A322115 -id:A322115 - cp's OEIS frontend

A007719 Number of independent polynomial invariants of symmetric matrix of order n.

1, 2, 4, 11, 30, 95, 328, 1211, 4779, 19902, 86682, 393072, 1847264, 8965027, 44814034, 230232789, 1213534723, 6552995689, 36207886517, 204499421849, 1179555353219, 6942908667578, 41673453738272, 254918441681030, 1588256152307002, 10073760672179505

Offset: 0

Colin Mallows

Also, number of connected multigraphs with n edges (allowing loops) and any number of nodes.

Also the number of non-isomorphic connected multiset partitions of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018

			From _Gus Wiseman_, Jul 18 2018: (Start)
Non-isomorphic representatives of the a(3) = 11 connected multiset partitions of {1, 1, 2, 2, 3, 3}:
  (112233),
  (1)(12233), (12)(1233), (112)(233), (123)(123),
  (1)(2)(1233), (1)(12)(233), (1)(23)(123), (12)(13)(23),
  (1)(2)(3)(123), (1)(2)(13)(23).
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..50
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) Table 63.

Row sums of A322115.

Cf. A002905, A007716, A007717, A007719, A020555, A050535, A053419, A076864, A191970, A316972, A316974.

mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++,
  c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {};
  For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
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];
Kq[q_, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];
RowSumMats[n_, m_, k_] := Module[{s = 0}, Do[s += permcount[q]* SeriesCoefficient[ Exp[Sum[Kq[q, t, k]/t x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];
A007717 = Table[Print[n]; RowSumMats[n, 2 n, 2], {n, 0, 20}];
Join[{1}, EULERi[Rest[A007717]]] (* Jean-François Alcover, Oct 29 2018, using Andrew Howroyd's code for A007717 *)

Inverse Euler transform of A007717.

a(0)=1 added by Alberto Tacchella, Jun 20 2011

a(7)-a(25) from Franklin T. Adams-Watters, Jun 21 2011

A333397 Array read by antidiagonals: T(n,k) is the number of connected k-regular multigraphs on n unlabeled nodes, loops allowed, n >= 0, k >= 0.

Original entry on oeis.org

1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 2, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 1, 3, 4, 5, 1, 0, 0, 1, 0, 3, 0, 10, 0, 1, 0, 0, 1, 1, 4, 9, 26, 28, 17, 1, 0, 0, 1, 0, 4, 0, 47, 0, 97, 0, 1, 0, 0, 1, 1, 5, 17, 91, 291, 639, 359, 71, 1, 0, 0, 1, 0, 5, 0, 149, 0, 2789, 0, 1635, 0, 1, 0, 0

Offset: 0

Andrew Howroyd, Mar 18 2020

This sequence can be derived from A167625 by inverse Euler transform.

			Array begins:
=========================================================
n\k | 0 1 2  3    4     5        6       7          8
----+----------------------------------------------------
  0 | 1 1 1  1    1     1        1       1          1 ...
  1 | 1 0 1  0    1     0        1       0          1 ...
  2 | 0 1 1  2    2     3        3       4          4 ...
  3 | 0 0 1  0    4     0        9       0         17 ...
  4 | 0 0 1  5   10    26       47      91        149 ...
  5 | 0 0 1  0   28     0      291       0       1934 ...
  6 | 0 0 1 17   97   639     2789   12398      44821 ...
  7 | 0 0 1  0  359     0    35646       0    1631629 ...
  8 | 0 0 1 71 1635 40264   622457 8530044   89057367 ...
  9 | 0 0 1  0 8296     0 14019433       0 6849428873 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..405 (first 28 antidiagonals)

Columns k=3..8 (with interspersed 0's for odd k) are: A005967, A085549, A129430, A129432, A129434, A129436.

Cf. A167625 (not necessarily connected), A322115 (not necessarily regular), A328682 (loopless), A333330.

Column k is the inverse Euler transform of column k of A167625.

A322137 Number of labeled connected graphs with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 1, 3, 17, 140, 1524, 20673, 336259, 6382302, 138525780, 3384988809, 91976158434, 2751122721402, 89833276321440, 3179852538140115, 121287919647418118, 4959343701136929850, 216406753768138678671, 10037782414506891597734, 493175891246093032826160

Offset: 0

Gus Wiseman, Nov 27 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200
P. S. Kolesnikov and B. K. Sartayev, On the special identities of Gelfand--Dorfman algebras, arXiv:2105.13815 [math.RA], 2021.
Gus Wiseman, The a(4) = 140 labeled connected graphs with 4 edges.

