Stirling numbers of the second kind
GPTKB entity
Statements (31)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
integer sequence |
| 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 |
combinatorics
|
| gptkbp:generatingFunction |
sum_{n=0}^{∞} S(n, k) * x^n / n! = (e^{x} - 1)^k / k!
|
| https://www.w3.org/2000/01/rdf-schema#label |
Stirling numbers of the second kind
|
| 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
|