A125702 -id:A125702 - cp's OEIS frontend

A321155 Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with density -1 <= k < n-2.

1, 2, 1, 3, 2, 1, 6, 6, 4, 1, 10, 14, 11, 4, 1, 22, 38, 38, 20, 6, 1, 42, 94, 111, 72, 28, 6, 1, 94, 250, 348, 278, 138, 42, 8, 1, 203, 648, 1044, 992, 596, 226, 56, 8, 1, 470, 1728, 3192, 3538, 2536, 1192, 370, 76, 10, 1

Offset: 1

Gus Wiseman, Oct 29 2018

The density of a multiset partition of weight n with e parts and v vertices is n - e - v. The weight of a multiset partition is the sum of sizes of its parts.

			Triangle begins:
    1
    2    1
    3    2    1
    6    6    4    1
   10   14   11    4    1
   22   38   38   20    6    1
   42   94  111   72   28    6    1
   94  250  348  278  138   42    8    1
  203  648 1044  992  596  226   56    8    1
  470 1728 3192 3538 2536 1192  370   76   10    1
Non-isomorphic representatives of the connected multiset partitions counted in row 5:
{1,2,3,4,5}         {1,2,3,4,4}       {1,2,2,3,3}     {1,1,2,2,2}   {1,1,1,1,1}
{1,4},{2,3,4}       {1,2},{2,3,3}     {1,2,3,3,3}     {1,2,2,2,2}
{4},{1,2,3,4}       {1,3},{2,3,3}     {1,1},{1,2,2}   {1},{1,1,1,1}
{2},{1,3},{2,3}     {2},{1,2,3,3}     {1},{1,2,2,2}   {1,1},{1,1,1}
{2},{3},{1,2,3}     {2,3},{1,2,3}     {1,2},{1,2,2}
{3},{1,3},{2,3}     {3},{1,2,3,3}     {1,2},{2,2,2}
{3},{3},{1,2,3}     {3,3},{1,2,3}     {2},{1,1,2,2}
{1},{2},{2},{1,2}   {1},{1},{1,2,2}   {2},{1,2,2,2}
{2},{2},{2},{1,2}   {1},{1,2},{2,2}   {2,2},{1,2,2}
{1},{1},{1},{1},{1} {1},{2},{1,2,2}   {1},{1},{1,1,1}
                    {2},{1,2},{1,2}   {1},{1,1},{1,1}
                    {2},{1,2},{2,2}
                    {2},{2},{1,2,2}
                    {1},{1},{1},{1,1}

First column is A125702. Row sums are A007718.

Cf. A007716, A030019, A052888, A056156, A134954, A147878, A317631, A319557, A320921.

A245883 Number of distinct chromatic polynomials among all connected graphs on n nodes.

Original entry on oeis.org

1, 1, 2, 5, 14, 50, 231, 1650, 21121, 584432

Offset: 1

Travis Hoppe and Anna Petrone, Aug 05 2014

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 polynomial is given by chi_G(x) = Sum_p (x)k, where the sum is over all stable partitions of G, k is the length (number of blocks) of p, and (x)_k is the falling factorial x(x-1)(x-2)...(x-k+1). - _Gus Wiseman, Nov 24 2018

			From _Gus Wiseman_, Nov 24 2018: (Start)
The a(4) = 5 chromatic polynomials:
  -6x + 11x^2 - 6x^3 + x^4
  -4x +  8x^2 - 5x^3 + x^4
  -2x +  5x^2 - 4x^3 + x^4
  -3x +  6x^2 - 4x^3 + x^4
   -x +  3x^2 - 3x^3 + x^4
(End)

Travis Hoppe and Anna Petrone, Encyclopedia of Finite Graphs
T. Hoppe and A. Petrone, Integer sequence discovery from small graphs, arXiv preprint arXiv:1408.3644 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Chromatic Polynomial

Cf. A229048 (simple graphs, including disconnected ones, with unique chromatic polynomials).

Cf. A001187, A001349, A006125, A125702, A229048, A240936, A245883, A277203, A321911, A322011.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
falling[x_,k_]:=Product[(x-i),{i,0,k-1}];
chromPoly[g_]:=Expand[Sum[falling[x,Length[stn]],{stn,spsu[Select[Subsets[Union@@g],Select[DeleteCases[g,{_}],Function[ed,Complement[ed,#]=={}]]=={}&],Union@@g]}]];
simpleSpans[n_]:=simpleSpans[n]=If[n==0,{{}},Union@@Table[If[#=={},Union[ine,{{n}}],Union[Complement[ine,List/@#],{#,n}&/@#]]&/@Subsets[Range[n-1]],{ine,simpleSpans[n-1]}]];
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[Union[chromPoly/@Select[simpleSpans[n],Length[csm[#]]==1&]]],{n,5}] (* Gus Wiseman, Nov 24 2018 *)

A321911 Number of distinct chromatic symmetric functions of simple connected graphs with n vertices.

