A330282 -id:A330282 - cp's OEIS frontend

A083323 a(n) = 3^n - 2^n + 1.

1, 2, 6, 20, 66, 212, 666, 2060, 6306, 19172, 58026, 175100, 527346, 1586132, 4766586, 14316140, 42981186, 129009092, 387158346, 1161737180, 3485735826, 10458256052, 31376865306, 94134790220, 282412759266, 847255055012

Offset: 0

Paul Barry, Apr 27 2003

Binomial transform of A000225 (if this starts 1,1,3,7....).
Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x, or 1) x = y. - Ross La Haye, Jan 10 2008
Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if either 0) x is not a subset of y and y is not a subset of x and x and y are disjoint, or 1) x equals y. Then a(n) = |R|. - Ross La Haye, Mar 19 2009

			From _Gus Wiseman_, Dec 10 2019: (Start)
Also the number of achiral set-systems on n vertices, where a set-system is achiral if it is not changed by any permutation of the covered vertices. For example, the a(0) = 1 through a(3) = 20 achiral set-systems are:
  0  0    0           0
     {1}  {1}         {1}
          {2}         {2}
          {12}        {3}
          {1}{2}      {12}
          {1}{2}{12}  {13}
                      {23}
                      {123}
                      {1}{2}
                      {1}{3}
                      {2}{3}
                      {1}{2}{3}
                      {1}{2}{12}
                      {1}{3}{13}
                      {2}{3}{23}
                      {12}{13}{23}
                      {1}{2}{3}{123}
                      {12}{13}{23}{123}
                      {1}{2}{3}{12}{13}{23}
                      {1}{2}{3}{12}{13}{23}{123}
BII-numbers of these set-systems are A330217. Fully chiral set-systems are A330282, with covering case A330229.
(End)

M. H. Albert, M. D. Atkinson, and V. Vatter, Inflations of geometric grid classes: three case studies, arXiv:1209.0425 [math.CO], 2012.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
Jay Pantone, The Enumeration of Permutations Avoiding 3124 and 4312, arXiv:1309.0832 [math.CO], 2013.
Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Cf. A134319, A028243, A000079.

Cf. A000612, A003238, A330098, A330234.

List([0..30], n -> 3^n-2^n+1); # G. C. Greubel, Feb 13 2019

[3^n-2^n+1: n in [0..30]]; // G. C. Greubel, Feb 13 2019

LinearRecurrence[{6,-11,6}, {1,2,6}, 30] (* G. C. Greubel, Feb 13 2019 *)

a(n)=3^n-2^n+1 \\ Charles R Greathouse IV, Oct 07 2015

[3^n-2^n+1 for n in range(30)] # G. C. Greubel, Feb 13 2019

G.f.: (1-4*x+5*x^2)/((1-x)*(1-2*x)*(1-3*x)).
E.g.f.: exp(3*x) - exp(2*x) + exp(x).
Row sums of triangle A134319. - Gary W. Adamson, Oct 19 2007
a(n) = 2*StirlingS2(n+1,3) + StirlingS2(n+1,2) + 1. - Ross La Haye, Jan 10 2008
a(n) = Sum_{k=0..n}(binomial(n,k)*A255047(k)). - Yuchun Ji, Feb 23 2019

A330229 Number of fully chiral set-systems covering n vertices.

Original entry on oeis.org

1, 1, 2, 42, 21336

Offset: 0

Gus Wiseman, Dec 08 2019

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the vertices gives a different representative.

			The a(3) = 42 set-systems:
  {1}{2}{13}    {1}{2}{12}{13}    {1}{2}{12}{13}{123}
  {1}{2}{23}    {1}{2}{12}{23}    {1}{2}{12}{23}{123}
  {1}{3}{12}    {1}{3}{12}{13}    {1}{3}{12}{13}{123}
  {1}{3}{23}    {1}{3}{13}{23}    {1}{3}{13}{23}{123}
  {2}{3}{12}    {2}{3}{12}{23}    {2}{3}{12}{23}{123}
  {2}{3}{13}    {2}{3}{13}{23}    {2}{3}{13}{23}{123}
  {1}{12}{23}   {1}{2}{13}{123}
  {1}{13}{23}   {1}{2}{23}{123}
  {2}{12}{13}   {1}{3}{12}{123}
  {2}{13}{23}   {1}{3}{23}{123}
  {3}{12}{13}   {2}{3}{12}{123}
  {3}{12}{23}   {2}{3}{13}{123}
  {1}{12}{123}  {1}{12}{23}{123}
  {1}{13}{123}  {1}{13}{23}{123}
  {2}{12}{123}  {2}{12}{13}{123}
  {2}{23}{123}  {2}{13}{23}{123}
  {3}{13}{123}  {3}{12}{13}{123}
  {3}{23}{123}  {3}{12}{23}{123}

The non-covering version is A330282.
Costrict (or T_0) covering set-systems are A059201.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic, fully chiral multiset partitions are A330227.
Fully chiral partitions are counted by A330228.
Fully chiral covering set-systems are A330229.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000612, A055621, A083323, A283877, A319559, A319564, A330224, A330234.

graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&Length[graprms[#]]==n!&]],{n,0,3}]

Binomial transform is A330282.

A330294 Number of non-isomorphic fully chiral set-systems on n vertices.

