A056391 -id:A056391 - cp's OEIS frontend

A293181 Irregular triangle read by rows: T(n,k) is the number of k-partitions of {1..2n} that are invariant under a permutation consisting of n 2-cycles (1 <= k <= 2n).

1, 1, 1, 3, 2, 1, 1, 7, 10, 9, 3, 1, 1, 15, 38, 53, 34, 18, 4, 1, 1, 31, 130, 265, 261, 195, 80, 30, 5, 1, 1, 63, 422, 1221, 1700, 1696, 1016, 515, 155, 45, 6, 1, 1, 127, 1330, 5369, 10143, 13097, 10508, 6832, 2926, 1120, 266, 63, 7, 1

Offset: 1

Andrew Howroyd, Oct 01 2017

See A002872 for detailed description.
T(m,k) is the number of achiral color patterns in a row or loop of length 2m using exactly k different colors. Two color patterns are equivalent if we permute the colors. - Robert A. Russell, Apr 24 2018
T(n,k) = coefficient of x^k for A(2,n)(x) in Gilbert and Riordan's article. - Robert A. Russell, Jun 14 2018

			Triangle begins:
  1,   1;
  1,   3,    2,    1;
  1,   7,   10,    9,     3,     1;
  1,  15,   38,   53,    34,    18,     4,    1;
  1,  31,  130,  265,   261,   195,    80,   30,    5,    1;
  1,  63,  422, 1221,  1700,  1696,  1016,  515,  155,   45,   6,  1;
  1, 127, 1330, 5369, 10143, 13097, 10508, 6832, 2926, 1120, 266, 63, 7, 1;
  ...
For T(2,2)=3, the row patterns are AABB, ABAB, and ABBA.  The loop patterns are AAAB, AABB, and ABAB. - _Robert A. Russell_, Apr 24 2018

Alois P. Heinz, Rows n = 1..100, flattened (first 30 rows from Andrew Howroyd)
Ira Gessel, What is the number of achiral color patterns for a row of n colors containing k different colors?, mathoverflow, Jan 30 2018.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Row sums are A002872.
Maximum row values are A002873.
Number of achiral color patterns of length odd n in A140735.
Column k=3 gives A056182.

(* Ach[n, k] is the number of achiral color patterns for a row or loop of n
  colors containing k different colors *)
Ach[n_, k_] := Ach[n, k] = Which[0==k, Boole[0==n], 1==k, Boole[n>0],
  OddQ[n], Sum[Binomial[(n-1)/2, i] Ach[n-1-2i, k-1], {i, 0, (n-1)/2}],
  True, Sum[Binomial[n/2-1, i] (Ach[n-2-2i, k-1]
  + 2^i Ach[n-2-2i, k-2]), {i, 0, n/2-1}]]
Table[Ach[n, k], {n, 2, 14, 2}, {k, 1, n}] // Flatten
(* Robert A. Russell, Feb 06 2018 *)
Table[Drop[MatrixPower[Table[Switch[j-i, 0, i-1, 1, 1, 2, 1, _, 0],
  {i, 1, 2n+1}, {j, 1, 2n+1}], n][[1]], 1], {n, 1, 10}] // Flatten
(* Robert A. Russell, Apr 14 2018 *)
Aeven[m_, k_] := Aeven[m, k] = If[m>0, k Aeven[m-1, k] + Aeven[m-1, k-1]
  + Aeven[m-1, k-2], Boole[m == 0 && k == 0]]
Table[Aeven[m, k], {m, 1, 10}, {k, 1, 2m}] // Flatten (* Robert A. Russell, Apr 24 2018 *)

\\ see A056391 for Polya enumeration functions
T(n,k) = 2*NonequivalentStructsExactly(CylinderPerms(2,n),k) - stirling(2*n,k,2);

seq(n)={Vec(serlaplace(exp(y*(exp(x + O(x*x^n))-1)+(1/2)*y^2*(exp(2*x + O(x*x^n))-1))) - 1)}
{my(T=seq(10)); for(n=1, #T, for(k=1, 2*n, print1(polcoeff(T[n], k), ", ")); print)} \\ Andrew Howroyd, Jan 31 2018

T(n,k) = coefficient of t^k x^n/n! in exp(t*(exp(x)-1)+(1/2)*t^2*(exp(2*x)-1)). - Ira M. Gessel, Jan 30 2018
T(m,k) = [m>0]*(k*T(m-1,k)+T(m-1,k-1)+T(m-1,k-2)) + [m==0]*[k==0]. - Robert A. Russell, Apr 24 2018
Conjecture: T(n,k) = R(n,k)-R(n,k-1), with R(n,k) = Sum_{m=0..k} m^n*A000085(m)*A038205(k-m)/(m!*(k-m)!). - Mikhail Kurkov, Jun 26 2018

A056323 Number of reversible string structures with n beads using a maximum of four different colors.

