Stirling numbers of the second kind

GPTKB entity

Predicate	Object
gptkbp:instanceOf	gptkb:integer_sequence gptkb:mathematical_concept
gptkbp:application	enumerative combinatorics polynomial interpolation counting set partitions
gptkbp:category	integer sequences combinatorial numbers
gptkbp:defines	number of ways to partition a set of n objects into k non-empty unlabeled subsets
gptkbp:designer	gptkb:Stirling_numbers_of_the_first_kind
gptkbp:explicitFormula	S(n, k) = (1/k!) * sum_{j=0}^{k} (-1)^{k-j} * binomial(k, j) * j^n
gptkbp:field	gptkb:combinatorics
gptkbp:generatingFunction	sum_{n=0}^{∞} S(n, k) * x^n / n! = (e^{x} - 1)^k / k!
gptkbp:initialCondition	S(0, 0) = 1 S(0, k) = 0 for k > 0 S(n, 0) = 0 for n > 0
gptkbp:matrixRepresentation	gptkb:Stirling_matrix_of_the_second_kind
gptkbp:namedAfter	gptkb:James_Stirling
gptkbp:OEIS	gptkb:A008277
gptkbp:property	S(n, 1) = 1 for all n > 0 S(n, 2) = 2^{n-1} - 1 S(n, k) = 0 if k > n S(n, n) = 1 for all n ≥ 0 S(n, n-1) = binomial(n, 2)
gptkbp:recurrence	S(n, k) = k * S(n-1, k) + S(n-1, k-1)
gptkbp:relatedTo	gptkb:Bell_numbers gptkb:Stirling_numbers_of_the_first_kind
gptkbp:symbol	S(n, k)
gptkbp:bfsParent	gptkb:Bell_numbers gptkb:Bell_polynomials
gptkbp:bfsLayer	6
https://www.w3.org/2000/01/rdf-schema#label	Stirling numbers of the second kind