A003216 -id:A003216 - cp's OEIS frontend

A000595 Number of binary relations on n unlabeled points.

1, 2, 10, 104, 3044, 291968, 96928992, 112282908928, 458297100061728, 6666621572153927936, 349390545493499839161856, 66603421985078180758538636288, 46557456482586989066031126651104256, 120168591267113007604119117625289606148096, 1152050155760474157553893461743236772303142428672

Offset: 0

N. J. A. Sloane

Number of orbits under the action of permutation group S(n) on n X n {0,1} matrices. The action is defined by f.M(i,j)=M(f(i),f(j)).

Equivalently, the number of digraphs on n unlabeled nodes with loops allowed but no more than one arc with the same start and end node. - Andrew Howroyd, Oct 22 2017

			From _Gus Wiseman_, Jun 17 2019: (Start)
Non-isomorphic representatives of the a(2) = 10 relations:
  {}
  {1->1}
  {1->2}
  {1->1, 1->2}
  {1->1, 2->1}
  {1->1, 2->2}
  {1->2, 2->1}
  {1->1, 1->2, 2->1}
  {1->1, 1->2, 2->2}
  {1->1, 1->2, 2->1, 2->2}
(End)

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 76 (2.2.30)
M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sept. 15, 1955, pp. 14-22.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jean-François Alcover, Table of n, a(n) for n = 0..50 (a(0)-a(37) from Charles R. Greathouse IV)
Edward A. Bender and E. Rodney Canfield, Enumeration of connected invariant graphs, Journal of Combinatorial Theory, Series B 34.3 (1983): 268-278. See p. 274.
Peter J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Alberto Casagrande, Carla Piazza, and Alberto Policriti, Is hyper-extensionality preservable under deletions of graph elements?, Preprint 2015.
Alexander Clow, Alfie Davies, and Neil Anderson McKay, Constructing All Birthday 3 Games as Digraphs, arXiv:2505.06206 [math.CO], 2025. See p. 13.
Matthew Dabkowski, N. Fan, and R. Breiger, Exploratory blockmodeling for one-mode, unsigned, deterministic networks using integer programming and structural equivalence, Social Networks, Volume 47, October 2016, Pages 93-106.
Robert L. Davis, The number of structures of finite relations, Proc. Amer. Math. Soc. 4 (1953), 486-495.
Thomas M. A. Fink, Emmanuel Barillot, and Sebastian E. Ahnert, Dynamics of network motifs, 2006.
Frank Harary, Edgar M. Palmer, Robert W. Robinson, and Allen J. Schwenk, Enumeration of graphs with signed points and lines, J. Graph Theory 1 (1977), no. 4, 295-308.
Sergiy Kozerenko, On the abstract properties of Markov graphs for maps on trees, Mathematical Bilten 41:2 (2017), pp. 5-21.
M. D. McIlroy, Calculation of numbers of structures of relations on finite sets, Massachusetts Institute of Technology, Research Laboratory of Electronics, Quarterly Progress Reports, No. 17, Sep. 15, 1955, pp. 14-22. [Annotated scanned copy]
W. Oberschelp, Kombinatorische Anzahlbestimmungen in Relationen, Math. Ann., 174 (1967), 53-78.
Götz Pfeiffer, Counting Transitive Relations, Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.2.
Samuel Reid, On Generalizing a Temporal Formalism for Game Theory to the Asymptotic Combinatorics of S5 Modal Frames, arXiv preprint arXiv:1305.0064 [math.LO], 2013.
Marko Riedel, Counting nonisomorphic binary relations (includes Maple code).
R. W. Robinson, Notes - "A Present for Neil Sloane"
R. W. Robinson, Notes - computer printout
J. M. Tangen and N. J. A. Sloane, Correspondence, 1976-1976
Leopold Travis, Graphical Enumeration: A Species-Theoretic Approach, arXiv:math/9811127 [math.CO], 1998.
Gus Wiseman, Non-isomorphic representatives of the a(3) = 104 digraphs.
Index entries for sequences related to binary matrices

Cf. A000088, A000273, A001173, A001174, A002416, A003087, A003216, A002724.