Original entry on oeis.org

1, 1, 2, 4, 11, 31, 107, 379, 1451, 5611, 22187, 87979, 350891, 1400491, 5597867, 22379179, 89500331, 357952171, 1431743147, 5726775979, 22906841771, 91626580651, 366505274027, 1466017950379, 5864067607211

Offset: 0

Marks R. Nester

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure. Thus aabc, cbaa and bbac are all considered to be identical.
Number of set partitions of an unoriented row of n elements with four or fewer nonempty subsets. - Robert A. Russell, Oct 28 2018
There are nonrecursive formulas, generating functions, and computer programs for A124303 and A305750, which can be used in conjunction with the formula. - Robert A. Russell, Oct 28 2018
From Allan Bickle, Jun 02 2022: (Start)
a(n) is the number of (unlabeled) 4-paths with n+6 vertices. (A 4-path with order n at least 6 can be constructed from a 5-clique by iteratively adding a new 4-leaf (vertex of degree 4) adjacent to an existing 4-clique containing an existing 4-leaf.)
Recurrences appear in the papers by Bickle, Eckhoff, and Markenzon et al. (End)

			For a(4)=11, the 7 achiral patterns are AAAA, AABB, ABAB, ABBA, ABCA, ABBC, and ABCD. The 4 chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Allan Bickle, How to Count k-Paths, J. Integer Sequences, 25 (2022) Article 22.5.6.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
J. Eckhoff, Extremal interval graphs, J. Graph Theory 17 1 (1993), 117-127.
L. Markenzon, O. Vernet, and P. R. da Costa Pereira, A clique-difference encoding scheme for labelled k-path graphs, Discrete Appl. Math. 156 (2008), 3216-3222.
Index entries for linear recurrences with constant coefficients, signature (5,0,-20,16).

Cf. A032121.
Column 4 of A320750.
Cf. A124303 (oriented), A320934 (chiral), A305750 (achiral).
The numbers of unlabeled k-paths for k = 2..7 are given in A005418, A001998, A056323, A056324, A056325, and A345207, respectively.
The sequences above converge to A103293(n+1).

Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k=4; Table[Sum[StirlingS2[n,j]+Ach[n,j],{j,0,k}]/2,{n,0,40}] (* Robert A. Russell, Oct 28 2018 *)
LinearRecurrence[{5, 0, -20, 16}, {1, 1, 2, 4, 11}, 40] (* Robert A. Russell, Oct 28 2018 *)

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.
For n > 0, a(n) = (16 + (-2)^n + 15*2^n + 4^n)/48. - Colin Barker, Nov 24 2012
G.f.: (1 - 4x - 3x^2 + 14x^3 - 5x^4) / ((1-x)*(1-4x)*(1-4x^2)). - Colin Barker, Nov 24 2012 [Adapted to offset 0 by Robert A. Russell, Nov 09 2018]
From Robert A. Russell, Oct 28 2018: (Start)
a(n) = (A124303(n) + A305750(n)) / 2.
a(n) = A124303(n) - A320934(n) = A320934(n) + A305750(n).
a(n) = Sum_{j=0..k} (S2(n,j) + Ach(n,j)) / 2, where k=4 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
a(n) = A000007(n) + A057427(n) + A056326(n) + A056327(n) + A056328(n). (End)

a(0)=1 prepended by Robert A. Russell, Nov 09 2018

A056325 Number of reversible string structures with n beads using a maximum of six different colors.

