A275549 Number of classes of endofunctions of [n] under reversal.

1, 1, 3, 18, 136, 1625, 23436, 412972, 8390656, 193739769, 5000050000, 142656721086, 4458051717120, 151437584670385, 5556003465485760, 218946946471875000, 9223372039002259456, 413620131002462320337, 19673204037747448432896, 989209827833222327690890

Offset: 0

Author

Olivier Gérard, Aug 01 2016

Keywords

• nonn
• easy

Comments

f and g are in the same class if function g(i) = f(n+1-i) for all i.

Decomposition by class size

.

n 1 2

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

1 1 0

2 2 1

3 9 9

4 16 120

5 125 1500

6 216 23220

7 2401 410571

.

Demonstration for the formula: the classes are either of size 1 or 2.

The classes of size 1 is for functions invariant by reversal. They are specified by half their values, including one more if n is odd. Their number is n^(ceiling(n/2)).

So the number of classes under this symmetry is half (the number of functions + the number of classes of size 1).

a(n) is the number of unoriented length n strings with a maximum of n colors. - Andrew Howroyd, Sep 13 2019

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossrefs

Main diagonal of A277504.

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. 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

Cf. A275558 Classes under rotation, complement and reversal

Cf. A078707 Endofunctions symmetric around their middle (stable by reversal).

Programs

PARI

a(n) = {(n^n + n^((n+1)\2))/2} \\ Andrew Howroyd, Sep 13 2019

Formula

a(n) = (n^n+n^ceiling(n/2))/2.