A275558 Number of classes of endofunctions of [n] under rotation, complement to n+1 and reversal.

1, 1, 2, 6, 31, 195, 2182, 30100, 529674, 10778125, 250155012, 6484839306, 185757443582, 5824538174455, 198428907905336, 7298232189810696, 288230385949610020, 12165298000307625609, 546477890436083284338, 26031837576091248872110, 1310720000028416000168044

Offset: 0

Author

Olivier Gérard, Aug 05 2016

Keywords

• nonn

Comments

Classes can be of size 1,2,4, n, 2n or 4n.

.

n 1 2 4 n 2n 4n

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

1 1

2 0 2

3 1 1 4

4 0 4 4 0 17 6

5 1 2 0 0 72 120

6 0 6 6 30 410 1730

7 1 3 0 0 1368 28728

.

For n odd, the constant function (n+1)/2 is the only stable by rotation, complement and reversal. So #c1=1.

For n even, there is no stable function, so #c1=0, but constant functions are grouped two by two making n/2 classes of size 2. Functions alternating a value and its complement are also grouped two by two, making another n/2 classes. This gives #c2=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. A275552 Classes under translation and complement

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

Programs

PARI

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

Extensions

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