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A050213
Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=5.
3
24, 120, 720, 5040, 40320, 362880, 72576, 3628800, 1330560, 39916800, 20338560, 479001600, 303937920, 6227020800, 4643084160, 87178291200, 73721007360, 1743565824, 1307674368000, 1224694598400, 69742632960, 20922789888000
OFFSET
5,1
COMMENTS
Generalizes Stirling numbers of the first kind.
REFERENCES
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.
LINKS
Eric Weisstein's World of Mathematics, Permutation Cycle.
EXAMPLE
Triangle begins:
05: 24;
06: 120;
07: 720;
08: 5040;
09: 40320;
10: 362880, 72576;
11: 3628800, 1330560;
12: 39916800, 20338560;
MAPLE
b:= proc(n) option remember; expand(`if`(n=0, 1, add(
b(n-i)*x*binomial(n-1, i-1)*(i-1)!, i=5..n)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=5..20); # Alois P. Heinz, Sep 25 2016
MATHEMATICA
b[n_] := b[n] = Expand[If[n == 0, 1, Sum[b[n - i]*x*Binomial[n - 1, i - 1]* (i - 1)!, {i, 5, n}]]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][ b[n]];
T /@ Range[5, 20] // Flatten (* Jean-François Alcover, Dec 08 2019, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabf
EXTENSIONS
Offset changed from 1 to 5 by Alois P. Heinz, Sep 25 2016
STATUS
approved