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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.

%I A050213 #23 Feb 16 2025 08:32:40
%S A050213 24,120,720,5040,40320,362880,72576,3628800,1330560,39916800,20338560,
%T A050213 479001600,303937920,6227020800,4643084160,87178291200,73721007360,
%U A050213 1743565824,1307674368000,1224694598400,69742632960,20922789888000
%N A050213 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=5.
%C A050213 Generalizes Stirling numbers of the first kind.
%D A050213 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.
%H A050213 Alois P. Heinz, <a href="/A050213/b050213.txt">Rows n = 5..300, flattened</a>
%H A050213 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PermutationCycle.html">Permutation Cycle.</a>
%e A050213 Triangle begins:
%e A050213 05:       24;
%e A050213 06:      120;
%e A050213 07:      720;
%e A050213 08:     5040;
%e A050213 09:    40320;
%e A050213 10:   362880,    72576;
%e A050213 11:  3628800,  1330560;
%e A050213 12: 39916800, 20338560;
%p A050213 b:= proc(n) option remember; expand(`if`(n=0, 1, add(
%p A050213       b(n-i)*x*binomial(n-1, i-1)*(i-1)!, i=5..n)))
%p A050213     end:
%p A050213 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
%p A050213 seq(T(n), n=5..20);  # _Alois P. Heinz_, Sep 25 2016
%t A050213 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 A050213 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][ b[n]];
%t A050213 T /@ Range[5, 20] // Flatten (* _Jean-François Alcover_, Dec 08 2019, after _Alois P. Heinz_ *)
%Y A050213 Cf. A008275, A008306, A050211, A050212.
%K A050213 nonn,tabf
%O A050213 5,1
%A A050213 _Eric W. Weisstein_
%E A050213 Offset changed from 1 to 5 by _Alois P. Heinz_, Sep 25 2016