unsigned Stirling numbers of the first kind
GPTKB entity
Statements (23)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
integer sequence |
| gptkbp:application |
gptkb:algebra
enumerative combinatorics polynomial expansions |
| gptkbp:generating_function |
x(x+1)...(x+n-1) = sum_{k=0}^n c(n, k) x^k
|
| https://www.w3.org/2000/01/rdf-schema#label |
unsigned Stirling numbers of the first kind
|
| gptkbp:initial_condition |
c(0, 0) = 1
c(0, k) = 0 for k > 0 c(n, 0) = 0 for n > 0 |
| gptkbp:matrixRepresentation |
triangular matrix
|
| gptkbp:namedAfter |
gptkb:James_Stirling
|
| gptkbp:notation |
[n k]
c(n, k) |
| gptkbp:numberOfRooms |
number of permutations of n elements with exactly k cycles
|
| gptkbp:OEIS_sequence |
A132393
|
| gptkbp:property |
always non-negative
|
| gptkbp:recurrence |
c(n+1, k) = n * c(n, k) + c(n, k-1)
|
| gptkbp:relatedTo |
gptkb:Stirling_numbers_of_the_first_kind
signed Stirling numbers of the first kind |
| gptkbp:sum_property |
sum_{k=1}^n c(n, k) = n!
|
| gptkbp:bfsParent |
gptkb:Stirling_numbers_of_the_first_kind
|
| gptkbp:bfsLayer |
8
|