OFFSET
0,3
COMMENTS
f and g are in the same class if function g(i) = f(n+1-i) for all i.
Decomposition by class size
.
n 1 2
---------------
1 1 0
2 2 1
3 9 9
4 16 120
5 125 1500
6 216 23220
7 2401 410571
.
Demonstration for the formula: the classes are either of size 1 or 2.
The classes of size 1 is for functions invariant by reversal. They are specified by half their values, including one more if n is odd. Their number is n^(ceiling(n/2)).
So the number of classes under this symmetry is half (the number of functions + the number of classes of size 1).
a(n) is the number of unoriented length n strings with a maximum of n colors. - Andrew Howroyd, Sep 13 2019
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
FORMULA
a(n) = (n^n+n^ceiling(n/2))/2.
PROG
(PARI) a(n) = {(n^n + n^((n+1)\2))/2} \\ Andrew Howroyd, Sep 13 2019
CROSSREFS
Main diagonal of A277504.
Cf. A000312 All endofunctions
Cf. A000169 Classes under translation mod n
Cf. A001700 Classes under sort
Cf. A056665 Classes under rotation
Cf. A168658 Classes under complement to n+1
Cf. A130293 Classes under translation and rotation
Cf. A081721 Classes under rotation and reversal
Cf. A275550 Classes under reversal and complement
Cf. A275551 Classes under translation and reversal
Cf. A275552 Classes under translation and complement
Cf. A275553 Classes under translation, complement and reversal
Cf. A275554 Classes under translation, rotation and complement
Cf. A275555 Classes under translation, rotation and reversal
Cf. A275556 Classes under translation, rotation, complement and reversal
Cf. A275557 Classes under rotation and complement
Cf. A275558 Classes under rotation, complement and reversal
Cf. A078707 Endofunctions symmetric around their middle (stable by reversal).
KEYWORD
nonn,easy
AUTHOR
Olivier Gérard, Aug 01 2016
STATUS
approved