A001187 -id:A001187 - cp's OEIS frontend

A127097 Triangle T(n,m) = A127093 * A126988 read by rows.

1, 5, 2, 10, 0, 3, 21, 10, 0, 4, 26, 0, 0, 0, 5, 50, 20, 15, 0, 0, 6, 50, 0, 0, 0, 0, 0, 7, 85, 42, 0, 20, 0, 0, 0, 8, 91, 0, 30, 0, 0, 0, 0, 0, 9, 130, 52, 0, 0, 25, 0, 0, 0, 0, 10, 122, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 210, 100, 63, 40, 0, 30, 0, 0, 0, 0, 0, 12, 170, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gary W. Adamson, Jan 05 2007

Multiply the infinite lower triangular matrices A127093 and A126988.

			First few rows of the triangle are:
1;
5, 2;
10, 0, 3;
21, 10, 0, 4;
26, 0, 0, 0, 5;
50, 20, 15, 0, 0, 6;
50, 0, 0, 0, 0, 0, 7;
...

Cf. A126988, A000203, A127098, A001187, A001001 (row sums), A127093.

A127093 := proc(n,m) if n mod m = 0 then m; else 0 ; fi; end:
A126988 := proc(n,k) if n mod k = 0 then n/k; else 0; fi; end:
A127097 := proc(n,m) add( A127093(n,j)*A126988(j,m),j=m..n) ; end:
for n from 1 to 15 do for m from 1 to n do printf("%d,",A127097(n,m)) ; od: od: # R. J. Mathar, Aug 18 2009

T(n,m) = sum_{j=m..n} A127093(n,j)*A126988(j,m).

T(n,1) = A001157(n).

A-numbers corrected by R. J. Mathar, Aug 18 2009

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

A321980 Row n gives the chromatic symmetric function of the n-path, expanded in terms of elementary symmetric functions and ordered by Heinz number.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 4, 2, 2, 0, 0, 5, 3, 7, 1, 0, 0, 0, 6, 10, 4, 6, 2, 0, 4, 0, 0, 0, 0, 7, 5, 13, 17, 6, 0, 11, 4, 1, 0, 0, 0, 0, 0, 0, 8, 6, 16, 12, 0, 22, 16, 8, 12, 20, 2, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 9, 7, 19, 27, 0, 31, 10, 9, 21, 0, 58, 16, 12, 9, 0

Offset: 1

Gus Wiseman, Nov 23 2018

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The chromatic symmetric function is given by X_G = Sum_p m(t(p)) where the sum is over all stable partitions of G, t(p) is the integer partition whose parts are the block-sizes of p, and m is augmented monomial symmetric functions (see A321895).
All terms are nonnegative [Stanley].

			Triangle begins:
  1
  2  0
  3  1  0
  4  2  2  0  0
  5  3  7  1  0  0  0
  6 10  4  6  2  0  4  0  0  0  0
  7  5 13 17  6  0 11  4  1  0  0  0  0  0  0
  8  6 16 12  0 22 16  8 12 20  2  0  0  6  0  0  0  0  0  0  0  0
For example, row 6 gives: X_P6 = 6e(6) + 10e(42) + 4e(51) + 6e(33) + 2e(222) + 4e(321).

Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.
Richard P. Stanley, Graph colorings and related symmetric functions: ideas and applications, Discrete Mathematics 193 (1998), 267-286.
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.

Row sums are A000079.

Cf. A000569, A001187, A001349, A006125, A056239, A229048, A240936, A245883, A277203, A321911, A321918, A321914, A321979, A321981, A321982.

A321982 Row n gives the chromatic symmetric function of the n-ladder, expanded in terms of elementary symmetric functions and ordered by Heinz number.

Original entry on oeis.org

2, 0, 12, 2, 0, 0, 0, 54, 26, 16, 0, 2, 0, 0, 0, 0, 0, 0, 216, 120, 168, 84, 0, 24, 40, 32, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 810, 648, 822, 56, 240, 870, 280, 282, 120, 24, 0, 266, 232, 0, 48, 0, 54, 0, 48, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 23 2018

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The chromatic symmetric function is given by X_G = Sum_p m(t(p)) where the sum is over all stable partitions of G, t(p) is the integer partition whose parts are the block-sizes of p, and m is augmented monomial symmetric functions (see A321895).
The n-ladder has 2*n vertices and looks like:
  o-o-o-   -o
  | | | ... |
  o-o-o-   -o
Conjecture: All terms are nonnegative (verified up to the 5-ladder).

