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A050212 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=4.

%I A050212 #22 Feb 16 2025 08:32:40
%S A050212 6,24,120,720,5040,1260,40320,18144,362880,223776,3628800,2756160,
%T A050212 39916800,35307360,1247400,479001600,476910720,38918880,6227020800,
%U A050212 6822541440,889945056,87178291200,103440879360,18478684224
%N A050212 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=4.
%C A050212 Generalizes Stirling numbers of the first kind.
%D A050212 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.
%H A050212 Alois P. Heinz, <a href="/A050212/b050212.txt">Rows n = 4..300, flattened</a>
%H A050212 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PermutationCycle.html">Permutation cycle.</a>
%e A050212 Triangle begins:
%e A050212 :        6;
%e A050212 :       24;
%e A050212 :      120;
%e A050212 :      720;
%e A050212 :     5040,     1260;
%e A050212 :    40320,    18144;
%e A050212 :   362880,   223776;
%e A050212 :  3628800,  2756160;
%e A050212 : 39916800, 35307360, 1247400;
%p A050212 b:= proc(n) option remember; expand(`if`(n=0, 1, add(
%p A050212       b(n-i)*x*binomial(n-1, i-1)*(i-1)!, i=4..n)))
%p A050212     end:
%p A050212 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
%p A050212 seq(T(n), n=4..20);  # _Alois P. Heinz_, Sep 25 2016
%t A050212 b[n_] := b[n] = Expand[If[n==0, 1, Sum[b[n-i] x Binomial[n-1, i-1] (i-1)!, {i, 4, n}]]];
%t A050212 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, Exponent[p, x]}]][b[n]];
%t A050212 Table[T[n], {n, 4, 20}] // Flatten (* _Jean-François Alcover_, Dec 07 2019, after _Alois P. Heinz_ *)
%Y A050212 Cf. A008275, A008306, A050211, A050213.
%K A050212 nonn,tabf
%O A050212 4,1
%A A050212 _Eric W. Weisstein_
%E A050212 Offset changed from 1 to 4 by _Alois P. Heinz_, Sep 25 2016