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%I #12 Sep 06 2022 10:29:09
%S 0,1,3,8,3,30,20,144,90,40,840,504,420,5760,3360,2688,1260,45360,
%T 25920,20160,18144,403200,226800,172800,151200,72576,3991680,2217600,
%U 1663200,1425600,1330560,43545600,23950080,17740800,14968800,13685760,6652800,518918400
%N Decomposition of Stirling's S(n,2) based on associated numeric partitions.
%C These values also appear in a wider context when counting elements of finite groups by cycle structure. For example, the alternating group on four symbols has 12 elements; eight associated with the partition 3+1, three associated with 2+2 and the identity associated with 1+1+1+1. The cross-referenced sequences are all associated with similar numeric partitions and "M2" weights.
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 831.
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F From _Sean A. Irvine_, Sep 05 2022: (Start)
%F T(1,1) = 0.
%F T(n,k) = n! / (k * (n-k)) for 1 <= k < n/2.
%F T(2n,n) = (2*n)! / (2*n^2).
%F (End)
%e Triangle begins:
%e 0;
%e 1;
%e 3;
%e 8, 3;
%e 30, 20;
%e 144, 90, 40;
%e 840, 504, 420;
%e ...
%Y Cf. A000012, A000035, A000027, A004526, A022003, A008619, A000217, A007997, A001399, A011765 A008620, A027656, A002620, A000292, A008627.
%K nonn,tabf
%O 1,3
%A _Alford Arnold_, Jan 11 2001
%E More terms from _Sean A. Irvine_, Sep 05 2022
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