A275553 -id:A275553 - cp's OEIS frontend

A056665 Number of equivalence classes of n-valued Post functions of 1 variable under action of complementing group C(1,n).

1, 3, 11, 70, 629, 7826, 117655, 2097684, 43046889, 1000010044, 25937424611, 743008623292, 23298085122493, 793714780783770, 29192926025492783, 1152921504875290696, 48661191875666868497, 2185911559749720272442, 104127350297911241532859

Offset: 1

Vladeta Jovovic, Aug 09 2000

Diagonal of arrays defined in A054630 and A054631.
Given n colors, a(n) = number of necklaces with n beads and 1 up to n colors effectively assigned to them (super_labeled: which also generates n different monochrome necklaces). - Wouter Meeussen, Aug 09 2002
Number of endofunctions on a set with n objects up to cyclic permutation (rotation). E.g., for n = 3, the 11 endofunctions are 1,1,1; 2,2,2; 3,3,3; 1,1,2; 1,2,2; 1,1,3; 1,3,3; 2,2,3; 2,3,3; 1,2,3; and 1,3,2. - Franklin T. Adams-Watters, Jan 17 2007
Also number of pre-necklaces in Sigma(n,n) (see Ruskey and others). - Peter Luschny, Aug 12 2012
From Olivier Gérard, Aug 01 2016: (Start)
Decomposition of the endofunctions by class size.
.
  n |  1   2   3   4    5     6       7
  --+----------------------------------
  1 |  1
  2 |  2   1
  3 |  3   0   8
  4 |  4   6   0  60
  5 |  5   0   0   0  624
  6 |  6  15  70   0    0  7735
  7 |  7   0   0   0    0     0  117648
.
The right diagonal gives the number of Lyndon Words or aperiodic necklaces, A075147. By multiplying each column by the corresponding size and summing, one gets A000312.
(End)

			The 11 necklaces for n=3 are (grouped by partition of 3): (RRR,GGG,BBB),(RRG,RGG, RRB,RBB, GGB,GBB), (RGB,RBG).

D. E. Knuth. Generating All Tuples and Permutations. The Art of Computer Programming, Vol. 4, Fascicle 2, 7.2.1.1. Addison-Wesley, 2005.

Alois P. Heinz, Table of n, a(n) for n = 1..200
M. A. Harrison and R. G. High, On the cycle index of a product of permutation groups, J. Combin. Theory, 4 (1968), 277-299.
F. Ruskey, C. Savage, and T. M. Y. Wang, Generating necklaces, Journal of Algorithms, 13(3), 414 - 430, 1992.
Index entries for sequences related to groups

Cf. A001323, A001324, A001372, A000312.
Diagonal of arrays defined in A054630, A054631 and A075195.
Cf. A075147 Aperiodic necklaces, a subset of this sequence.
Cf. A000169 Classes under translation mod n
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. A228640.

with(numtheory):
a:= n-> add(phi(d)*n^(n/d), d=divisors(n))/n:
seq(a(n), n=1..25);  # Alois P. Heinz, Jun 18 2013

Table[Fold[ #1+EulerPhi[ #2] n^(n/#2)&, 0, Divisors[n]]/n, {n, 7}]

a(n) = sum(k=1,n,n^gcd(k,n)) / n; \\ Joerg Arndt, Mar 19 2017

# This algorithm counts all n-ary n-tuples (a_1,..,a_n) such that the string a_1...a_n is preprime. It is algorithm F in Knuth 7.2.1.1.
def A056665_list(n):
    C = []
    for m in (1..n):
        a = [0]*(n+1); a[0]=-1;
        j = 1; count = 0
        while(true):
            if m%j == 0 : count += 1;
            j = n
            while a[j] >= m-1 : j -= 1
            if j == 0 : break
            a[j] += 1
            for k in (j+1..n): a[k] = a[k-j]
        C.append(count)
    return C

def A056665(n): return sum(euler_phi(d)*n^(n//d)//n for d in divisors(n))
[A056665(n) for n in (1..18)] # Peter Luschny, Aug 12 2012

a(n) = Sum_{d|n} phi(d)*n^(n/d)/n.
a(n) ~ n^(n-1). - Vaclav Kotesovec, Sep 11 2014
a(n) = (1/n) * Sum_{k=1..n} n^gcd(k,n). - Joerg Arndt, Mar 19 2017
a(n) = [x^n] -Sum_{k>=1} phi(k)*log(1 - n*x^k)/k. - Ilya Gutkovskiy, Mar 21 2018
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = (1/n)*Sum_{k=1..n} n^(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)).
a(n) = (1/n)*A228640(n). (End)

A081721 Number of bracelets of n beads in up to n colors.