NSeq := function ( n ) return Sum(List(ConjugacyClasses(SymmetricGroup(n)), c -> (2^Length(Orbits(Group(Representative(c)), CartesianProduct([1..n],[1..n]), OnTuples))) * Size(c)))/Factorial(n); end; # Dan Hoey, May 04 2001

Join[{1,2}, Table[CycleIndex[Join[PairGroup[SymmetricGroup[n],Ordered], Permutations[Range[n^2-n+1,n^2]],2],s] /. Table[s[i]->2, {i,1,n^2-n}], {n,2,7}]] (* Geoffrey Critzer, Nov 02 2011 *)
permcount[v_] := Module[{m=1, s=0, k=0, t}, For[i=1, i <= Length[v], i++, t = v[[i]]; k = If[i>1 && t == v[[i-1]], k+1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v];
a[n_] := (s=0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]}]; s/n!);
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 08 2018, after Andrew Howroyd *)
dinorm[m_]:=If[m=={},{},If[Union@@m!=Range[Max@@Flatten[m]],dinorm[m/.Apply[Rule,Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}],{1}]],First[Sort[dinorm[m,1]]]]];
dinorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#1>=aft&]}]},Union@@(dinorm[#1,aft+1]&)/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0}],{par,First/@Position[mx,Max[mx]]}]]]];
Table[Length[Union[dinorm/@Subsets[Tuples[Range[n],2]]]],{n,0,3}] (* Gus Wiseman, Jun 17 2019 *)

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v) = {sum(i=2, #v, sum(j=1, i-1, 2*gcd(v[i],v[j]))) + sum(i=1, #v, v[i])}
a(n) = {my(s=0); forpart(p=n, s+=permcount(p)*2^edges(p)); s/n!} \\ Andrew Howroyd, Oct 22 2017

from itertools import product
from math import prod, factorial, gcd
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A000595(n): return int(sum(Fraction(1<Chai Wah Wu, Jul 02 2024

a(n) = sum {1*s_1+2*s_2+...=n} (fixA[s_1, s_2, ...] / (1^s_1*s_1!*2^s_2*s_2!*...)) where fixA[s_1, s_2, ...] = 2^sum {i, j>=1} (gcd(i, j)*s_i*s_j). - Christian G. Bower, Jan 05 2004

a(n) ~ 2^(n^2)/n! [McIlroy, 1955]. - Vaclav Kotesovec, Dec 19 2016

More terms from Vladeta Jovovic, Feb 07 2000

Still more terms from Dan Hoey, May 04 2001

A057864 Number of simple traceable graphs on n nodes.

Original entry on oeis.org

1, 1, 2, 5, 18, 91, 734, 10030, 248427, 11482572, 1000231510

Offset: 1

Eric W. Weisstein

Number of undirected graphs on n nodes possessing a Hamiltonian path (not circuit).

F. Hüffner, tinygraph, software for generating integer sequences based on graph properties, version f0eaa32.
Eric Weisstein's World of Mathematics, Traceable Graph
Wikipedia, Hamiltonian path
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.
Gus Wiseman, Non-isomorphic representatives of the a(5) = 18 unlabeled simple graphs containing a Hamiltonian path.

Main diagonal of A309524.
The labeled case is A326206.
The directed case is A326221 (with loops).
Unlabeled simple graphs not containing a Hamiltonian path are A283420.
Unlabeled simple graphs containing a Hamiltonian cycle are A003216.
Cf. A000088, A006125, A246446, A283420, A326205, A326217.

A000088(n) = a(n) + A283420(n). - Gus Wiseman, Jun 17 2019

a(8) and a(9) from Eric W. Weisstein, Jun 04 2004
a(10) from Eric W. Weisstein, May 27 2009
a(11) added using tinygraph by Falk Hüffner, Jan 19 2016

A283420 Number of simple (not necessarily connected) untraceable graphs on n nodes.

Original entry on oeis.org

0, 1, 2, 6, 16, 65, 310, 2316, 26241, 522596, 18766354

Offset: 1

Eric W. Weisstein, May 14 2017

Eric Weisstein's World of Mathematics, Untraceable Graph
Wikipedia, Hamiltonian path
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.
Gus Wiseman, The a(5) = 16 unlabeled simple graphs not containing a Hamiltonian path.

