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

0,3

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

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.

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}]

Prog

(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

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)

Sequence in context: A277502 A231039 A224799 * A243250 A088189 A228609

Adjacent sequences: A275547 A275548 A275549 * A275551 A275552 A275553

Keyword

nonn,easy

Author

Olivier Gérard, Aug 01 2016

Extensions

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

Status

approved