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A186526 Number T(n,k) of permutations on n elements with exactly k 3-cycles; triangle read by rows.

%I A186526 #37 May 20 2020 16:07:09
%S A186526 1,1,2,4,2,16,8,80,40,520,160,40,3640,1120,280,29120,8960,2240,259840,
%T A186526 87360,13440,2240,2598400,873600,134400,22400,28582400,9609600,
%U A186526 1478400,246400,343235200,114329600,19219200,1971200,246400,4462057600,1486284800,249849600,25625600,3203200,62468806400,20807987200,3497894400,358758400,44844800,936987251200,312344032000,52019968000,5829824000,448448000,44844800
%N A186526 Number T(n,k) of permutations on n elements with exactly k 3-cycles; triangle read by rows.
%C A186526 Triangle T(n,k) with 0<=k<=floor(n/3) gives the number of permutations in the symmetric group Sn that have exactly k cycles of length 3. The sum of T(n,k) over all k equals n!.
%D A186526 Arratia, R. and Tavaré, S. (1992). The cycle structure of random permutations. Ann. Probab. 20 1567-1591.
%H A186526 Alois P. Heinz, <a href="/A186526/b186526.txt">Rows n = 0..250, flattened</a>
%F A186526 T(n,k) = (n!(1/3)^k)/k!*sum((-1/3)^j/j!, j=0..(m-k)) where m=floor(n/3).
%F A186526 E.g.f.: exp(x^3/3*(y-1))/(1-x). - _Geoffrey Critzer_, Aug 26 2012.
%e A186526 For n=4 and k=1, T(4,1)=8 since there are 8 permutations on 4 elements with 1 cycle of length 3, namely, (abc)(d), (acb)(d), (abd)(c), (adb)(c), (acd)(b), (adc)(b), (bcd)(a), and (bdc)(a).
%e A186526 Triangle T(n,k) begins:
%e A186526 :     1;
%e A186526 :     1;
%e A186526 :     2;
%e A186526 :     4,    2;
%e A186526 :    16,    8;
%e A186526 :    80,   40;
%e A186526 :   520,  160,   40;
%e A186526 :  3640, 1120,  280;
%e A186526 : 29120, 8960, 2240;
%e A186526 : ...
%p A186526 seq(seq(n!*(1/3)^x/x!*sum((-1/3)^j/j!,j=0..(floor(n/3)-x)),x=0..floor(n/3)),n=0..15);
%p A186526 # second Maple program:
%p A186526 b:= proc(n) option remember; expand(`if`(n=0, 1, add(b(n-i)*
%p A186526       `if`(i=3, x, 1)*binomial(n-1, i-1)*(i-1)!, i=1..n)))
%p A186526     end:
%p A186526 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n)):
%p A186526 seq(T(n), n=0..15);  # _Alois P. Heinz_, Sep 25 2016
%t A186526 nn = 8; Range[0, nn]! CoefficientList[
%t A186526    Series[Exp[x^3/3 (y - 1)]/(1 - x), {x, 0, nn}], {x, y}] // Grid
%Y A186526 Cf. A000142, A057693, A008290, A114320.
%K A186526 nonn,tabf
%O A186526 0,3
%A A186526 _Dennis P. Walsh_, Feb 23 2011