A006024 -id:A006024 - cp's OEIS frontend

A079307 Number of n-node labeled mating digraphs whose complements are connected.

1, 2, 47, 3537, 990504, 1052216730, 4368243719790, 71899754628285990, 4719127231162959826680, 1237680394434222243386835240, 1297992518900121521819851167607200

Offset: 1

Vladeta Jovovic, Feb 09 2003

nonn

Cf. A006024, A006025, A079306.

A259763 Number of symmetric n X n (0,1)-matrices with pairwise distinct rows and columns.

Original entry on oeis.org

1, 2, 6, 44, 716, 24416, 1680224, 229468288, 61820527104, 32848197477760, 34502874046006912, 71850629135663531776, 297429744309497638961920, 2452504520881914016303901696, 40340635076928240671195746599936, 1324981038432182976845483456362661888, 86953044949519288083916385603832568137728

Offset: 0

Max Alekseyev, Jul 04 2015

nonn

G. C. Greubel, Table of n, a(n) for n = 0..80
MathOverflow, Counting matrices of special types, 2015.

Cf. A088310, A006024

Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763

[(&+[StirlingFirst(n,k)*2^Binomial(k+1,2): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Nov 04 2018

Table[Sum[StirlingS1[n,k]*2^Binomial[k+1,2], {k,0,n}], {n,0,20}] (* G. C. Greubel, Nov 04 2018*)

A259763(n) = sum(k=1,n, stirling(n,k,1) * 2^(k*(k+1)/2) );

a(n) = Sum_{k=0..n} Stirling1(n,k) * 2^(k*(k+1)/2).

a(0)=1 prepended by Alois P. Heinz, Jul 12 2015

A079306 Number of n-node labeled mating graphs whose complements are connected.

Original entry on oeis.org

1, 0, 3, 19, 462, 18268, 1410394, 206677954, 58152577504, 31715883698744, 33827568741818376, 71066571962356168856, 295645506683051855529248, 2444503529745123468127635920, 40269655263141217619540592737168

Offset: 1

Vladeta Jovovic, Feb 09 2003

nonn

Cf. A006024, A006025, A079307.

A006025 Number of labeled mating digraphs with n nodes.

Original entry on oeis.org

1, 1, 3, 54, 3750, 1009680, 1058347920, 4375678520640, 71934792452208000, 4719774805970453006400, 1237727595442264073683462080, 1298006134163762816201615178698880, 5444432200219729912412940250057668378240

Offset: 0

Simon Plouffe

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..50
R. C. Read, The enumeration of mating-type graphs, Report CORR 89-38, Dept. Combinatorics and Optimization, Univ. Waterloo, 1989.

Cf. A006024.

a[0] = 1; a[n_] := Sum[StirlingS1[n , k]*2^(k^2 - k), {k, 0, n}];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Aug 16 2019 *)

a(n)=n!*polcoeff(sum(k=0,n,2^(k*(k-1))*log(1+x+x*O(x^n))^k/k!),n) \\ Paul D. Hanna, May 20 2009

a(n) = Sum_{k=0..n} Stirling1(n, k)*2^(k^2-k). - Vladeta Jovovic, Feb 11 2003

E.g.f.: Sum_{n>=0} 2^(n*(n-1))*log(1+x)^n/n!. - Paul D. Hanna, May 20 2009

More terms from Vladeta Jovovic, Feb 11 2003

a(0)=1 prepended by Andrew Howroyd, Sep 09 2018

A129584 Number of unlabeled bi-point-determining graphs: graphs in which no two vertices have the same neighborhoods or the same augmented neighborhoods (the augmented neighborhood of a vertex is the neighborhood of the vertex union the vertex itself).

Original entry on oeis.org

1, 0, 0, 1, 6, 36, 324, 5280, 156088, 8415760

Offset: 1

Ji Li (vieplivee(AT)hotmail.com), May 07 2007

This is the unlabeled case of bi-point-determining graphs, which are basically graphs that are both point-determining (no two vertices have the same neighborhoods) and co-point-determining (graphs whose complements are point-determining)

Ira M. Gessel and Ji Li, Enumeration of point-determining Graphs, Journal of Combinatorial Theory, Series A 118 (2011) 591-612.

Cf. graphs: labeled A006125, unlabeled A000568; connected graphs: labeled A001187, unlabeled A001349; point-determining graphs: labeled A006024, unlabeled A004110; connected point-determining graphs: labeled A092430, unlabeled A004108; connected co-point-determining graphs: labeled A079306, unlabeled A004108; bi-point-determining graphs: labeled A129583, unlabeled A129584; connected bi-point-determining graphs: labeled A129585, unlabeled A129586; phylogenetic trees: labeled A000311, unlabeled A000669.

A369283 Triangle read by rows: T(n,k) is the number of labeled point-determining graphs with n nodes and k edges, n >= 0, 0 <= k <= n*(n - 1)/2.