Cf. A000664, A002905, A007718, A013922, A054923, A057500, A191646, A275421, A291842 (planar case), A322114, A322115.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[Subsets[Range[n+1],{2}],{n}],And[Union@@#==Range[Max@@Union@@#],Length[csm[#]]==1]&]],{n,6}]

Connected(v)={my(u=vector(#v));for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1,k)*v[k]*u[n-k])); u}
seq(n)={Vec(vecsum(Connected(vector(2*n, j, (1 + x + O(x*x^n))^binomial(j,2)))))} \\ Andrew Howroyd, Nov 28 2018

Terms a(8) and beyond from Andrew Howroyd, Nov 28 2018

A322147 Regular triangle read by rows where T(n,k) is the number of labeled connected graphs with loops with n edges and k vertices, 1 <= k <= n+1.

Original entry on oeis.org

1, 1, 1, 0, 2, 3, 0, 1, 10, 16, 0, 0, 12, 79, 125, 0, 0, 6, 162, 847, 1296, 0, 0, 1, 179, 2565, 11436, 16807, 0, 0, 0, 116, 4615, 47100, 185944, 262144, 0, 0, 0, 45, 5540, 121185, 987567, 3533720, 4782969, 0, 0, 0, 10, 4720, 220075, 3376450, 23315936, 76826061, 100000000

Offset: 0

Gus Wiseman, Nov 28 2018

			Triangle begins:
  1
  1     1
  0     2     3
  0     1    10    16
  0     0    12    79   125
  0     0     6   162   847  1296
  0     0     1   179  2565 11436 16807

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Row sums are A322151. Last column is A000272.
Column sums are A062740.
Cf. A000664, A007718, A007719, A054923, A191646, A275421, A321254, A322114, A322115, A322137.

multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[If[n==0,1,Length[Select[Subsets[multsubs[Range[k],2],{n}],And[Union@@#==Range[k],Length[csm[#]]==1]&]]],{n,0,6},{k,1,n+1}]

Connected(v)={my(u=vector(#v)); for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1, k)*v[k]*u[n-k])); u}
M(n)={Mat([Col(p, -(n+1)) | p<-Connected(vector(2*n, j, (1 + x + O(x*x^n) )^binomial(j+1,2)))[1..n+1]])}
{ my(T=M(10)); for(n=1, #T, print(T[n,][1..n])) } \\ Andrew Howroyd, Nov 29 2018

Terms a(28) and beyond from Andrew Howroyd, Nov 29 2018

A322151 Number of labeled connected graphs with loops with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 2, 5, 27, 216, 2311, 30988, 499919, 9431026, 203743252, 4960335470, 134382267082, 4009794148101, 130668970606412, 4617468180528235, 175867725701333896, 7182126650899080024, 313063334893103361130, 14507460736615554141354, 712192629608088061633746

Offset: 0

Gus Wiseman, Nov 28 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Row sums of A322147. The unlabeled version is A191970.

Cf. A000664, A002905, A007718, A013922, A054923, A057500, A191646, A291842 (planar case), A321254, A322114, A322115.

multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[Subsets[multsubs[Range[n+1],2],{n}],And[Union@@#==Range[Max@@Union@@#],Length[csm[#]]==1]&]],{n,5}]

Connected(v)={my(u=vector(#v)); for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1, k)*v[k]*u[n-k])); u}
seq(n)={Vec(vecsum(Connected(vector(2*n, j, (1 + x + O(x*x^n))^binomial(j+1,2)))))} \\ Andrew Howroyd, Nov 28 2018

Terms a(7) and beyond from Andrew Howroyd, Nov 28 2018

A322152 Number of labeled connected multigraphs with loops with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 2, 7, 39, 314, 3359, 45000, 725269, 13670256, 295099184, 7179749707, 194399095705, 5797793490859, 188855813757729, 6671188010874785, 254007814638737649, 10370334196814589256, 451923738493729293016, 20937747226064522726151, 1027666505638118490940059

Offset: 0

Gus Wiseman, Nov 28 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..200

Row sums of A322148. The unlabeled version is A007719.

