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Number of classes of endofunctions of [n] under vertical translation mod n, rotation and reversal.
13

%I #9 Oct 01 2017 16:59:24

%S 1,1,2,4,16,77,730,8578,132422,2394795,50031012,1179054376,

%T 30959574248,896082610429,28346986843640,973097619619654,

%U 36028798243701780,1431211529242786625,60719765604009463866,2740193429053744941868,131072000002841600036024

%N Number of classes of endofunctions of [n] under vertical translation mod n, rotation and reversal.

%C Because of the interaction between the two symmetries indexed by n, classes can be of size from n up to 2*n^2.

%C n possible class sizes

%C -----------------------------------

%C 1 1

%C 2 2

%C 3 3, 6, 9

%C 4 4, 8, 16, 32

%C 5 5, 10, 25, 50

%C 6 6, 12, 18, 24, 36, 72

%C 7 7, 14, 49, 98

%C but classes of size 2*n^2 account for the bulk of a(n).

%C n number of classes

%C -----------------------------------

%C 1 1

%C 2 2

%C 3 1, 1, 2

%C 4 2, 3, 8, 3

%C 5 1, 2, 24, 50

%C 6 2, 4, 10, 2, 136, 576

%C 7 1, 3, 342, 8232

%H Andrew Howroyd, <a href="/A275555/b275555.txt">Table of n, a(n) for n = 0..100</a>

%o (PARI) \\ see A056391 for Polya enumeration functions

%o a(n) = NonequivalentSorts(DihedralPerms(n), CyclicPerms(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. A275552 Classes under translation and complement

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

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

%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

%O 0,3

%A _Olivier GĂ©rard_, Aug 05 2016

%E Terms a(8) and beyond from _Andrew Howroyd_, Sep 30 2017