			Triangle begins:
    2   0
   12   2   0   0   0
   54  26  16   0   2   0   0   0   0   0   0
  216 120 168  84   0  24  40  32   0   0   2   0   0   [+9 more zeros]
For example, row 3 gives: X_L3 = 54e(6) + 26e(42) + 16e(51) + 2e(222).

Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, Advances in Math. 111 (1995), 166-194.
Richard P. Stanley, Graph colorings and related symmetric functions: ideas and applications, Discrete Mathematics 193 (1998), 267-286.
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.

Row sums are A109808.

Cf. A000569, A001187, A006125, A056239, A229048, A240936, A245883, A277203, A321911, A321918, A321914, A321979, A321980, A321981.

A322396 Number of unlabeled simple connected graphs with n vertices whose bridges are all leaves, meaning at least one end of any bridge is an endpoint of the graph.

Original entry on oeis.org

1, 1, 1, 2, 5, 18, 98, 779, 10589, 255790, 11633297, 1004417286, 163944008107, 50324877640599, 29001521193534445, 31396727025729968365, 63969154112074956299242, 245871360738448777028919520, 1787330701747389106609369225312, 24636017249593067184544456944967278

Offset: 0

Gus Wiseman, Dec 06 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..25
Eric Weisstein's World of Mathematics, Graph Bridge
Eric Weisstein's World of Mathematics, Endpoint
Gus Wiseman, The a(5) = 18 simple connected graphs whose bridges are all leaves.

Cf. A001187, A006125, A007146, A013922, A054921, A095983, A322338, A322394, A322395.

\\ See A004115 for graphsSeries and A339645 for combinatorial species functions.
bridgelessGraphs(n)={my(gc=sLog(graphsSeries(n)), gcr=sPoint(gc)); sSolve( gc + gcr^2/2 - sRaise(gcr,2)/2, x*sv(1)*sExp(gcr) )}
cycleIndexSeries(n)={1+sSubstOp(bridgelessGraphs(n), symGroupSeries(n))}
NumUnlabeledObjsSeq(cycleIndexSeries(15)) \\ Andrew Howroyd, Dec 31 2020

a(6)-a(10) from Andrew Howroyd, Dec 08 2018

Terms a(11) and beyond from Andrew Howroyd, Dec 31 2020

A327199 Number of labeled simple graphs with n vertices whose edge-set is not connected.

Original entry on oeis.org

1, 1, 1, 1, 4, 56, 1031, 27189, 1165424, 89723096, 13371146135, 3989665389689, 2388718032951812, 2852540291841718752, 6768426738881535155247, 31870401029679493862010949, 297787425565749788134314214272

Offset: 0

Gus Wiseman, Sep 01 2019

nonn

Also graphs with non-spanning edge-connectivity 0.

			The a(4) = 4 edge-sets: {}, {12,34}, {13,24}, {14,23}.

Column k = 0 of A327148.
The covering case is A327070.
The unlabeled version is A327235.
Cf. A001187, A006129, A322395, A326787, A327075, A327076, A327079, A327129, A327200, A327201, A327231, A327236.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[csm[#]]!=1&]],{n,0,5}]

Binomial transform of A327070.

A327364 Number of labeled simple graphs with n vertices, a connected edge-set, and at least one endpoint (vertex of degree 1).

Original entry on oeis.org

0, 0, 1, 6, 46, 655, 17991, 927416, 89009740, 16020407709, 5468601546685, 3578414666656214, 4529751815161579194, 11175105490563109463875, 54043272967471942825421219, 514566625051705610110588073460, 9677104749727084630538798805505880