Original entry on oeis.org

1, 2, 3, 10, 899

Offset: 0

Gus Wiseman, Dec 10 2019

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the covered vertices gives a different representative.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 10 set-systems:
  0  0    0        0
     {1}  {1}      {1}
          {2}{12}  {2}{12}
                   {1}{3}{23}
                   {2}{13}{23}
                   {3}{23}{123}
                   {2}{3}{13}{23}
                   {1}{3}{23}{123}
                   {2}{13}{23}{123}
                   {2}{3}{13}{23}{123}

The labeled version is A330282.
Partial sums of A330295 (the covering case).
Unlabeled costrict (or T_0) set-systems are A326946.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Fully chiral partitions are A330228.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000612, A016031, A055621, A083323, A283877, A319637, A330098, A330231, A330232, A330234.

A330295 Number of non-isomorphic fully chiral set-systems covering n vertices.

Original entry on oeis.org

1, 1, 1, 7, 889

Offset: 0

Gus Wiseman, Dec 10 2019

A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the covered vertices gives a different representative.

			Non-isomorphic representatives of the a(0) = 1 through a(3) = 7 set-systems:
  0  {1}  {1}{12}  {1}{2}{13}
                   {1}{12}{23}
                   {1}{12}{123}
                   {1}{2}{12}{13}
                   {1}{2}{13}{123}
                   {1}{12}{23}{123}
                   {1}{2}{12}{13}{123}

The labeled version is A330229.
First differences of A330294 (the non-covering case).
Unlabeled costrict (or T_0) set-systems are A326946.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Fully chiral partitions are A330228.
Fully chiral factorizations are A330235.
MM-numbers of fully chiral multisets of multisets are A330236.
Cf. A000612, A016031, A055621, A083323, A283877, A319637, A330098, A330231, A330232, A330234, A330282.

A330345 Number of labeled simple graphs with n vertices whose covered portion has exactly two automorphisms.

Original entry on oeis.org

0, 0, 1, 6, 42, 700, 16995

Offset: 0

Gus Wiseman, Dec 12 2019

			The a(4) = 42 graphs:
  {12}  {12,13}  {12,13,24}  {12,13,14,23}
  {13}  {12,14}  {12,13,34}  {12,13,14,24}
  {14}  {12,23}  {12,14,23}  {12,13,14,34}
  {23}  {12,24}  {12,14,34}  {12,13,23,24}
  {24}  {13,14}  {12,23,34}  {12,13,23,34}
  {34}  {13,23}  {12,24,34}  {12,14,23,24}
        {13,34}  {13,14,23}  {12,14,24,34}
        {14,24}  {13,14,24}  {12,23,24,34}
        {14,34}  {13,23,24}  {13,14,23,34}
        {23,24}  {13,24,34}  {13,14,24,34}
        {23,34}  {14,23,24}  {13,23,24,34}
        {24,34}  {14,23,34}  {14,23,24,34}

The unlabeled version is A330344.
The covering case is A330297.
Covering simple graphs are A006129.
Graphs with exactly two automorphisms are A330297 (labeled covering), A330344 (unlabeled), A330345 (labeled), A330346 (unlabeled covering), A241454 (unlabeled connected).
Cf. A006125, A143543, A330098, A330228, A330230, A330282, A330343.

graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[graprms[#]]==Length[Union@@#]!/2&]],{n,0,4}]

A330343 Number of labeled fully chiral simple graphs (also called identity or asymmetric graphs) covering n vertices.

Original entry on oeis.org

1, 0, 0, 0, 0, 5760, 766080, 149022720, 48990251520, 28928242022400, 32147584690636800, 69035206021583155200

Offset: 1

Gus Wiseman, Dec 12 2019

In a fully chiral graph, every permutation of the vertices gives a different representative, so the only automorphism is the identity.

The unlabeled version is A003400.
Identity trees are A004111.
Covering simple graphs are A006129.
Full chiral integer partitions are A330228.
Fully chiral factorizations are A330235.
Fully chiral set-systems are A330229 (labeled covering), A330282 (labeled), A330294 (unlabeled), A330295 (unlabeled covering).
Graphs with exactly two automorphisms are A330297 (labeled covering), A330344 (unlabeled), A330345 (labeled), A330346 (unlabeled covering), A241454 (unlabeled connected).
Cf. A006125, A016031, A124059, A143543, A330098, A330224, A330226, A330227, A330230, A330231, A330236.

graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[graprms[#]]==n!&]],{n,5}] (* brute force, not for computation *)

a(n) = n! * A003400(n).

cp's OEIS Frontend

A083323 a(n) = 3^n - 2^n + 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

GAP

Magma

Mathematica

PARI

Sage

Formula

A330229 Number of fully chiral set-systems covering n vertices.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A330294 Number of non-isomorphic fully chiral set-systems on n vertices.

Views

Author

Keywords

Comments

Examples

Crossrefs

A330295 Number of non-isomorphic fully chiral set-systems covering n vertices.

Views

Author

Keywords

Comments

Examples

Crossrefs

A330345 Number of labeled simple graphs with n vertices whose covered portion has exactly two automorphisms.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A330343 Number of labeled fully chiral simple graphs (also called identity or asymmetric graphs) covering n vertices.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula