A275550 Number of classes of endofunctions of [n] under reversal and complement to n+1.

1, 1, 2, 10, 72, 819, 11772, 206572, 4196352, 96871525, 2500050000, 71328400806, 2229026605056, 75718793541895, 2778001759096256, 109473473278652344, 4611686020574871552, 206810065502975099529

Offset: 0

Author

Olivier Gérard, Aug 01 2016

Keywords

• nonn
• easy

Comments

Possible classes size are 1,2,4

n 1 2 4

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

1 1 0 0

2 0 2 0

3 1 5 4

4 0 16 56

5 1 74 744

6 0 216 11556

7 1 1371 205200.

Classes of size 2 can be further decomposed by whether the function is stable by reversal or stable by (reversal and complement).

n 2 2-r 2-rc

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

1 0 0 0

2 2 1 1

3 5 4 1

4 16 8 8

5 74 62 12

6 216 108 108

7 1371 1200 171.

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

Cf. A275558 Classes under rotation, complement and reversal

Cf. A192396 floor(((k+1)^n-(1+(-1)^k)/2)/2)

Cf. A275574 (2-r classes)

Programs

Mathematica

Table[1/8 (1+(-1)^(1+n)+2 n^n+n^Floor[n/2] (3+(-1)^(n+1) (-1+n)+n)),{n,1,17}]

PARI

a(n) = (1+(-1)^(n+1)+2*n^n+(3+((-1)^(n+1))*(n-1)+n)*n^(floor(n/2)) )/8; \\ Andrew Howroyd, Sep 30 2017

Formula

a(n) = (1+(-1)^(n+1)+2*n^n+(3+((-1)^(n+1))*(n-1)+n)*n^(floor(n/2)) )/8.

Classes of size 2: (2 (-1 + (-1)^n) + n^floor(n/2)*(3 + ((-1)^(1 + n))* (-1 + n) + n))/4.

Extensions

Duplicate a(7) removed by Andrew Howroyd, Sep 30 2017