A192553 Sums of powers of permutations of length n.

1, 2, 5, 15, 51, 197, 850, 3897, 19461, 104264

Offset: 1

Author

Olivier Gérard, Jul 04 2011

Keywords

• nonn

Comments

Permutations of length n can be represented by an ordered list of the first n integers. Summing vertically the lists corresponding to the successive powers of a permutation up to its order ord, gives a new list of n integers from ord to n*ord. a(n) gives the number of different such lists. a(n) is related to the Bell numbers and majored by them, as permutations with the same cycle decomposition will have the same order and only a different ordering of powers.

Examples

			{2, 1, 3, 6, 4, 5} is a permutation of length 6 and order 6. Its successive powers are
{2, 1, 3, 6, 4, 5},
{1, 2, 3, 5, 6, 4},
{2, 1, 3, 4, 5, 6},
{1, 2, 3, 6, 4, 5},
{2, 1, 3, 5, 6, 4},
{1, 2, 3, 4, 5, 6}. Their sum is
{9, 9, 18, 30, 30, 30} or
{8, 7, 15, 26, 25, 24} if one does not includes the identity.
There are 197 different sums for n=6.
		

Crossrefs

Cf. A000110, A051625.

Formula

a(n) <= Bell(n)