Cf. A000088 (number of simple graphs on n vertices).
Cf. A057864 (number of simple traceable graphs on n vertices).
Cf. A283421 (number of simple connected untraceable graphs on n vertices).
The labeled case is A326205.
The directed case is A326224 (with loops).
Unlabeled simple graphs not containing a Hamiltonian cycle are A246446.
Cf. A003216, A326206, A326217, A326221.

a(n) = A000088(n) - A057864(n).

A246446 Number of nonhamiltonian graphs with n nodes.

Original entry on oeis.org

0, 2, 3, 8, 26, 108, 661, 6150, 97585, 2700050, 135841840, 12568984762, 2179513027405

Offset: 1

Eric W. Weisstein, Aug 26 2014

Jan Goedgebeur, Barbara Meersman, Carol T. Zamfirescu, Graphs with few Hamiltonian Cycles, arXiv:1812.05650 [math.CO], 2018.
Eric Weisstein's World of Mathematics, Nonhamiltonian Graph
Wikipedia, Hamiltonian path

Cf. A000088 (number of simple graphs on n nodes).
Cf. A003216 (number of Hamiltonian graphs on n nodes).
Cf. A126149 (number of connected nonhamiltonian graphs on n nodes).
The labeled case is A326207.
The directed case is A326223 (with loops) or A326222 (without loops).
Unlabeled simple graphs not containing a Hamiltonian path are A283420.
Cf. A006125, A057864, A326205, A326206.

A000088 = Cases[Import["https://oeis.org/A000088/b000088.txt", "Table"], {, }][[All, 2]];
A003216 = Cases[Import["https://oeis.org/A003216/b003216.txt", "Table"], {, }][[All, 2]];
a[n_] := A000088[[n + 1]] - A003216[[n]];
a /@ Range[13] (* Jean-François Alcover, Jan 02 2020 *)

a(n) = A000088(n) - A003216(n).

a(12) from formula by Falk Hüffner, Aug 13 2017

a(13) added by Jan Goedgebeur, May 07 2019

A326204 Number of Hamiltonian labeled n-vertex digraphs (with loops).

Original entry on oeis.org

0, 2, 4, 120, 19104

Offset: 0

Gus Wiseman, Jun 14 2019

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

			The a(2) = 4 digraph edge-sets:
  {12,21}
  {11,12,21}
  {12,21,22}
  {11,12,21,22}

Wikipedia, Hamiltonian path

The unlabeled case is A326226.
The case without loops is A326219.
The undirected case (without loops) is A326208.
Non-Hamiltonian digraphs are A326220.
Digraphs containing a Hamiltonian path are A326214.
Cf. A000595, A002416, A003024, A003216, A057864.

Table[Length[Select[Subsets[Tuples[Range[n],2]],FindHamiltonianCycle[Graph[Range[n],DirectedEdge@@@#]]!={}&]],{n,0,4}] (* Mathematica 8.0+. Warning: Using HamiltonianGraphQ instead of FindHamiltonianCycle returns a(4) = 19200 which is incorrect *)

A326225 Number of Hamiltonian unlabeled n-vertex digraphs (without loops).

Original entry on oeis.org

0, 1, 1, 4, 61, 3725, 844141, 626078904

Offset: 0

Gus Wiseman, Jun 15 2019

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

			Non-isomorphic representatives of the a(3) = 4 digraph edge-sets:
  {12,23,31}
  {12,13,21,32}
  {12,13,21,23,31}
  {12,13,21,23,31,32}

The labeled case is A326219.
The case with loops is A326226.
The undirected case is A003216.
Non-Hamiltonian unlabeled digraphs (without loops) are A326222.
Cf. A000595, A002416, A003087, A003216, A053763, A246446, A326204, A326218, A326223.

a(5)-a(7) from Sean A. Irvine, Jun 16 2019

A326206 Number of n-vertex labeled simple graphs containing a Hamiltonian path.

Original entry on oeis.org

0, 0, 1, 4, 34, 633, 23368, 1699012, 237934760, 64558137140, 34126032806936, 35513501049012952

Offset: 0

Gus Wiseman, Jun 14 2019

A path is Hamiltonian if it passes through every vertex exactly once.