Cf. A000272, A000664, A007718, A191646, A191970, A322114, A322115, A322147, A322151.

multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[Length[Select[multsubs[multsubs[Range[n+1],2],n],And[Union@@#==Range[Max@@Union@@#],Length[csm[#]]==1]&]],{n,5}]

Connected(v)={my(u=vector(#v)); for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1, k)*v[k]*u[n-k])); u}
seq(n)={Vec(vecsum(Connected(vector(2*n, j, 1/(1 - x + O(x*x^n))^binomial(j+1,2)))))} \\ Andrew Howroyd, Nov 28 2018

Terms a(7) and beyond from Andrew Howroyd, Nov 28 2018

A327615 Irregular triangle read by rows: T(n,k) is the number of unlabeled multigraphs with loops allowed and n edges covering k vertices, n >= 1, 1 <= k <= 2*n.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 1, 5, 8, 6, 2, 1, 1, 8, 19, 25, 16, 7, 2, 1, 1, 11, 40, 73, 73, 47, 19, 7, 2, 1, 1, 15, 77, 194, 263, 232, 133, 58, 20, 7, 2, 1, 1, 19, 132, 454, 835, 951, 719, 397, 164, 61, 20, 7, 2, 1, 1, 24, 217, 984, 2385, 3507, 3365, 2306, 1177, 490, 175, 62, 20, 7, 2, 1

Offset: 1

Andrew Howroyd, Oct 23 2019

Covering k vertices means there are no vertices of degree zero.

			Triangle begins:
  1,  1;
  1,  3,   2,   1;
  1,  5,   8,   6,   2,   1;
  1,  8,  19,  25,  16,   7,   2,   1;
  1, 11,  40,  73,  73,  47,  19,   7,   2,  1;
  1, 15,  77, 194, 263, 232, 133,  58,  20,  7,  2, 1;
  1, 19, 132, 454, 835, 951, 719, 397, 164, 61, 20, 7, 2, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..650 (rows 1..25)

Row sums are A007717.
Columns k=2..3 are A024206, A327728.
Cf. A136564, A290428, A309936, A322115.

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, t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i], v[j])); t(v[i]*v[j]/g)^g )) * prod(i=1, #v, my(c=v[i]); t(c)^((c+1)\2)*if(c%2, 1, t(c/2)))}
C(n,m)={my(s=O(x*x^m)); forpart(p=n, s+=permcount(p)/edges(p, i->1-x^i+O(x*x^m))); Col(s/n!)}
T(m) = {my(n=2*m, A=Mat(vector(n+1, n, C(n-1,m)))); A[2..m+1, 2..n+1]-A[2..m+1, 1..n]}
{ my(A=T(8)); for(n=1, matsize(A)[1], print(A[n, 1..2*n])) }

T(n,k) = A290428(n,k) - A290428(n,k-1).

A360862 Triangle read by rows: T(n,k) is the number of unlabeled connected multigraphs with n edges on k nodes and degree >= 3 at each node, loops allowed, n >= 2, 1 <= k <= floor(2*n/3).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 7, 5, 1, 10, 20, 5, 1, 14, 48, 36, 1, 18, 99, 153, 30, 1, 23, 181, 481, 277, 17, 1, 28, 303, 1239, 1451, 323, 1, 34, 479, 2811, 5572, 2946, 193, 1, 40, 726, 5805, 17607, 17343, 3806, 71, 1, 47, 1055, 11148, 48401, 77708, 36872, 3188, 1, 54, 1492, 20219, 120018, 288476, 243007, 54386, 1496

Offset: 2

Andrew Howroyd, Feb 24 2023

Terms may be computed using the tools geng, vcolg and multig in nauty with some additional processing to check the degrees of nodes.

			Triangle begins:
  1;
  1,  2;
  1,  4;
  1,  7,    5;
  1, 10,   20,     5;
  1, 14,   48,    36;
  1, 18,   99,   153,     30;
  1, 23,  181,   481,    277,     17;
  1, 28,  303,  1239,   1451,    323;
  1, 34,  479,  2811,   5572,   2946,    193;
  1, 40,  726,  5805,  17607,  17343,   3806,    71;
  1, 47, 1055, 11148,  48401,  77708,  36872,  3188;
  1, 54, 1492, 20219, 120018, 288476, 243007, 54386, 1496;
  ...

Brendan McKay and Adolfo Piperno, nauty and Traces

Column 2 is A014616.
Row sums are A360863.
Diagonal sums are A360864.
Cf. A322115, A327615, A360866 (loopless).

A360870 Triangle read by rows: T(n,k) is the number of unlabeled connected multigraphs with n edges on k nodes, no cut-points and degree >= 3 at each node, loops allowed, n >= 2, 1 <= k <= floor(2*n/3).

