A285793 - OEIS

A285793

Sum T(n,k) of the k-th entries in all cycles of all permutations of [n]; triangle T(n,k), n>=1, 1<=k<=n, read by rows.

1, 4, 2, 18, 13, 5, 96, 83, 43, 18, 600, 582, 342, 192, 84, 4320, 4554, 2874, 1824, 1068, 480, 35280, 39672, 26232, 17832, 11784, 7080, 3240, 322560, 382248, 261288, 185688, 131256, 88920, 54360, 25200, 3265920, 4044240, 2834640, 2078640, 1534320, 1110960, 765360, 473760, 221760

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OFFSET

1,2

COMMENTS

Each cycle is written with the smallest element first and cycles are arranged in increasing order of their first elements.

LINKS

Table of n, a(n) for n=1..45.

Wikipedia, Permutation

FORMULA

T(n,1) = n * n!.

T(n,n) = floor((n-1)!*(n+2)/2).

EXAMPLE

T(3,2) = 13 because the sum of the second entries in all cycles of all permutations of [3] ((123), (132), (12)(3), (13)(2), (1)(23), (1)(2)(3)) is 2+3+2+3+3+0 = 13.

Triangle T(n,k) begins:

: 1;

: 4, 2;

: 18, 13, 5;

: 96, 83, 43, 18;

: 600, 582, 342, 192, 84;

: 4320, 4554, 2874, 1824, 1068, 480;

: 35280, 39672, 26232, 17832, 11784, 7080, 3240;

: 322560, 382248, 261288, 185688, 131256, 88920, 54360, 25200;

CROSSREFS

Columns k=1-2 give: A001563, A285795.

Main diagonal and first lower diagonal give: A038720(n-1) (for n>1), A286175.