A229048 -id:A229048 - cp's OEIS frontend

A277203 Number of distinct chromatic symmetric functions realizable by a graph on n vertices.

1, 2, 4, 11, 33, 146, 939, 10932

Offset: 1

Sam Heil and Caleb Ji, Oct 04 2016

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). - Gus Wiseman, Nov 21 2018

			For n = 3, under the p basis, the CSF's are: p_{1, 1, 1}, p_{1, 1, 1} - p_{2, 1}, p_{1, 1, 1} - 2p_{2, 1} + p_{3}, p_{1, 1, 1} - 3p_{2, 1} + 2p_{3}.
From _Gus Wiseman_, Nov 21 2018: (Start)
The a(4) = 11 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)
                           3m(211) + m(1111)
          m(22) +          3m(211) + m(1111)
                   m(31) + 3m(211) + m(1111)
         2m(22) +          4m(211) + m(1111)
          m(22) +  m(31) + 4m(211) + m(1111)
         2m(22) + 2m(31) + 5m(211) + m(1111)
  m(4) + 3m(22) + 4m(31) + 6m(211) + m(1111)
(End)

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.

Cf. A000088, A000110, A000569, A006125, A229048, A240936, A245883, A277204, A277205, A321750, A321751, A321895, A321911.

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]}]];
Table[Length[Union[chromSF/@simpleSpans[n]]],{n,6}] (* Gus Wiseman, Nov 21 2018 *)

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}]

A321979 Number of e-positive simple labeled graphs on n vertices.

Original entry on oeis.org

1, 1, 2, 8, 60, 899

Offset: 0

Gus Wiseman, Nov 23 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 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). A graph is e-positive if, in the expansion of its chromatic symmetric function in terms of elementary symmetric functions, all coefficients are nonnegative.

			The 4 non-e-positive simple labeled graphs on 4 vertices are:
  {{1,2},{1,3},{1,4}}
  {{1,2},{2,3},{2,4}}
  {{1,3},{2,3},{3,4}}
  {{1,4},{2,4},{3,4}}

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.
Richard P. Stanley and John R. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, Journal of Combinatorial Theory Series A 62-2 (1993), 261-279.
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.
Gus Wiseman, The a(4) = 60 e-positive simple labeled graphs.

Cf. A000569, A006125, A229048, A240936, A277203, A321895, A321911, A321918, A321914, A321931, A321980, A321981, A321982.

A322064 Number of ways to choose a stable partition of a simple connected graph with n vertices.

Original entry on oeis.org

1, 1, 1, 7, 141, 6533, 631875, 123430027, 48659732725, 39107797223409, 64702785181953175, 221636039917857648631, 1575528053913118966200441, 23249384407499950496231003021, 711653666389829384034090082068939, 45128328085994437067694854477617868995

Offset: 0

Gus Wiseman, Nov 25 2018

nonn

A stable partition of a graph is a set partition of the vertices where no non-singleton edge has both ends in the same block.

			The a(3) = 7 stable partitions. The simple connected graph is on top, and below is a list of all its stable partitions.
  {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}}

Andrew Howroyd, Table of n, a(n) for n = 0..75

Row sums of A322278.

Cf. A000110, A000569, A001187, A006125, A048143, A229048, A240936, A245883, A277203, A321911, A321979, A322063, A322065.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
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[Subsets[Complement[Subsets[Range[n],{2}],Union@@Subsets/@stn]],And[Union@@#==Range[n],Length[csm[#]]==1]&]],{stn,sps[Range[n]]}],{n,5}]

\\ See A322278 for M.
seq(n)={concat([1], (M(n)*vectorv(n,i,1))~)} \\ Andrew Howroyd, Dec 01 2018

Terms a(7) and beyond from Andrew Howroyd, Dec 01 2018

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.

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

A321981 Row n gives the chromatic symmetric function of the n-girder, expanded in terms of elementary symmetric functions and ordered by Heinz number.

Original entry on oeis.org

1, 2, 0, 6, 0, 0, 16, 0, 2, 0, 0, 40, 12, 2, 0, 0, 0, 0, 96, 16, 44, 6, 0, 0, 0, 0, 0, 0, 0, 224, 136, 66, 52, 2, 4, 0, 2, 0, 0, 0, 0, 0, 0, 0, 512, 384, 208, 96, 30, 178, 0, 18, 30, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1152, 1024, 584, 522, 138, 588, 102

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-girder has n vertices and looks like:
  2-4-6-     -n
  |\|\|\ ... \|
  1-3-5-     n-1
Conjecture: All terms are nonnegative (verified up to n = 10). This is a special case of Stanley and Stembridge's poset-chain conjecture.

			Triangle begins:
    1
    2   0
    6   0   0
   16   0   2   0   0
   40  12   2   0   0   0   0
   96  16  44   6   0   0   0   0   0   0   0
  224 136  66  52   2   4   0   2   0   0   0   0   0   0   0
For example, row 6 gives: X_G6 = 96e(6) + 6e(33) + 16e(42) + 44e(51).

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.
Richard P. Stanley and John R. Stembridge, On immanants of Jacobi-Trudi matrices and permutations with restricted position, Journal of Combinatorial Theory Series A 62-2 (1993), 261-279.
Gus Wiseman, Enumeration of paths and cycles and e-coefficients of incomparability graphs, arXiv:0709.0430 [math.CO], 2007.

Row sums are A025192.

Cf. A000569, A001187, A006125, A056239, A229048, A240936, A245883, A277203, A321750, A321911, A321918, A321914, A321979, A321980, A321982.

A321994 Number of different chromatic symmetric functions of hypertrees on n vertices.

Original entry on oeis.org

1, 1, 2, 4, 9, 22, 59, 165

Offset: 1

Gus Wiseman, Nov 24 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 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).

Stanley conjectured that the number of distinct chromatic symmetric functions of trees with n vertices is equal to A000055, i.e., the chromatic symmetric function distinguishes between trees. It has been proven for trees with up to 25 vertices. If it is true in general, does the chromatic symmetric function also distinguish between hypertrees, meaning this sequence would be equal to A035053?

Jeremy L. Martin, The uniqueness problem for chromatic symmetric functions of trees (2015)
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.

Cf. A000055, A000569, A001187, A001349, A006125, A035053, A229048, A240936, A245883, A277203, A030019, A321750, A321895, A321911.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{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]]]]]]]]];
density[c_]:=Total[(Length[#]-1&)/@c]-Length[Union@@c];
hyall[n_]:=Select[stableSets[Select[Subsets[Range[n]],Length[#]>1&],Or[SubsetQ[#1,#2],Length[Intersection[#1,#2]]>1]&],And[Union@@#==Range[n],Length[csm[#]]==1,density[#]==-1]&];
chromSF[g_]:=Sum[m[Sort[Length/@stn,Greater]],{stn,spsu[Select[Subsets[Union@@g],Select[DeleteCases[g,{_}],Function[ed,Complement[ed,#]=={}]]=={}&],Union@@g]}];
Table[Length[Union[chromSF/@If[n==1,{{{1}}},hyall[n]]]],{n,5}]

cp's OEIS Frontend

A277203 Number of distinct chromatic symmetric functions realizable by a graph on n vertices.

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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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A321979 Number of e-positive simple labeled graphs on n vertices.

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A322064 Number of ways to choose a stable partition of a simple connected graph with n vertices.

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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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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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A321981 Row n gives the chromatic symmetric function of the n-girder, expanded in terms of elementary symmetric functions and ordered by Heinz number.

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