A322064 -id:A322064 - cp's OEIS frontend

A322278 Triangle read by rows: T(n,k) is the number of k-colored connected graphs on n labeled nodes up to permutation of the colors.

1, 0, 1, 0, 3, 4, 0, 19, 84, 38, 0, 195, 2470, 3140, 728, 0, 3031, 108390, 307390, 186360, 26704, 0, 67263, 7192444, 42747460, 52630060, 18926544, 1866256, 0, 2086099, 726782784, 9030799218, 20784069600, 14401134944, 3463311488, 251548592

Offset: 1

Andrew Howroyd, Dec 01 2018

Equivalently, the number of ways to choose a stable partition of a simple connected graph on n labeled nodes with k parts. See A322064 for the definition of stable partition.

			Triangle begins:
  1;
  0,     1;
  0,     3,       4;
  0,    19,      84,       38;
  0,   195,    2470,     3140,      728;
  0,  3031,  108390,   307390,   186360,    26704;
  0, 67263, 7192444, 42747460, 52630060, 18926544, 1866256;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Row sums are A322064.
Columns k=2..4 are A001832(for n > 1), A322330, A322331.
Right diagonal is A001187.
Cf. A058843, A058875, A322279, A322280.

M(n, K=n)={
  my(p=sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n));
  my(q=sum(j=0, n, x^j*2^binomial(j, 2)) + O(x*x^n));
  my(W=vector(K, k, Col(serlaplace(log(serconvol(q, p^k))))));
  Mat(vector(K, k, sum(i=1, k, (-1)^(k-i)*binomial(k,i)*W[i])/k!));
}
my(T=M(7)); for(n=1, #T, print(T[n, 1..n]))

T(n,k) = (1/k!)*Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*A322279(n,j).

A207864 Number of n X 2 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

Original entry on oeis.org

1, 4, 34, 500, 10900, 322768, 12297768, 580849872, 33093252880, 2227152575552, 174131286983712, 15604440074084672, 1584856558077903168, 180712593036822482176, 22946861101272125055616, 3222156375409363475703040

Offset: 1

R. H. Hardin, Feb 21 2012

nonn

From Gus Wiseman, Mar 01 2019: (Start)
Also the number of stable partitions of the n-ladder graph. A stable partition of a graph is a set partition of the vertices where no edge has both ends in the same block. The n-ladder has 2n vertices and looks like:
  o-o-o-   -o
  | | | ... |
  o-o-o-   -o
(End)

			Some solutions for n=5:
  0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1   0 1
  1 0   1 0   1 2   1 2   1 0   1 0   1 2   1 0   1 0   1 0
  0 1   0 1   0 1   0 1   2 1   0 1   0 1   0 2   2 1   0 1
  1 2   1 0   1 0   1 3   3 0   2 0   3 2   2 1   1 0   1 2
  0 1   0 1   2 1   2 4   1 2   0 1   0 1   0 2   0 1   2 0

R. H. Hardin, Table of n, a(n) for n = 1..61

Column 2 of A207868.

Cf. A000110, A000569, A109808, A229048, A240936, A321750, A321979, A321980, A321982, A322064.

Table[Expand[x*(x-1)*(x^2-3*x+3)^(n-1)]/.x^k_.->BellB[k],{n,20}] (* Gus Wiseman, Mar 01 2019 *)

It appears that the sequence terms are given by the Dobinski-type formula a(n+1) = (1/e) * Sum_{k>=0} (1+k+k^2)^n/k!. - Peter Bala, Mar 12 2012

Apply x^n -> B(n) to the polynomial chi(n) = x (x - 1) (x^2 - 3 x + 3)^(n - 1), where B = A000110. - Gus Wiseman, Mar 01 2019

A322330 Number of 3-colored connected graphs on n labeled nodes up to permutation of the colors.

Original entry on oeis.org

4, 84, 2470, 108390, 7192444, 726782784, 112795368970, 27132558531330, 10196751602156944, 6022337098348167564, 5612248139616665602510, 8274349264629020203315230, 19333678744046195877906230404, 71675537050405087142116150917624, 421915518251999125756688245906168690

Offset: 3

Andrew Howroyd, Dec 03 2018

nonn

Equivalently, the number of ways to choose a stable partition of a simple connected graph on n labeled nodes with 3 parts. See A322064 for the definition of stable partition.

Andrew Howroyd, Table of n, a(n) for n = 3..50

Column k=3 of A322278.
Cf. A058873 (not necessarily connected), A322064.
Cf. A001832, A322331.

\\ See A322278 for M.
{ my(N=20); M(N,3)[3..N, 3]~ }

A322331 Number of 4-colored connected graphs on n labeled nodes up to permutation of the colors.