Original entry on oeis.org

1, 3, 10, 55, 377, 4291, 60028, 1058058, 21552969, 500280022, 12969598086, 371514016094, 11649073935505, 396857785692525, 14596464294191704, 576460770691256356, 24330595997127372497, 1092955780817066765469, 52063675152021153895330, 2621440000054016000176044

Offset: 1

N. J. A. Sloane, based on information supplied by Gary W. Adamson, Apr 05 2003

nonn

T(n,n), T given in A081720.
From Olivier Gérard, Aug 01 2016: (Start)
Number of classes of functions of [n] to [n] under rotation and reversal.
.
Classes can be of size between 1 and 2n
depending on divisibility properties of n.
.
n   1   2   3   4    5         n     2n
----------------------------------------
1   1
2   2   1
3   3   0   6                         1
4   4   6   0                 30     15
5   5   0   0                120    252
6   6  15  30                725   3515
7   7   0   0               2394  57627
.
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..200

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. 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
Row sums of partition array A213941.
Main diagonal of A321791.

Table[CycleIndex[DihedralGroup[n],s]/.Table[s[i]->n,{i,1,n}],{n,1,20}] (* Geoffrey Critzer, Jun 18 2013 *)
t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n + 1)/2))/(2*n)]); a[n_] := t[n, n]; Array[a, 20] (* Jean-François Alcover, Nov 02 2017, after Maple code for A081720 *)

a(n) ~ n^(n-1) / 2. - Vaclav Kotesovec, Mar 18 2017

Name changed by Olivier Gérard, Aug 05 2016

Name revised by Álvar Ibeas, Apr 20 2018

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

Original entry on oeis.org

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

Offset: 0

Olivier Gérard, Aug 01 2016

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

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

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).

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

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

A130293 Number of necklaces of n beads with up to n colors, with cyclic permutation {1,..,n} of the colors taken to be equivalent.

Original entry on oeis.org

1, 2, 5, 20, 129, 1316, 16813, 262284, 4783029, 100002024, 2357947701, 61917406672, 1792160394049, 56693913450992, 1946195068379933, 72057594071484456, 2862423051509815809, 121439531097819321972, 5480386857784802185957, 262144000000051200072048, 13248496640331026150086281

Offset: 1

Wouter Meeussen, Aug 06 2007, Aug 14 2007

nonn

From Olivier Gérard, Aug 01 2016: (Start)
Equivalent to the definition: number of classes of endofunctions of [n] under rotation and translation mod n.
Classes can be of size between n and n^2 depending on divisibility properties of n.
  n   n   2n   3n  ...  n^2
  --------------------------
  1   1
  2   2
  3   3                   2
  4   4    2             14
  5   5    0            124
  6   6    6   22      1282
  7   7    0          16806
For prime n, the only possible class sizes are n and n^2, the classes of size n are the n arithmetical progression modulo n so #(c-n)=n, #(c-n^2)=(n^n - n*n)/n^2 = n^(n-2)-1 and a(n) = n^(n-2)+n-1.
(End)

			The 5 necklaces for n=3 are: 000, 001, 002, 012 and 021.

Stefano Spezia, Table of n, a(n) for n = 1..350
Darij Grinberg and Peter Mao, Necklaces over a group with identity product, arXiv:2405.08937 [math.CO], 2024. See p. 22.

Cf. A002075-A002076.
Cf. A081720.
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.

tor8={};ru8=Thread[ i_ ->Table[ Mod[i+k,8],{k,8}]];Do[idi=IntegerDigits[k,8,8];try= Function[w, First[temp=Union[Join @@(Table[RotateRight[w,k],{k,8}]/.#&)/@ ru8]]][idi];If[idi===try, tor8=Flatten[ {tor8,{{Length[temp],idi}}},1] ],{k,0,8^8-1}];
a[n_]:=Sum[d EulerPhi[d]n^(n/d),{d,Divisors[n]}]/n^2; Array[a,21] (* Stefano Spezia, May 21 2024 *)

a(n) = sumdiv(n, d, d*eulerphi(d)*n^(n/d))/n^2; \\ Michel Marcus, Aug 05 2016

a(n) = (1/n^2)*Sum_{d|n} d*phi(d)*n^(n/d). - Vladeta Jovovic, Aug 14 2007, Aug 24 2007

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

Original entry on oeis.org

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

Offset: 0

Olivier Gérard, Aug 05 2016

nonn

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.