Original entry on oeis.org

1, 1, 0, 1, 0, 3, 0, 1, 0, 0, 3, 16, 12, 0, 1, 0, 0, 15, 60, 130, 132, 140, 80, 30, 0, 1, 0, 0, 0, 15, 600, 1692, 3160, 4635, 4620, 3480, 2088, 885, 240, 60, 0, 1, 0, 0, 0, 105, 1260, 7665, 28042, 74280, 142380, 218960, 271404, 276150, 230860, 157710, 86250, 38752, 13524, 3360, 560, 105, 0, 1

Offset: 0

Andrew Howroyd, Jan 18 2024

Point-determining graphs are also called mating graphs.

			Triangle begins:
 [0] 1;
 [1] 1;
 [2] 0, 1;
 [3] 0, 3,  0,  1;
 [4] 0, 0,  3, 16,  12,    0,    1;
 [5] 0, 0, 15, 60, 130,  132,  140,   80,   30,    0,    1;
 [6] 0, 0,  0, 15, 600, 1692, 3160, 4635, 4620, 3480, 2088, 885, 240, 60, 0, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1350 (rows 0..20)
Ira M. Gessel and Ji Li, Enumeration of point-determining graphs, J. Combinatorial Theory Ser. A 118 (2011), 591-612.

Row sums are A006024.

Cf. A102579, A102596, A368987 (unlabeled).

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(p,t) = {prod(i=2, #p, prod(j=1, i-1, t(p[i]*p[j])))}
row(n) = {my(s=0); forpart(p=n, s += permcount(p)*(-1)^(n-#p)*edges(p, w->1 + x^w)); Vecrev(s)}

Sum_{k>=0} 2^k*T(n,k) = A102596(n).

Sum_{k>=0} 3^k*T(n,k) = A102579(n).

A129583 Number of labeled bi-point-determining graphs with n vertices.

Original entry on oeis.org

1, 1, 0, 0, 12, 312, 13824, 1147488, 178672128, 52666091712, 29715982846848, 32452221242518272, 69259424722321036032, 291060255757818125657088, 2421848956937579216663491584, 40050322614433939228627991906304, 1319551659023608317386779165849208832

Offset: 0

Ji Li (vieplivee(AT)hotmail.com), May 07 2007

A bi-point determining graph is a graph in which no two vertices have the same neighborhoods or the same augmented neighborhoods (the augmented neighborhood of a vertex is the neighborhood of the vertex union the vertex itself).

R. C. Read, The Enumeration of Mating-Type Graphs. Report CORR 89-38, Dept. Combinatorics and Optimization, Univ. Waterloo, 1989.

Andrew Howroyd, Table of n, a(n) for n = 0..50
Ira M. Gessel and Ji Li, Enumeration of point-determining Graphs, Journal of Combinatorial Theory, Series A 118 (2011) 591-612.

Cf. graphs: labeled A006125, unlabeled A000568; connected graphs: labeled A001187, unlabeled A001349; point-determining graphs: labeled A006024, unlabeled A004110; connected point-determining graphs: labeled A092430, unlabeled A004108; connected co-point-determining graphs: labeled A079306, unlabeled A004108; bi-point-determining graphs: labeled A129583, unlabeled A129584; connected bi-point-determining graphs: labeled A129585, unlabeled A129586; phylogenetic trees: labeled A000311, unlabeled A000669.

seq(n)={my(g=sum(k=0, n, 2^binomial(k,2)*x^k/k!) + O(x*x^n)); Vec(serlaplace(subst(g, x, 2*log(1+x+O(x*x^n))-x)))} \\ Andrew Howroyd, May 06 2021

E.g.f.: G(2*log(1+x)-x) where G(x) is the e.g.f. of A006125.

a(0)=1 prepended and terms a(13) and beyond from Andrew Howroyd, May 06 2021

A129585 Number of labeled connected bi-point-determining graphs with n vertices (see A129583).

Original entry on oeis.org

1, 1, 0, 0, 12, 252, 12312, 1061304, 170176656, 51134075424, 29204599254624, 32130964585236096, 68873851786953047040, 290164895151435531345024, 2417786648013402212500060416, 40014055814155246577685250570752, 1318911434129029730677931158374449664

Offset: 0

Ji Li (vieplivee(AT)hotmail.com), May 07 2007

The calculation of connected bi-point-determining graphs is carried out by examining the connected components of bi-point-determining graphs. For more details, see reference.

Andrew Howroyd, Table of n, a(n) for n = 0..50
Ira M. Gessel and Ji Li, Enumeration of point-determining Graphs, Journal of Combinatorial Theory, Series A 118 (2011) 591-612.