Original entry on oeis.org

1, 1, 2, 4, 11, 32, 117, 467, 2135, 10480, 55091, 301633, 1704115, 9819216, 57365191, 338134521, 2005134639, 11937364184, 71254895955, 426063226937, 2550552314219, 15280103807200, 91588104196415, 549159428968825

Offset: 0

Marks R. Nester

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure. Thus aabc, cbaa and bbac are all considered to be identical.
Number of set partitions of an unoriented row of n elements with six or fewer nonempty subsets. - Robert A. Russell, Oct 28 2018
There are nonrecursive formulas, generating functions, and computer programs for A056273 and A305752, which can be used in conjunction with the first formula. - Robert A. Russell, Oct 28 2018
From Allan Bickle, Jun 23 2022: (Start)
a(n) is the number of (unlabeled) 6-paths with n+8 vertices. (A 6-path with order n at least 8 can be constructed from a 6-clique by iteratively adding a new 6-leaf (vertex of degree 6) adjacent to an existing 6-clique containing an existing 6-leaf.)
Recurrences appear in the papers by Bickle, Eckhoff, and Markenzon et al. (End)

			For a(4)=11, the 7 achiral patterns are AAAA, AABB, ABAB, ABBA, ABCA, ABBC, and ABCD.  The 4 chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Colin Barker, Table of n, a(n) for n = 0..1000
Allan Bickle, How to Count k-Paths, J. Integer Sequences, 25 (2022) Article 22.5.6.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
J. Eckhoff, Extremal interval graphs, J. Graph Theory 17 1 (1993), 117-127.
L. Markenzon, O. Vernet, and P. R. da Costa Pereira, A clique-difference encoding scheme for labelled k-path graphs, Discrete Appl. Math. 156 (2008), 3216-3222.
Index entries for linear recurrences with constant coefficients, signature (16,-84,84,685,-2140,180,7200,-8244,-4176,11664,-5184).

Cf. A056308.
Column 6 of A320750.
Cf. A056273 (oriented), A320936 (chiral), A305752 (achiral).
The numbers of unlabeled k-paths for k = 2..7 are given in A005418, A001998, A056323, A056324, A056325, and A345207, respectively.
The sequences above converge to A103293(n+1).

Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k=6; Table[Sum[StirlingS2[n,j]+Ach[n,j],{j,0,k}]/2,{n,0,40}] (* Robert A. Russell, Oct 28 2018 *)
LinearRecurrence[{16, -84, 84, 685, -2140, 180, 7200, -8244, -4176, 11664, -5184}, {1, 1, 2, 4, 11, 32, 117, 467, 2135, 10480, 55091, 301633}, 40] (* Robert A. Russell, Oct 28 2018 *)

Vec((1 - 15*x + 70*x^2 - 28*x^3 - 654*x^4 + 1479*x^5 + 783*x^6 - 5481*x^7 + 3512*x^8 + 4640*x^9 - 5922*x^10 + 1530*x^11) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 6*x)*(1 - 2*x^2)*(1 - 3*x^2)*(1 - 6*x^2)) + O(x^30)) \\ Colin Barker, Apr 15 2020

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.
From Robert A. Russell, Oct 28 2018: (Start)
a(n) = (A056273(n) + A305752(n)) / 2.
a(n) = A056273(n) - A320936(n) = A320936(n) + A305752(n).
a(n) = Sum_{j=0..k} (S2(n,j) + Ach(n,j)) / 2, where k=6 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
a(n) = A000007(n) + A057427(n) + A056326(n) + A056327(n) + A056328(n) + A056329(n) + A056330(n).
(End)
From Colin Barker, Mar 24 2020: (Start)
G.f.: (1 - 15*x + 70*x^2 - 28*x^3 - 654*x^4 + 1479*x^5 + 783*x^6 - 5481*x^7 + 3512*x^8 + 4640*x^9 - 5922*x^10 + 1530*x^11) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 6*x)*(1 - 2*x^2)*(1 - 3*x^2)*(1 - 6*x^2)).
a(n) = 16*a(n-1) - 84*a(n-2) + 84*a(n-3) + 685*a(n-4) - 2140*a(n-5) + 180*a(n-6) + 7200*a(n-7) - 8244*a(n-8) - 4176*a(n-9) + 11664*a(n-10) - 5184*a(n-11) for n>11.
(End)
From Allan Bickle, Jun 23 2022: (Start)
a(n) = (1/1440)*6^n + (1/96)*4^n + (1/36)*3^n + (3/32)*2^n + (19/360)*6^(n/2) + (1/9)*3^(n/2) + (1/8)*2^(n/2) + 17/60 for n > 0 even;
a(n) = (1/1440)*6^n + (1/96)*4^n + (1/36)*3^n + (3/32)*2^n + (13/720)*6^((n+1)/2) + (1/18)*3^((n+1)/2) + (1/16)*2^((n+1)/2) + 17/60 for n > 0 odd. (End)

Another term from Robert A. Russell, Oct 29 2018

a(0)=1 prepended by Robert A. Russell, Nov 09 2018

A056449 a(n) = 3^floor((n+1)/2).