Offset: 0

Gus Wiseman, Sep 04 2019

nonn

			The a(4) = 46 edge-sets:
  {12}  {12,13}  {12,13,14}  {12,13,14,23}
  {13}  {12,14}  {12,13,24}  {12,13,14,24}
  {14}  {12,23}  {12,13,34}  {12,13,14,34}
  {23}  {12,24}  {12,14,23}  {12,13,23,24}
  {24}  {13,14}  {12,14,34}  {12,13,23,34}
  {34}  {13,23}  {12,23,24}  {12,14,23,24}
        {13,34}  {12,23,34}  {12,14,24,34}
        {14,24}  {12,24,34}  {12,23,24,34}
        {14,34}  {13,14,23}  {13,14,23,34}
        {23,24}  {13,14,24}  {13,14,24,34}
        {23,34}  {13,23,24}  {13,23,24,34}
        {24,34}  {13,23,34}  {14,23,24,34}
                 {13,24,34}
                 {14,23,24}
                 {14,23,34}
                 {14,24,34}

Andrew Howroyd, Table of n, a(n) for n = 0..50

The covering case is A327362.
Graphs with endpoints are A245797.
Graphs with connected edge-set are A287689.
Connected graphs with bridges are A327071.
Covering graphs with endpoints are A327227.
Cf. A001187, A059167, A141580, A322395, A327148, A327335, A327369.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],Length[csm[#]]==1&&Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,5}]

seq(n)={my(x=x + O(x*x^n)); Vec(serlaplace(exp(x)*(-x^2/2 + log(sum(k=0, n, 2^binomial(k, 2)*x^k/k!)) - log(sum(k=0, n, 2^binomial(k, 2)*(x*exp(-x))^k/k!)))), -(n+1))} \\ Andrew Howroyd, Sep 11 2019

Binomial transform of A327362.

Terms a(7) and beyond from Andrew Howroyd, Sep 11 2019

A370318 Number of labeled simple graphs with n vertices and the same number of edges as covered vertices, such that the edge set is connected.

Original entry on oeis.org

0, 0, 0, 1, 19, 307, 5237, 99137, 2098946, 49504458, 1291570014, 37002273654, 1156078150969, 39147186978685, 1428799530304243, 55933568895261791, 2338378885159906196, 103995520598384132516, 4903038902046860966220, 244294315694676224001852, 12827355456239840407125363

Offset: 0

Gus Wiseman, Feb 18 2024

nonn

The case of an empty edge set is excluded.

Andrew Howroyd, Table of n, a(n) for n = 0..100

The covering case is A057500, which is also the covering case of A370317.
This is the connected case of A367862, covering A367863.
A001187 counts connected graphs, A001349 unlabeled.
A006125 counts graphs, A000088 unlabeled.
A006129 counts covering graphs, A002494 unlabeled.
A062734 counts connected graphs by edge count.
A133686 = graphs satisfy strict AoC, connected A129271, covering A367869.
A143543 counts simple labeled graphs by number of connected components.
A367867 = graphs contradict strict AoC, connected A140638, covering A367868.
Cf. A001429, A006649, A061540, A116508, A323818, A367916, A368951, A369197.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n], {2}]],Length[#]==Length[Union@@#] && Length[csm[#]]==1&]],{n,0,5}]

\\ Compare A370317; use A057500 for efficiency.
a(n)=n!*polcoef(polcoef(exp(x*y + O(x*x^n))*(-x+log(sum(k=0, n, (1 + y + O(y*y^n))^binomial(k, 2)*x^k/k!, O(x*x^n)))), n), n) \\ Andrew Howroyd, Feb 19 2024

Binomial transform of A057500 (if the null graph is not connected).

a(n) = n!*[x^n][y^n] exp(x*y)*(-x + log(Sum_{k>=0} (1 + y)^binomial(k, 2)*x^k/k!)). - Andrew Howroyd, Feb 19 2024

cp's OEIS Frontend

A127097 Triangle T(n,m) = A127093 * A126988 read by rows.

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

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

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

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A321980 Row n gives the chromatic symmetric function of the n-path, expanded in terms of elementary symmetric functions and ordered by Heinz number.

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A321982 Row n gives the chromatic symmetric function of the n-ladder, expanded in terms of elementary symmetric functions and ordered by Heinz number.

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A322396 Number of unlabeled simple connected graphs with n vertices whose bridges are all leaves, meaning at least one end of any bridge is an endpoint of the graph.

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A327199 Number of labeled simple graphs with n vertices whose edge-set is not connected.

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A327364 Number of labeled simple graphs with n vertices, a connected edge-set, and at least one endpoint (vertex of degree 1).