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A050211 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=3.

%I A050211 #30 Aug 15 2025 22:03:46
%S A050211 2,6,24,120,40,720,420,5040,3948,40320,38304,2240,362880,396576,50400,
%T A050211 3628800,4419360,859320,39916800,53048160,13665960,246400,479001600,
%U A050211 684478080,216339552,9609600,6227020800,9464307840,3501834336
%N A050211 Triangle of number of permutations of {1, 2, ..., n} having exactly k cycles, each of which is of length >=r for r=3.
%C A050211 Generalizes Stirling numbers of the first kind
%D A050211 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 257.
%H A050211 Alois P. Heinz, <a href="/A050211/b050211.txt">Rows n = 3..200, flattened</a>
%H A050211 S. Brassesco, M. A. Méndez, <a href="http://arxiv.org/abs/1002.3894">The asymptotic expansion for the factorial and Lagrange inversion formula</a>, arXiv:1002.3894v1 [math.CA], 2010.
%H A050211 G. Nemes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Nemes/nemes2.html">On the Coefficients of the Asymptotic Expansion of n!</a>, J. Int. Seq. 13 (2010), 10.6.6.
%H A050211 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PermutationCycle.html">Permutation Cycle.</a>
%F A050211 From _Peter Bala_, Sep 06 2011: (Start)
%F A050211 E.g.f.: (1-t)^(-u)*exp(-u*(t+t^2/2)) - 1 = (2*u)*t^3/3!+(6*u)*t^4/4!+(24*u)*t^5/5!+(120*u+40*u^2)*t^6/6!+(720*u+420*u^2)*t^7/7!+....
%F A050211 E.g.f. for column k: 1/k!*(-log(1-x)-x-x^2/2)^k.
%F A050211 Starting at row 3, row lengths are 1, 1, 1, 2, 2, 2, 3, 3, 3, ....
%F A050211 Recurrence: T(n,k) = (n-1)*T(n-1,k) + (n-1)*(n-2)*T(n-3,k-1). (End)
%e A050211 Table begins:
%e A050211    n\k |      u     u^2    u^3
%e A050211   = = = = = = = = = = = = = = =
%e A050211     3  |      2
%e A050211     4  |      6
%e A050211     5  |     24
%e A050211     6  |    120     40
%e A050211     7  |    720    420
%e A050211     8  |   5040   3948
%e A050211     9  |  40320  38304    2240
%e A050211    ...
%p A050211 b:= proc(n) option remember; expand(`if`(n=0, 1, add(
%p A050211       b(n-i)*x*binomial(n-1, i-1)*(i-1)!, i=3..n)))
%p A050211     end:
%p A050211 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
%p A050211 seq(T(n), n=3..15);  # _Alois P. Heinz_, Sep 25 2016
%t A050211 t[n_ /; n >= 3, k_ /; k >= 1] := t[n, k] = (n - 1)*t[n - 1, k] + (n - 2)*(n - 1)*t[n - 3, k - 1] ; t[_, _] = 0; t[3, 1] = 2; Flatten[ Table[t[n, k], {n, 3, 15}, {k, 1, Floor[n/3]}]] (* _Jean-François Alcover_, Nov 05 2012, after _Peter Bala_ *)
%Y A050211 Cf. A008275, A008306, A050212, A050213.
%K A050211 nonn,tabf
%O A050211 3,1
%A A050211 _Eric W. Weisstein_