A275555 Number of classes of endofunctions of [n] under vertical translation mod n, rotation and reversal.

1, 1, 2, 4, 16, 77, 730, 8578, 132422, 2394795, 50031012, 1179054376, 30959574248, 896082610429, 28346986843640, 973097619619654, 36028798243701780, 1431211529242786625, 60719765604009463866, 2740193429053744941868, 131072000002841600036024

Offset: 0

Author

Olivier Gérard, Aug 05 2016

Keywords

• nonn

Comments

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

n possible class sizes

-----------------------------------

1 1

2 2

3 3, 6, 9

4 4, 8, 16, 32

5 5, 10, 25, 50

6 6, 12, 18, 24, 36, 72

7 7, 14, 49, 98

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

n number of classes

-----------------------------------

1 1

2 2

3 1, 1, 2

4 2, 3, 8, 3

5 1, 2, 24, 50

6 2, 4, 10, 2, 136, 576

7 1, 3, 342, 8232

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. A275552 Classes under translation and complement

Cf. A275553 Classes under translation, complement and reversal

Cf. A275554 Classes under translation, rotation and complement

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

PARI

\\ see A056391 for Polya enumeration functions
a(n) = NonequivalentSorts(DihedralPerms(n), CyclicPerms(n)); \\ Andrew Howroyd, Sep 30 2017

Extensions

Terms a(8) and beyond from Andrew Howroyd, Sep 30 2017