Cf. graphs: labeled A006125, unlabeled A000568; connected graphs: labeled A001187, unlabeled A001349; point-determining graphs: labeled A006024, unlabeled A004110; connected point-determining graphs: labeled A092430, unlabeled A004108; connected co-point-determining graphs: labeled A079306, unlabeled A004108; bi-point-determining graphs: labeled A129583, unlabeled A129584; connected bi-point-determining graphs: labeled A129585, unlabeled A129586; phylogenetic trees: labeled A000311, unlabeled A000669.

max = 15; f[x_] := x + Log[ Sum[ 2^Binomial[n, 2]*((2*Log[1 + x] - x)^n/n!), {n, 0, max}]/(1 + x)]; A129585 = Drop[ CoefficientList[ Series[ f[x], {x, 0, max}], x]*Range[0, max]!, 1](* Jean-François Alcover, Jan 13 2012, after e.g.f. *)

seq(n)={my(g=sum(k=0, n, 2^binomial(k,2)*x^k/k!) + O(x*x^n)); Vec(serlaplace(1+x+log(subst(g, x, 2*log(1+x+O(x*x^n))-x)/(1+x))))} \\ Andrew Howroyd, May 06 2021

E.g.f.: 1 + x + log((Sum_{n>=0} 2^binomial(n,2)*(2*log(1+x)-x)^n/n!)/(1+x)). - Goran Kilibarda, Vladeta Jovovic, May 09 2007

E.g.f.: 1 + x + log(B(x)/(1+x)) where B(x) is the e.g.f. of A129583. - Andrew Howroyd, May 06 2021

More terms from Goran Kilibarda, Vladeta Jovovic, May 09 2007

a(0)=1 prepended and terms a(16) and beyond from Andrew Howroyd, May 06 2021

A129586 Number of unlabeled connected bi-point-determining graphs (see A129583).

Original entry on oeis.org

1, 0, 0, 1, 5, 31, 293, 4986, 151096, 8264613, 812528493, 144251345591, 46649058611515, 27744159658789435, 30603223477819571330, 63039669933956074333128, 243839768084859914114367906, 1779006737976575676931317142360, 24571827603944282248499044846893618

Offset: 1

Ji Li (vieplivee(AT)hotmail.com), May 07 2007

The calculation of the number of connected bi-point-determining graphs is carried out by examining the connected components of bi-point-determining graphs. For more details, see linked paper "Enumeration of point-determining Graphs".

Martin Rubey, Table of n, a(n) for n = 1..29
Florian Fürnsinn, Moritz Gangl, and Martin Rubey, Unexpectedly, a symmetry on unlabeled graphs, arXiv:2505.03665 [math.CO], 2025. See p. 18.
Ira M. Gessel and Ji Li, Enumeration of point-determining Graphs, Journal of Combinatorial Theory, Series A 118 (2011) 591-612.

Cf. graphs: labeled A006125, unlabeled A000568; connected graphs: labeled A001187, unlabeled A001349; point-determining graphs: labeled A006024, unlabeled A004110; connected point-determining graphs: labeled A092430, unlabeled A004108; connected co-point-determining graphs: labeled A079306, unlabeled A004108; bi-point-determining graphs: labeled A129583, unlabeled A129584; connected bi-point-determining graphs: labeled A129585, unlabeled A129586; phylogenetic trees: labeled A000311, unlabeled A000669.

151096 and 8264613 from Vladeta Jovovic, May 10 2007

a(n) for n >= 11 from Martin Rubey, May 08 2025

A007833 Number of point-labeled reduced two-graphs with n nodes.

Original entry on oeis.org

1, 0, 1, 1, 28, 448, 18788, 1419852, 207249896, 58206408344, 31725488477648, 33830818147141904, 71068681534173472576, 295648155633330113713344, 2444510010072634827916776064, 40269686339597630128483872278656, 1323732128140903183968664175047409152

Offset: 1

Peter J. Cameron

nonn

Also number of (n-1)-node labeled mating graphs without isolated nodes, cf. A006024. - Vladeta Jovovic, Mar 23 2004

Michael De Vlieger, Table of n, a(n) for n = 1..83
P. J. Cameron, Counting two-graphs related to trees, Elec. J. Combin., Vol. 2, #R4.

Cf. A092430 (connected).

Array[Sum[StirlingS1[#, k] 2^((k - 1) (k - 2)/2), {k, #}] &, 15] (* Michael De Vlieger, Feb 03 2018 *)

a(n) = Sum_{k=1..n} s(n, k) * 2^((k-1) * (k-2) / 2) where s(n, k) are the Stirling numbers of the first kind. - Sean A. Irvine, Feb 03 2018

cp's OEIS Frontend

A079307 Number of n-node labeled mating digraphs whose complements are connected.

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A259763 Number of symmetric n X n (0,1)-matrices with pairwise distinct rows and columns.

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A079306 Number of n-node labeled mating graphs whose complements are connected.

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A006025 Number of labeled mating digraphs with n nodes.

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A129584 Number of unlabeled bi-point-determining graphs: graphs in which no two vertices have the same neighborhoods or the same augmented neighborhoods (the augmented neighborhood of a vertex is the neighborhood of the vertex union the vertex itself).

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A369283 Triangle read by rows: T(n,k) is the number of labeled point-determining graphs with n nodes and k edges, n >= 0, 0 <= k <= n*(n - 1)/2.

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A129583 Number of labeled bi-point-determining graphs with n vertices.

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PARI

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A129585 Number of labeled connected bi-point-determining graphs with n vertices (see A129583).

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Mathematica

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A129586 Number of unlabeled connected bi-point-determining graphs (see A129583).

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