Given n colors, a(n) = number of necklaces with n beads and 1 up to n colors effectively assigned to them (super_labeled: which also generates n different monochrome necklaces). -
Wouter Meeussen, Aug 09 2002
Number of endofunctions on a set with n objects up to cyclic permutation (rotation). E.g., for n = 3, the 11 endofunctions are 1,1,1; 2,2,2; 3,3,3; 1,1,2; 1,2,2; 1,1,3; 1,3,3; 2,2,3; 2,3,3; 1,2,3; and 1,3,2. -
Franklin T. Adams-Watters, Jan 17 2007
Also number of pre-necklaces in Sigma(n,n) (see Ruskey and others). -
Peter Luschny, Aug 12 2012
Decomposition of the endofunctions by class size.
.
n | 1 2 3 4 5 6 7
--+----------------------------------
1 | 1
2 | 2 1
3 | 3 0 8
4 | 4 6 0 60
5 | 5 0 0 0 624
6 | 6 15 70 0 0 7735
7 | 7 0 0 0 0 0 117648
.
The right diagonal gives the number of Lyndon Words or aperiodic necklaces,
A075147. By multiplying each column by the corresponding size and summing, one gets
A000312.
(End)
