A168658 -id:A168658 - cp's OEIS frontend

A275554 Number of classes of endofunctions of [n] under vertical translation mod n, rotation and complement to n+1.

1, 1, 2, 3, 14, 65, 680, 8407, 131416, 2391515, 50006040, 1178973851, 30958827996, 896080197025, 28346960490560, 973097534189967, 36028797169965112, 1431211525754907905, 60719765554419645244, 2740193428892401092979, 131072000000281600209176

Offset: 0

Olivier Gérard, Aug 02 2016

nonn

Because of the interaction between the two symmetries indexed by n, classes can be of size from n up to 2*n^2.
.
n  possible class sizes
-------------------------------
1  1
2  2
3  3, 6,                   18
4  4, 8,      16,          32
5  5, 10,                  50
6  6, 12, 18, 24,     36,  72
7  7, 14,                  98
.
but classes of size 2*n^2 account for the bulk of a(n).
n  number of classes
-----------------------------------
1  1
2  2
3  1, 1,                   1
4  2, 3,       4,          5
5  1, 2,                   62
6  2, 4,   2,  2,     48,  622
7  1, 3,                   8403

Andrew Howroyd, Table of n, a(n) for n = 0..100

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. 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

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

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

A275555 Number of classes of endofunctions of [n] under vertical translation mod n, rotation and reversal.

Original entry on oeis.org

1, 1, 2, 4, 16, 77, 730, 8578, 132422, 2394795, 50031012, 1179054376, 30959574248, 896082610429, 28346986843640, 973097619619654, 36028798243701780, 1431211529242786625, 60719765604009463866, 2740193429053744941868, 131072000002841600036024

Offset: 0

Olivier Gérard, Aug 05 2016

nonn

Because of the interaction between the two symmetries indexed by n, classes can be of size from n up to 2*n^2.
n  possible class sizes
-----------------------------------
1
2
3, 6,   9
4, 8,      16,               32
5, 10,         25,           50
6, 12, 18, 24,     36,       72
7, 14,                 49,   98
but classes of size 2*n^2 account for the bulk of a(n).
n  number of classes
-----------------------------------
1
2
1, 1,   2
2, 3,       8,               3
1, 2,          24,           50
2, 4,  10,  2,    136,       576
1, 3,                 342,   8232

Andrew Howroyd, Table of n, a(n) for n = 0..100

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. A275556 Classes under translation, rotation, complement and reversal
Cf. A275557 Classes under rotation and complement
Cf. A275558 Classes under rotation, complement and reversal

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

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

A275556 Number of classes of endofunctions of [n] under vertical translation mod n, rotation, complement to n+1 and reversal.

Original entry on oeis.org

1, 1, 2, 3, 13, 45, 412, 4375, 66988, 1199038, 25033020, 589567451, 15480284910, 448042511917, 14173510363424, 486548852524671, 18014399792942108, 715605766365332673, 30359882832309625502, 1370096714607544395379, 65536000002956800104588

Offset: 0

Olivier Gérard, Aug 05 2016

nonn

Because of the interaction between the two symmetries indexed by n and the two involutions, classes can be of size from n up to 4*n^2.
.
n  possible class sizes
------------------------------------
1  1
2  2
3  3, 6,                   18
4  4, 8,      16,          32,   64
5  5, 10,                  50,  100
6  6, 12, 18, 24,     36,  72,  144
7  7, 14,                  98,  196
.
but classes of size 4*n^2 account for the bulk of a(n).
n  number of classes
------------------------------------
1  1
2  2
3  1, 1,                    1
4  2, 3,       4,           3,    1
5  1, 2,                   22,   20
6  2, 4,   2,  2,     28, 116,  258
7  1, 3,                  339, 4032

Andrew Howroyd, Table of n, a(n) for n = 0..100

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. A275557 Classes under rotation and complement
Cf. A275558 Classes under rotation, complement and reversal

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

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

A275557 Number of classes of endofunctions of [n] under rotation and complement to n+1.

Original entry on oeis.org

1, 1, 2, 6, 38, 315, 3932, 58828, 1049108, 21523445, 500010024, 12968712306, 371504436220, 11649042561247, 396857394156656, 14596463012746392, 576460752571867208, 24330595937833434249, 1092955779880370116836, 52063675148955620766430, 2621440000000512000336088

Offset: 0

Olivier Gérard, Aug 05 2016

nonn

Classes can be of size 1,2,4, n and 2n.
n   1  2  4      n     2n
--------------------------
1   1
2   0  2
3   1  1                4
4   0  4  4      2     28
5   1  2  0      0    312
6   0  6  6     70   3850
7   1  3  0      0  58824
For n odd, the constant function (n+1)/2 is the only stable by rotation and complement. 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.

Andrew Howroyd, Table of n, a(n) for n = 0..100

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. A275558 Classes under rotation, complement and reversal

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

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

A078707 Number of vectors of length n that are symmetric about the middle, where each element is drawn from a set of n distinct elements.

Original entry on oeis.org

1, 1, 2, 9, 16, 125, 216, 2401, 4096, 59049, 100000, 1771561, 2985984, 62748517, 105413504, 2562890625, 4294967296, 118587876497, 198359290368, 6131066257801, 10240000000000, 350277500542221, 584318301411328, 21914624432020321, 36520347436056576

Offset: 0

Mark Sterling, Dec 18 2002

			Examples added by _N. J. A. Sloane_, Jun 17 2014:
n=1: 1 (1).
n=2: 11, 22 (2).
n=3: 111X3, 121X6 (9).
n=4: 1111X4, 1221X12 (16).
n=5: 11111X5, 11211X20, 12221X20, 12121X20, 12321X60 (125).

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A004526, A243520.
Cf. A168658, A275549.
This is for Coxeter type B what A152291 is for Coxeter type A.

a:= n-> n^ceil(n/2): seq(a(n), n=0..30);  # Alois P. Heinz, Jul 23 2014

Join[{1}, Table[n^Ceiling[n/2], {n, 30}]] (* Wesley Ivan Hurt, Jan 15 2017 *)

PARI
```
for(n=1,22,print1(n^((n+n%2)/2),","))
```

a(n) = n^(floor((n+1)/2)) = n^ceiling(n/2).

Extended by Klaus Brockhaus, Dec 19 2002

a(0)=1 inserted by Alois P. Heinz, Jul 23 2014

cp's OEIS Frontend

A275554 Number of classes of endofunctions of [n] under vertical translation mod n, rotation and complement to n+1.

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A275555 Number of classes of endofunctions of [n] under vertical translation mod n, rotation and reversal.

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A275556 Number of classes of endofunctions of [n] under vertical translation mod n, rotation, complement to n+1 and reversal.

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A275557 Number of classes of endofunctions of [n] under rotation and complement to n+1.

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A078707 Number of vectors of length n that are symmetric about the middle, where each element is drawn from a set of n distinct elements.

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