Original entry on oeis.org

38, 3140, 307390, 42747460, 9030799218, 3012940879620, 1628920258500230, 1451200592494754420, 2152262350514389189978, 5344908165470797467243700, 22297912999366719508496874990, 156537595118740106754291705258180, 1850935702258755131781978373277937218

Offset: 4

Andrew Howroyd, Dec 03 2018

nonn

Equivalently, the number of ways to choose a stable partition of a simple connected graph on n labeled nodes with 4 parts. See A322064 for the definition of stable partition.

Andrew Howroyd, Table of n, a(n) for n = 4..50

Column k=4 of A322278.
Cf. A058873 (not necessarily connected), A322064.
Cf. A001832, A322330.

\\ See A322278 for M.
{ my(N=20); M(N,4)[4..N, 4]~ }

A322063 Number of ways to choose a stable partition of an antichain of sets spanning n vertices.

Original entry on oeis.org

1, 1, 3, 25, 773, 160105

Offset: 0

Gus Wiseman, Nov 25 2018

A stable partition of a hypergraph or set system is a set partition of the vertices where no non-singleton edge has all its vertices in the same block.

			The a(3) = 25 stable partitions of antichains on 3 vertices. The antichain is on top, and below is a list of all its stable partitions.
  {1}{2}{3}      {1,2,3}        {1}{2,3}       {1,3}{2}       {1,2}{3}
  --------       --------       --------       --------       --------
  {{1,2,3}}      {{1},{2,3}}    {{1,2},{3}}    {{1},{2,3}}    {{1},{2,3}}
  {{1},{2,3}}    {{1,2},{3}}    {{1,3},{2}}    {{1,2},{3}}    {{1,3},{2}}
  {{1,2},{3}}    {{1,3},{2}}    {{1},{2},{3}}  {{1},{2},{3}}  {{1},{2},{3}}
  {{1,3},{2}}    {{1},{2},{3}}
  {{1},{2},{3}}
.
  {1,3}{2,3}     {1,2}{2,3}     {1,2}{1,3}     {1,2}{1,3}{2,3}
  --------       --------       --------       --------
  {{1,2},{3}}    {{1,3},{2}}    {{1},{2,3}}    {{1},{2},{3}}
  {{1},{2},{3}}  {{1},{2},{3}}  {{1},{2},{3}}

Cf. A000110, A000569, A006125, A006126, A229048, A240936, A277203, A321979, A322064, A322065.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
Table[Sum[Length[stableSets[Complement[Subsets[Range[n]],Union@@Subsets/@stn],SubsetQ]],{stn,sps[Range[n]]}],{n,5}]

A322065 Number of ways to choose a stable partition of a connected antichain of sets spanning n vertices.

Original entry on oeis.org

1, 1, 1, 11, 525, 146513

Offset: 0

Gus Wiseman, Nov 25 2018

A stable partition of a hypergraph or set system is a set partition of the vertices where no non-singleton edge has all its vertices in the same block.

			The a(3) = 11 stable partitions. The connected antichain is on top, and below is a list of all its stable partitions.
{1,2,3}        {1,3}{2,3}     {1,2}{2,3}     {1,2}{1,3}     {1,2}{1,3}{2,3}
--------       --------       --------       --------       --------
{{1},{2,3}}    {{1,2},{3}}    {{1,3},{2}}    {{1},{2,3}}    {{1},{2},{3}}
{{1,2},{3}}    {{1},{2},{3}}  {{1},{2},{3}}  {{1},{2},{3}}
{{1,3},{2}}
{{1},{2},{3}}

Cf. A000110, A001187, A006125, A048143, A229048, A240936, A245883, A277203, A321911, A321979, A322063, A322064.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
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[Sum[Length[Select[stableSets[Complement[Subsets[Range[n]],Union@@Subsets/@stn],SubsetQ],And[Union@@#==Range[n],Length[csm[#]]==1]&]],{stn,sps[Range[n]]}],{n,5}]

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A322278 Triangle read by rows: T(n,k) is the number of k-colored connected graphs on n labeled nodes up to permutation of the colors.

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A207864 Number of n X 2 nonnegative integer arrays with new values 0 upwards introduced in row major order and no element equal to any horizontal or vertical neighbor (colorings ignoring permutations of colors).

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A322330 Number of 3-colored connected graphs on n labeled nodes up to permutation of the colors.

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A322331 Number of 4-colored connected graphs on n labeled nodes up to permutation of the colors.

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A322063 Number of ways to choose a stable partition of an antichain of sets spanning n vertices.

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Mathematica

A322065 Number of ways to choose a stable partition of a connected antichain of sets spanning n vertices.

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