Original entry on oeis.org

1, 3, 3, 9, 9, 27, 27, 81, 81, 243, 243, 729, 729, 2187, 2187, 6561, 6561, 19683, 19683, 59049, 59049, 177147, 177147, 531441, 531441, 1594323, 1594323, 4782969, 4782969, 14348907, 14348907, 43046721, 43046721, 129140163, 129140163, 387420489, 387420489, 1162261467

Offset: 0

Marks R. Nester

One followed by powers of 3 with positive exponent, repeated. - Omar E. Pol, Jul 27 2009

Number of achiral rows of n colors using up to three colors. E.g., for a(3) = 9, the rows are AAA, ABA, ACA, BAB, BBB, BCB, CAC, CBC, and CCC. - Robert A. Russell, Nov 07 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (0,3).

Column k=3 of A321391.
Cf. A000007, A000244, A016116, A056391, A056453, A056454, A057427.
Essentially the same as A108411 and A162436.
Cf. A000244 (oriented), A032120 (unoriented), A032086(n>1) (chiral).

[3^Floor((n+1)/2): n in [0..40]]; // Vincenzo Librandi, Aug 16 2011

Riffle[3^Range[0, 20], 3^Range[20]] (* Harvey P. Dale, Jan 21 2015 *)
Table[3^Ceiling[n/2], {n,0,40}] (* or *)
LinearRecurrence[{0, 3}, {1, 3}, 40] (* Robert A. Russell, Nov 07 2018 *)

a(n)=3^floor((n+1)/2); \\ Joerg Arndt, Apr 23 2013

def A056449(n): return 3**(n+1>>1) # Chai Wah Wu, Oct 28 2024

G.f.: (1 + 3*x) / (1 - 3*x^2). - R. J. Mathar, Jul 06 2011 [Adapted to offset 0 by Robert A. Russell, Nov 07 2018]
a(n) = k^ceiling(n/2), where k = 3 is the number of possible colors. - Robert A. Russell, Nov 07 2018
a(n) = C(3,0)*A000007(n) + C(3,1)*A057427(n) + C(3,2)*A056453(n) + C(3,3)*A056454(n). - Robert A. Russell, Nov 08 2018
E.g.f.: cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x). - Stefano Spezia, Dec 31 2022

Edited by N. J. A. Sloane at the suggestion of Klaus Brockhaus, Jul 03 2009

a(0)=1 prepended by Robert A. Russell, Nov 07 2018

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

A275553 Number of classes of endofunctions of [n] under vertical translation mod n, complement to n+1 and reversal.

Original entry on oeis.org

1, 1, 2, 4, 24, 169, 2024, 29584, 525600, 10764961, 250030128, 6484436676, 185752964096, 5824523694025, 198428723433728, 7298231591777344, 288230377359679488, 12165297972404595841, 546477889989773968640, 26031837574639154232100, 1310720000002816000131072

Offset: 0

Olivier Gérard, Aug 05 2016

nonn

There are three size of classes : n, 2n, 4n.
n   c:n   c:2n   c:4n
----------------------------------
0   1
1   1
2   2
3   1     2      1
4   4     10     10
5   1     24     144
6   8     148    1868
7   1     342    29241
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, we have 2^(n/2) binary words which have mirror-symmetry
There are three types of classes of size of 2n (stable by reversal, stable by complement, stable by rc as in A275550).

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. 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), DihedralPerms(n)); \\ Andrew Howroyd, Sep 30 2017

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

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

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

cp's OEIS Frontend

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A056323 Number of reversible string structures with n beads using a maximum of four different colors.

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A056325 Number of reversible string structures with n beads using a maximum of six different colors.

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A056449 a(n) = 3^floor((n+1)/2).

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

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A275553 Number of classes of endofunctions of [n] under vertical translation mod n, complement to n+1 and reversal.

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

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