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. A275557 Classes under rotation and complement

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

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

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

Original entry on oeis.org

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

Offset: 0

Olivier Gérard, Aug 01 2016

Possible classes size are 1,2,4
n 1 2    4
-----------------
1 0    0
0 2    0
1 5    4
0 16   56
1 74   744
0 216  11556
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
-----------------
0    0    0
2    1    1
5    4    1
16   8    8
74   62   12
216  108  108
1371 1200 171.

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. 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)

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

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

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.

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

A275551 Number of classes of endofunctions of [n] under vertical translation mod n and reversal.

Original entry on oeis.org

1, 1, 2, 6, 36, 325, 3924, 58996, 1049088, 21526641, 500010000, 12968792826, 371504434176, 11649044974645, 396857394156608, 14596463098125000, 576460752571858944, 24330595941321312961, 1092955779880368226560, 52063675149116964615310, 2621440000000512000000000

Offset: 0

Olivier Gérard, Aug 02 2016

nonn

There are two size of classes, n or 2n.
n   c:n    c:2n   (c:n)/n  (c:2n)/n
 1
 1
 2
 3      3      1        1
 8      28     2        7
 25     300    5        60
 72     3852   12       642
 343    58653  49       8379

			a(2) = 2: 11, 12.
a(3) = 6: 111, 112, 113, 121, 123, 131.
a(4) = 36: 1111, 1112, 1113, 1114, 1121, 1122, 1123, 1124, 1131, 1132, 1133, 1134, 1141, 1142, 1143, 1212, 1213, 1214, 1221, 1223, 1224, 1231, 1234, 1241, 1242, 1243, 1312, 1313, 1323, 1324, 1331, 1334, 1341, 1412, 1423, 1441.

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. 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

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

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

A275552 Number of classes of endofunctions of [n] under vertical translation mod n and complement to n+1.

Original entry on oeis.org

1, 1, 2, 5, 36, 313, 3904, 58825, 1048640, 21523361, 500000256, 12968712301, 371504186368, 11649042561241, 396857386631168, 14596463012695313, 576460752303439872, 24330595937833434241, 1092955779869348331520, 52063675148955620766421, 2621440000000000000262144

Offset: 0

Olivier Gérard, Aug 02 2016

There are two size of classes, n or 2n.
.
n   c:n  c:2n   (c:2n)/4
0   1
1   1
2   2
3   1    4      1
4   8    28     7
5   1    312    78
6   32   3872   968
7   1    58824  14706
For n odd, only the set of n constant functions can have a member of their class equal to their complement, so c:n size is 1.
For n even, the c:n class is populated by binary words using k for 0 and n+1-k for 1. There are (2^n)/2 such words as the complement operation identifies them by pairs.
For n odd, c:2n(n) = (n^n - 1*n)/(2*n)
For n even, c:2n(n) = (n^n - 2^(n-1)*n)/(2*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. 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.

a[0] = 1; a[n_?OddQ] := 1 + (n^n - n)/(2n); a[n_?EvenQ] := 2^(n-1) + (n^n - 2^(n-1)*n)/(2n); Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 07 2017, translated from PARI *)

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

a(n) = 1 + (n^n - 1*n)/(2*n) if n is odd,

a(n) = 2^(n-1) + (n^n - 2^(n-1)*n)/(2*n) if n is even.

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

Original entry on oeis.org

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

cp's OEIS Frontend

A056665 Number of equivalence classes of n-valued Post functions of 1 variable under action of complementing group C(1,n).

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A081721 Number of bracelets of n beads in up to n colors.

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

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A130293 Number of necklaces of n beads with up to n colors, with cyclic permutation {1,..,n} of the colors taken to be equivalent.

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

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

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A275551 Number of classes of endofunctions of [n] under vertical translation mod n and reversal.

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