A058936 - OEIS

A058936

Decomposition of Stirling's S(n,2) based on associated numeric partitions.

0, 1, 3, 8, 3, 30, 20, 144, 90, 40, 840, 504, 420, 5760, 3360, 2688, 1260, 45360, 25920, 20160, 18144, 403200, 226800, 172800, 151200, 72576, 3991680, 2217600, 1663200, 1425600, 1330560, 43545600, 23950080, 17740800, 14968800, 13685760, 6652800, 518918400

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OFFSET

1,3

COMMENTS

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.

REFERENCES

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.

LINKS

Table of n, a(n) for n=1..38.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

FORMULA

From Sean A. Irvine, Sep 05 2022: (Start)

T(1,1) = 0.

T(n,k) = n! / (k * (n-k)) for 1 <= k < n/2.

T(2n,n) = (2*n)! / (2*n^2).

(End)

EXAMPLE

Triangle begins:

8, 3;

30, 20;

144, 90, 40;

840, 504, 420;