Eulerian and Stirling Numbers of First and Second Kind
Compute Eulerian numbers and Stirling numbers of the first and second kind, possibly vectorized for all k “at once”.
Stirling1(n, k)
Stirling2(n, k, method = c("lookup.or.store", "direct"))
Eulerian (n, k, method = c("lookup.or.store", "direct"))
Stirling1.all(n)
Stirling2.all(n)
Eulerian.all (n)n |
positive integer ( |
k |
integer in |
method |
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Eulerian numbers:
A(n,k) = the number of permutations of 1,2,...,n with exactly k
ascents (or exactly k descents).
Stirling numbers of the first kind:
s(n,k) = (-1)^n-k times
the number of permutations of 1,2,...,n with exactly k cycles.
Stirling numbers of the second kind:
S(n,k) is the number of ways of partitioning a set
of n elements into k non-empty subsets.
A(n,k), s(n,k) or S(n,k), respectively.
Eulerian.all(n) is the same as sapply(0:(n-1), Eulerian, n=n)
(for n > 0), Stirling1.all(n) is the same as sapply(1:n, Stirling1, n=n),
andStirling2.all(n) is the same as sapply(1:n, Stirling2, n=n),
but more efficient.
For typical double precision arithmetic,Eulerian*(n, *) overflow (to Inf) for n ≥ 172,Stirling1*(n, *) overflow (to +/-Inf) for
n ≥ 171, andStirling2*(n, *) overflow (to Inf) for n ≥ 220.
Eulerians:
NIST Digital Library of Mathematical Functions, 26.14: https://dlmf.nist.gov/26.14
Stirling numbers:
Abramowitz and Stegun 24,1,4 (p. 824-5 ; Table 24.4, p.835); Closed Form : p.824 "C."
NIST Digital Library of Mathematical Functions, 26.8: https://dlmf.nist.gov/26.8
Stirling1(7,2) Stirling2(7,3) Stirling1.all(9) Stirling2.all(9)
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