A275553 Number of classes of endofunctions of [n] under vertical translation mod n, complement to n+1 and reversal.

1, 1, 2, 4, 24, 169, 2024, 29584, 525600, 10764961, 250030128, 6484436676, 185752964096, 5824523694025, 198428723433728, 7298231591777344, 288230377359679488, 12165297972404595841, 546477889989773968640, 26031837574639154232100, 1310720000002816000131072

Offset: 0

Author

Olivier Gérard, Aug 05 2016

Keywords

• nonn

Comments

There are three size of classes : n, 2n, 4n.

n c:n c:2n c:4n

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

0 1

1 1

2 2

3 1 2 1

4 4 10 10

5 1 24 144

6 8 148 1868

7 1 342 29241

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, we have 2^(n/2) binary words which have mirror-symmetry

There are three types of classes of size of 2n (stable by reversal, stable by complement, stable by rc as in A275550).

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. 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

PARI

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

Extensions

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