Second kind S(n, k) when the parameter m equals to 1.
This computation is based on some interesting relation between the dimensions of the summands of the [E.sub.2]-page of the spectral sequence and some Stirling numbers
. Since these Euler characteristic have an exponential growth, we deduce an exponential growth of the dimension of the associated graded space of [E.sub.2].
After preliminaries they cover numerical methods for solving ordinary and partial fractional differential equations, efficient numerical methods, generalized Stirling numbers
and applications, fractional variational principles, continuous-time random walks (CTRW) and fractional diffusion models, and applications of CTRW to finance and economics.