F. Hüffner, tinygraph, software for generating integer sequences based on graph properties, version 9766535.
Wikipedia, Hamiltonian path
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.
Gus Wiseman, The a(4) = 34 simple graphs containing a Hamiltonian path

The unlabeled case is A057864.
The directed case is A326214 (with loops) or A326217 (without loops).
Simple graphs without a Hamiltonian path are A326205.
Simple graphs with a Hamiltonian cycle are A326208.
Cf. A003216, A006125, A057864, A283420.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],FindHamiltonianPath[Graph[Range[n],#]]!={}&]],{n,0,4}] (* Mathematica 10.2+ *)

A006125(n) = a(n) + A326205(n).

a(7)-a(11) added using tinygraph by Falk Hüffner, Jun 21 2019

A326208 Number of Hamiltonian labeled simple graphs with n vertices.

Original entry on oeis.org

0, 1, 0, 1, 10, 218, 10078, 896756, 151676112, 47754337568, 28229412456056, 31665593711174080

Offset: 0

Gus Wiseman, Jun 15 2019

A graph is Hamiltonian if it contains a cycle passing through every vertex exactly once.

F. Hüffner, tinygraph, software for generating integer sequences based on graph properties, version 9766535.
Wikipedia, Hamiltonian path
Gus Wiseman, The a(5) = 218 simple graphs containing a Hamiltonian cycle.

The unlabeled version is A003216.
The directed version is A326204 (with loops) or A326219 (without loops).
Simple graphs not containing a Hamiltonian cycle are A326207.
Simple graphs containing a Hamiltonian path are A326206.
Cf. A003216, A006125, A057864, A283420.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],FindHamiltonianCycle[Graph[Range[n],#]]!={}&]],{n,0,4}] (* Mathematica 8.0+ *)

A006125(n) = a(n) + A326207(n).

a(7)-a(11) added using tinygraph by Falk Hüffner, Jun 21 2019

A326226 Number of unlabeled n-vertex Hamiltonian digraphs (with loops).

Original entry on oeis.org

0, 2, 3, 24, 858

Offset: 0

Gus Wiseman, Jun 14 2019

A digraph is Hamiltonian if it contains a directed cycle passing through every vertex exactly once.

			Non-isomorphic representatives of the a(2) = 3 digraph edge-sets:
  {12,21}
  {11,12,21}
  {11,12,21,22}

Wikipedia, Hamiltonian path
Gus Wiseman, Non-isomorphic representatives of the a(3) = 24 Hamiltonian digraphs

The labeled case is A326204.
The case without loops is A326225.
The undirected case is A003216 (without loops) or A326215 (with loops).
Unlabeled non-Hamiltonian digraphs are A326223.
Unlabeled digraphs with a Hamiltonian path are A326221.
Cf. A000595, A002416, A003087, A057864, A246446, A326208.

dinorm[m_]:=If[m=={},{},If[Union@@m!=Range[Max@@Flatten[m]],dinorm[m/. Apply[Rule,Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}],{1}]],First[Sort[dinorm[m,1]]]]];
dinorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#1>=aft&]}]},Union@@(dinorm[#1,aft+1]&)/@Union[Table[Map[Sort,m/. {par+aft-1->aft,aft->par+aft-1},{0}],{par,First/@Position[mx,Max[mx]]}]]]];
Table[Length[Select[Union[dinorm/@Subsets[Tuples[Range[n],2]]],FindHamiltonianCycle[Graph[Range[n],DirectedEdge@@@#]]!={}&]],{n,0,4}] (* Mathematica 8.0+. Warning: Using HamiltonianGraphQ instead of FindHamiltonianCycle returns a(4) = 867 which is incorrect *)

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A034183 Erroneous version of A003216.

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A000595 Number of binary relations on n unlabeled points.

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A057864 Number of simple traceable graphs on n nodes.

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A283420 Number of simple (not necessarily connected) untraceable graphs on n nodes.

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A246446 Number of nonhamiltonian graphs with n nodes.

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A326204 Number of Hamiltonian labeled n-vertex digraphs (with loops).

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A326225 Number of Hamiltonian unlabeled n-vertex digraphs (without loops).

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A326206 Number of n-vertex labeled simple graphs containing a Hamiltonian path.

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A326208 Number of Hamiltonian labeled simple graphs with n vertices.

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