A275552 - OEIS

%I #13 Oct 07 2017 09:01:21

%S 1,1,2,5,36,313,3904,58825,1048640,21523361,500000256,12968712301,

%T 371504186368,11649042561241,396857386631168,14596463012695313,

%U 576460752303439872,24330595937833434241,1092955779869348331520,52063675148955620766421,2621440000000000000262144

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

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

%C .

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

%C 0   1

%C 1   1

%C 2   2

%C 3   1    4      1

%C 4   8    28     7

%C 5   1    312    78

%C 6   32   3872   968

%C 7   1    58824  14706

%C 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.

%C 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.

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

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

%H Andrew Howroyd, <a href="/A275552/b275552.txt">Table of n, a(n) for n = 0..100</a>

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

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

%t 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 *)

%o (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

%Y Cf. A000312 All endofunctions;

%Y Cf. A000169 Classes under translation mod n;

%Y Cf. A001700 Classes under sort;

%Y Cf. A056665 Classes under rotation;

%Y Cf. A168658 Classes under complement to n+1;

%Y Cf. A130293 Classes under translation and rotation;

%Y Cf. A081721 Classes under rotation and reversal;

%Y Cf. A275549 Classes under reversal;

%Y Cf. A275550 Classes under reversal and complement;

%Y Cf. A275551 Classes under translation and reversal;

%Y Cf. A275553 Classes under translation, complement and reversal;

%Y Cf. A275554 Classes under translation, rotation and complement;

%Y Cf. A275555 Classes under translation, rotation and reversal;

%Y Cf. A275556 Classes under translation, rotation, complement and reversal;

%Y Cf. A275557 Classes under rotation and complement;

%Y Cf. A275558 Classes under rotation, complement and reversal.

%K nonn,easy

%O 0,3

%A _Olivier Gérard_, Aug 02 2016