A050212 - OEIS

%I #21 Feb 04 2023 09:46:57

%S 6,24,120,720,5040,1260,40320,18144,362880,223776,3628800,2756160,

%T 39916800,35307360,1247400,479001600,476910720,38918880,6227020800,

%U 6822541440,889945056,87178291200,103440879360,18478684224

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

%C Generalizes Stirling numbers of the first kind.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.

%H Alois P. Heinz, <a href="/A050212/b050212.txt">Rows n = 4..300, flattened</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PermutationCycle.html">Permutation cycle.</a>

%e Triangle begins:

%e :        6;

%e :       24;

%e :      120;

%e :      720;

%e :     5040,     1260;

%e :    40320,    18144;

%e :   362880,   223776;

%e :  3628800,  2756160;

%e : 39916800, 35307360, 1247400;

%p b:= proc(n) option remember; expand(`if`(n=0, 1, add(

%p       b(n-i)*x*binomial(n-1, i-1)*(i-1)!, i=4..n)))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):

%p seq(T(n), n=4..20);  # _Alois P. Heinz_, Sep 25 2016

%t 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 T[n_] := Function[p, Table[Coefficient[p, x, i], {i, Exponent[p, x]}]][b[n]];

%t Table[T[n], {n, 4, 20}] // Flatten (* _Jean-François Alcover_, Dec 07 2019, after _Alois P. Heinz_ *)

%Y Cf. A008275, A008306, A050211, A050213.

%K nonn,tabf

%O 4,1

%A _Eric W. Weisstein_

%E Offset changed from 1 to 4 by _Alois P. Heinz_, Sep 25 2016