Original entry on oeis.org

1, 1, 2, 6, 20, 103, 759

Offset: 1

Gus Wiseman, Nov 21 2018

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 p 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 a(4) = 6 connected chromatic symmetric functions (m is the augmented monomial symmetric function basis):
                    m(1111)
           m(211) + m(1111)
          2m(211) + m(1111)
  m(22) + 2m(211) + m(1111)
  m(22) + 3m(211) + m(1111)
  m(31) + 3m(211) + m(1111)

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, A partition of connected graphs, Electronic J. Combinatorics 12, N1 (2005), 8pp. arXiv:math/0505155 [math.CO].

First differs from A079457 at a(7) = 759, A079457(7) = 706.

Cf. A000110, A001349, A001187, A125702, A147878, A191970, A229048, A240936, A245883, A277203, A317631, A317632, A320921, A321750, A321751, A321895.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
chromSF[g_]:=Sum[m[Sort[Length/@stn,Greater]],{stn,spsu[Select[Subsets[Union@@g],Select[DeleteCases[g,{_}],Function[ed,Complement[ed,#]=={}]]=={}&],Union@@g]}];
simpleSpans[n_]:=simpleSpans[n]=If[n==0,{{}},Union@@Table[If[#=={},Union[ine,{{n}}],Union[Complement[ine,List/@#],{#,n}&/@#]]&/@Subsets[Range[n-1]],{ine,simpleSpans[n-1]}]];
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[Union[chromSF/@Select[simpleSpans[n],Length[csm[#]]==1&]]],{n,6}]

A122086 Number of unlabeled free bicolored trees with n nodes (the colors are not interchangeable).

Original entry on oeis.org

2, 1, 2, 3, 6, 10, 22, 42, 94, 203, 470, 1082, 2602, 6270, 15482, 38525, 97258, 247448, 635910, 1645411, 4289010, 11245670, 29656148, 78595028, 209273780, 559574414, 1502130920, 4046853091, 10939133170, 29661655793

Offset: 1

N. J. A. Sloane, Oct 19 2006

nonn

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1978.

Andrew Howroyd, Table of n, a(n) for n = 1..500

Row sums of A122085.
Antidiagonal sums of A329054.
Same as A125702 except for n = 1.
Cf. A000055, A000081.

\\ here TreeGf is A000081 as g.f.
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={Vec(2*TreeGf(n) - TreeGf(n)^2)} \\ Andrew Howroyd, Nov 02 2019

For n even, a(n) = 2*A000055(n) - A000081(n/2), for n odd, a(n) = 2*A000055(n).

G.f.: 2*f(x) - f(x)^2 where f(x) is the g.f. of A000081. - Andrew Howroyd, Nov 02 2019

Edited by Christian G. Bower and Franklin T. Adams-Watters, Jan 05 2007

A125699 Table, T(n,k) is the number of connected categories with n morphisms and k objects.

Original entry on oeis.org

1, 2, 7, 1, 35, 6, 228, 28, 2, 2237, 159, 11, 31559, 1075, 77, 3, 1668997, 9389, 497, 24

Offset: 1

Franklin T. Adams-Watters and Christian G. Bower, Jan 05 2007

Connected in the sense that if the morphism direction and composition is ignored, resulting in a multigraph, that multigraph is connected.

			The table starts:
        1;
        2;
        7,    1;
       35,    6;
      228,   28,   2;
     2237,  159,  11;
    31559, 1075,  77,  3;
  1668997, 9389, 497, 24;
  ...

Ben Spitz, SmallCategories
Eric Weisstein's World of Mathematics, Category

Cf. A125696, A125698 (row sums), A110654 (row lengths), A058129 (column 1), A125700 (diagonal sums), A125702 (T(2n-1, n)).

a(14)-a(20) from Ben Spitz, Sep 02 2023

cp's OEIS Frontend

A321155 Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with density -1 <= k < n-2.

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A245883 Number of distinct chromatic polynomials among all connected graphs on n nodes.

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A321911 Number of distinct chromatic symmetric functions of simple connected graphs with n vertices.

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A122086 Number of unlabeled free bicolored trees with n nodes (the colors are not interchangeable).

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A125699 Table, T(n,k) is the number of connected categories with n morphisms and k objects.

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