A321895 -id:A321895 - 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 *)

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.

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.

A321931 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in M(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and M is augmented monomial symmetric functions.

Original entry on oeis.org

1, 1, 0, -1, 1, 1, 0, 0, -1, 1, 0, 2, -3, 1, 1, 0, 0, 0, 0, -1, 1, 0, 0, 0, -1, 0, 1, 0, 0, 2, -1, -2, 1, 0, -6, 3, 8, -6, 1, 1, 0, 0, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 2, -1, -2, 1, 0, 0, 0, 2, -2, -1, 0, 1, 0, 0, -6, 6, 5, -3, -3, 1, 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).

The augmented monomial symmetric functions are given by M(y) = c(y) * m(y) where c(y) = Product_i (y)_i! where (y)_i is the number of i's in y and m is monomial symmetric functions.

			Tetrangle begins (zeros not shown):
  (1):  1
.
  (2):   1
  (11): -1  1
.
  (3):    1
  (21):  -1  1
  (111):  2 -3  1
.
  (4):     1
  (22):   -1  1
  (31):   -1     1
  (211):   2 -1 -2  1
  (1111): -6  3  8 -6  1
.
  (5):      1
  (41):    -1  1
  (32):    -1     1
  (221):    2 -1 -2  1
  (311):    2 -2 -1     1
  (2111):  -6  6  5 -3 -3  1
  (11111): 24 30 20 15 20 10  1
For example, row 14 gives: M(32) = -p(5) + p(32).

Wikipedia, Symmetric polynomial

Row sums are A155972. This is a regrouping of the triangle A321895.

Cf. A008480, A056239, A124794, A124795, A215366, A318284, A318360, A319191, A319193, A321912-A321935.

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}]

A322011 Number of distinct chromatic symmetric functions of spanning hypergraphs (or antichain covers) on n vertices.

Original entry on oeis.org

1, 2, 5, 19, 121

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 the augmented monomial symmetric function basis (see A321895).

			The a(3) = 5 chromatic symmetric functions:
                  m(111)
          m(21) + m(111)
         2m(21) + m(111)
         3m(21) + m(111)
  m(3) + 3m(21) + m(111)

Cf. A000569, A006125, A229048, A240936, A245883, A277203, A030019, A321750, A321895, A321911, A321994.

chromSF[g_]:=Sum[m[Sort[Length/@stn,Greater]],{stn,spsu[Select[Subsets[Union@@g],Select[DeleteCases[g,{_}],Function[ed,Complement[ed,#]=={}]]=={}&],Union@@g]}];
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]]]];
hyps[n_]:=Select[stableSets[Rest[Subsets[Range[n]]],SubsetQ],Union@@#==Range[n]&];
Table[Length[Union[chromSF/@hyps[n]]],{n,5}]

A321899 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in F(u), where H is Heinz number, F is augmented forgotten symmetric functions, and p is power sum symmetric functions.

Original entry on oeis.org

1, 1, -1, 0, 1, 1, 1, 0, 0, -1, -1, 0, -1, 0, 0, 0, 0, 2, 3, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, -2, -1, -2, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 6, 3, 8, 6, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 20 2018

Row n has length A000041(A056239(n)).
The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
The augmented forgotten symmetric functions are given by F(y) = c(y) * f(y) where f is forgotten symmetric functions and c(y) = Product_i (y)_i!, where (y)_i is the number of i's in y.

			Triangle begins:
   1
   1
  -1   0
   1   1
   1   0   0
  -1  -1   0
  -1   0   0   0   0
   2   3   1
   1   1   0   0   0
   1   0   1   0   0
   1   0   0   0   0   0   0
  -2  -1  -2  -1   0
  -1   0   0   0   0   0   0   0   0   0   0
  -1  -1   0   0   0   0   0
  -1   0  -1   0   0   0   0
   6   3   8   6   1
   1   0   0   0   0   0   0   0   0   0   0   0   0   0   0
   2   1   2   1   0   0   0
For example, row 12 gives: F(211) = -2p(4) - p(22) - 2p(31) - p(211).

Wikipedia, Symmetric polynomial

Row sums are A130675, up to sign. Same as A321895, up to sign.

Cf. A005651, A008277, A008480, A048994, A056239, A124794, A124795, A319193, A321742-A321765.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Sum[(-1)^(Total[primeMS[m]]-PrimeOmega[m])*Product[(-1)^(Length[t]-1)*(Length[t]-1)!,{t,s}],{s,Select[sps[Range[PrimeOmega[n]]]/.Table[i->If[n==1,{},primeMS[n]][[i]],{i,PrimeOmega[n]}],Times@@Prime/@Total/@#==m&]}],{n,18},{m,Sort[Times@@Prime/@#&/@IntegerPartitions[Total[primeMS[n]]]]}]

cp's OEIS Frontend

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

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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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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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A321931 Tetrangle where T(n,H(u),H(v)) is the coefficient of p(v) in M(u), where u and v are integer partitions of n, H is Heinz number, p is power sum symmetric functions, and M is augmented monomial symmetric functions.

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

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A322011 Number of distinct chromatic symmetric functions of spanning hypergraphs (or antichain covers) on n vertices.

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A321899 Irregular triangle read by rows where T(H(u),H(v)) is the coefficient of p(v) in F(u), where H is Heinz number, F is augmented forgotten symmetric functions, and p is power sum symmetric functions.