A275552 Number of classes of endofunctions of [n] under vertical translation mod n and complement to n+1.

1, 1, 2, 5, 36, 313, 3904, 58825, 1048640, 21523361, 500000256, 12968712301, 371504186368, 11649042561241, 396857386631168, 14596463012695313, 576460752303439872, 24330595937833434241, 1092955779869348331520, 52063675148955620766421, 2621440000000000000262144

Offset: 0

Author

Olivier Gérard, Aug 02 2016

Keywords

• nonn
• easy

Comments

There are two size of classes, n or 2n.

.

n c:n c:2n (c:2n)/4

0 1

1 1

2 2

3 1 4 1

4 8 28 7

5 1 312 78

6 32 3872 968

7 1 58824 14706

For n odd, only the set of n constant functions can have a member of their class equal to their complement, so c:n size is 1.

For n even, the c:n class is populated by binary words using k for 0 and n+1-k for 1. There are (2^n)/2 such words as the complement operation identifies them by pairs.

For n odd, c:2n(n) = (n^n - 1*n)/(2*n)

For n even, c:2n(n) = (n^n - 2^(n-1)*n)/(2*n)

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..100

Crossrefs

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. A275549 Classes under reversal;

Cf. A275550 Classes under reversal and complement;

Cf. A275551 Classes under translation and reversal;

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.

Programs

Mathematica

a[0] = 1; a[n_?OddQ] := 1 + (n^n - n)/(2n); a[n_?EvenQ] := 2^(n-1) + (n^n - 2^(n-1)*n)/(2n); Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 07 2017, translated from PARI *)

PARI

a(n) = if(n%2, 1 + (n^n - 1*n)/(2*n), 2^(n-1) + (n^n - 2^(n-1)*n)/(2*n)); \\ Andrew Howroyd, Sep 30 2017

Formula

a(n) = 1 + (n^n - 1*n)/(2*n) if n is odd,

a(n) = 2^(n-1) + (n^n - 2^(n-1)*n)/(2*n) if n is even.