Original entry on oeis.org

1, 1, 2, 1, 4, 1, 7, 2, 1, 10, 8, 2, 1, 14, 19, 11, 1, 18, 40, 48, 7, 1, 23, 77, 154, 70, 5, 1, 28, 132, 421, 392, 71, 1, 34, 217, 1008, 1638, 690, 35, 1, 40, 340, 2210, 5623, 4548, 767, 16, 1, 47, 510, 4477, 16745, 22657, 8594, 566, 1, 54, 742, 8557, 44698, 92844, 64716, 11247, 226

Offset: 2

Andrew Howroyd, Feb 25 2023

Columns k >= 3 correspond to the 2-connected graphs.

Terms may be computed using the tools geng, vcolg and multig in nauty with some additional processing to check the degrees of nodes.

			Triangle begins:
  1;
  1,  2;
  1,  4;
  1,  7,   2;
  1, 10,   8,    2;
  1, 14,  19,   11;
  1, 18,  40,   48,     7;
  1, 23,  77,  154,    70,     5;
  1, 28, 132,  421,   392,    71;
  1, 34, 217, 1008,  1638,   690,    35;
  1, 40, 340, 2210,  5623,  4548,   767,    16;
  1, 47, 510, 4477, 16745, 22657,  8594,   566;
  1, 54, 742, 8557, 44698, 92844, 64716, 11247, 226;
  ...

Brendan McKay and Adolfo Piperno, nauty and Traces.

Column 2 is A014616.
Row sums are A360882.
Row sums except first column are A360871.
Cf. A046752, A322115, A360862, A360866.

A322148 Regular triangle where T(n,k) is the number of labeled connected multigraphs with loops with n edges and k vertices.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 1, 6, 16, 16, 1, 10, 51, 127, 125, 1, 15, 126, 574, 1347, 1296, 1, 21, 266, 1939, 8050, 17916, 16807, 1, 28, 504, 5440, 35210, 135156, 286786, 262144, 1, 36, 882, 13387, 125730, 736401, 2642122, 5368728, 4782969, 1, 45, 1452, 29854, 388190, 3239491, 17424610, 58925728, 115089813, 100000000

Offset: 0

Gus Wiseman, Nov 28 2018

			Triangle begins:
  1
  1     1
  1     3     3
  1     6    16    16
  1    10    51   127   125
  1    15   126   574  1347  1296
  1    21   266  1939  8050 17916 16807

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Row sums are A322152. Last column is A000272.

Cf. A007718, A191646, A191970, A275421, A321155, A322114, A322115, A322137, A322147.

multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
Table[If[n==0,1,Length[Select[multsubs[multsubs[Range[k],2],n],And[Union@@#==Range[k],Length[csm[#]]==1]&]]],{n,0,5},{k,1,n+1}]

Connected(v)={my(u=vector(#v)); for(n=1, #u, u[n]=v[n] - sum(k=1, n-1, binomial(n-1, k)*v[k]*u[n-k])); u}
M(n)={Mat([Col(p, -(n+1)) | p<-Connected(vector(2*n, j, 1/(1 - x + O(x*x^n) )^binomial(j+1, 2)))[1..n+1]])}
{ my(T=M(10)); for(n=1, #T, print(T[n,][1..n])) } \\ Andrew Howroyd, Nov 29 2018

Offset corrected and terms a(28) and beyond from Andrew Howroyd, Nov 29 2018

cp's OEIS Frontend

A007719 Number of independent polynomial invariants of symmetric matrix of order n.

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A333397 Array read by antidiagonals: T(n,k) is the number of connected k-regular multigraphs on n unlabeled nodes, loops allowed, n >= 0, k >= 0.

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A322137 Number of labeled connected graphs with n edges (the vertices are {1,2,...,k} for some k).

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A322147 Regular triangle read by rows where T(n,k) is the number of labeled connected graphs with loops with n edges and k vertices, 1 <= k <= n+1.

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A322151 Number of labeled connected graphs with loops with n edges (the vertices are {1,2,...,k} for some k).

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Mathematica

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A322152 Number of labeled connected multigraphs with loops with n edges (the vertices are {1,2,...,k} for some k).

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Mathematica

PARI

Extensions

A327615 Irregular triangle read by rows: T(n,k) is the number of unlabeled multigraphs with loops allowed and n edges covering k vertices, n >= 1, 1 <= k <= 2*n.

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A360862 Triangle read by rows: T(n,k) is the number of unlabeled connected multigraphs with n edges on k nodes and degree >= 3 at each node, loops allowed, n >= 2, 1 <= k <= floor(2*n/3).

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