A000569 -id:A000569 - cp's OEIS frontend

A029890 Number of odd graphical partitions.

1, 2, 7, 20, 70, 234, 832, 2956, 10759, 39394, 145892, 543564, 2038831, 7684116, 29092055, 110550260, 421495147, 1611662256

Offset: 1

TORSTEN.SILLKE(AT)LHSYSTEMS.COM

nonn

R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.

Index entries for sequences related to graphical partitions

Cf. A000569, A004250, A004251, A029889.

Calculated using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.

A029891 Number of even graphical partitions.

Original entry on oeis.org

1, 3, 7, 23, 70, 242, 832, 2983, 10759, 39482, 145892, 543877, 2038831, 7685211, 29092055, 110554267, 421495147, 1611676767

Offset: 1

TORSTEN.SILLKE(AT)LHSYSTEMS.COM

nonn

R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.

Index entries for sequences related to graphical partitions

Cf. A000569, A004250, A004251, A029889.

Calculated using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.

A174439 Partial sums of A001523.

Original entry on oeis.org

1, 2, 4, 8, 16, 31, 58, 105, 184, 314, 523, 853, 1365, 2149, 3332, 5097, 7701, 11505, 17009, 24907, 36147, 52027, 74304, 105352, 148355, 207575, 288673, 399157, 548926, 750996, 1022400, 1385374, 1868813, 2510181, 3357862, 4474187, 5939186

Offset: 0

Jonathan Vos Post, Mar 19 2010

The subsequence of primes begins: 2, 31, 523, 853, 24907, 52027, 1868813, ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A001523, A054250, A059618, A059623, A001522, A001524, A000569, A100505, A100506.

nmax = 41; A001523 = CoefficientList[Series[1 + Sum[(-1)^(k + 1)*x^(k*(k + 1)/2), {k, 1, nmax}] / QPochhammer[x]^2, {x, 0, nmax}], x]; s = 0; Table[s = s + A001523[[k]], {k, 1, nmax}] (* Vaclav Kotesovec, Dec 13 2015 *)

a(n) = Sum_{i=0..n} A001523(i).

a(n) ~ exp(2*Pi*sqrt(n/3))/(8*Pi*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Dec 13 2015

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}]

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}]

A029893 Number of graphical partitions with up to n parts (?).

Original entry on oeis.org

1, 2, 4, 10, 24, 68, 198, 656, 2112

Offset: 1

torsten.sillke(AT)lhsystems.com

R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.

T. M. Barnes and C. D. Savage, A recurrence for counting graphical partitions, Electronic J. Combinatorics, 2 (1995).
Index entries for sequences related to graphical partitions

Cf. A000569, A004250, A004251, A029889.

A possible duplicate of A028506.

Calculated using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.

A069831 Number of graphical partitions of simple Eulerian graphs (partitions given by the degrees of vertices of simple (no loops or multiple edges) graphs having only vertices of even degrees) having n edges.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 3, 3, 5, 8, 10, 13, 16, 22, 29, 36, 45, 61, 74, 95, 118, 152, 183, 232, 279, 354, 422, 524, 627, 780, 926, 1134, 1355, 1651, 1958, 2366, 2809, 3372, 3988, 4757, 5628, 6678, 7874, 9283, 10964, 12861, 15130, 17686, 20799, 24209, 28389

Offset: 0

Roland Bacher, Apr 23 2002

nonn

			a(1)=a(2)=0 since Eulerian graphs having 1 or 2 edges are not simple. The triangle is the unique Eulerian graph having 3 edges and no isolated vertices, thus showing a(3)=1.

Sean A. Irvine, Table of n, a(n) for n = 0..100
Sean A. Irvine, Java program (github)

Cf. A000569.

A321177 Heinz numbers of integer partitions that are the vertex-degrees of some set system with no singletons.

Original entry on oeis.org

1, 4, 8, 12, 16, 18, 24, 27, 32, 36, 40

Offset: 1

Gus Wiseman, Oct 29 2018

A set system is a finite set of finite nonempty sets.

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			Each term paired with its Heinz partition and a realizing set system:
  1:       (): {}
  4:     (11): {{1,2}}
  8:    (111): {{1,2,3}}
  12:   (211): {{1,2},{1,3}}
  16:  (1111): {{1,2,3,4}}
  18:   (221): {{1,2},{1,2,3}}
  24:  (2111): {{1,2},{1,3,4}}
  27:   (222): {{1,2},{1,3},{2,3}}
  32: (11111): {{1,2,3,4,5}}
  36:  (2211): {{1,2},{1,2,3,4}}
  40:  (3111): {{1,2},{1,3},{1,4}}

Cf. A000070, A000569, A056239, A112798, A283877, A306005, A318361, A320922, A320923, A320924, A320925, A321176.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
hyp[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,UnsameQ@@#,Min@@Length/@#>1]&];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[20],!hyp[nrmptn[#]]=={}&]

A321184 Number of integer partitions of n that are the vertex-degrees of some multiset of nonempty sets, none of which is a proper subset of any other, with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 7, 6, 15, 15, 30

Offset: 0

Gus Wiseman, Oct 29 2018

			The a(2) = 1 through a(8) = 15 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)
               (211)   (11111)  (222)     (3211)     (332)
               (1111)           (321)     (22111)    (422)
                                (2211)    (31111)    (431)
                                (3111)    (211111)   (2222)
                                (21111)   (1111111)  (3221)
                                (111111)             (3311)
                                                     (4211)
                                                     (22211)
                                                     (32111)
                                                     (41111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
The a(6) = 7 integer partitions together with a realizing multi-antichain of each (the parts of the partition count the appearances of each vertex in the multi-antichain):
      (33): {{1,2},{1,2},{1,2}}
     (321): {{1,2},{1,2},{1,3}}
    (3111): {{1,2},{1,3},{1,4}}
     (222): {{1,2,3},{1,2,3}}
    (2211): {{1,2,3},{1,2,4}}
   (21111): {{1,2},{1,3,4,5}}
  (111111): {{1,2,3,4,5,6}}

Cf. A000070, A000569, A006126, A096827, A147878, A209816, A283877, A318360, A319719, A319721, A320799, A320921, A321176.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
multanti[m_]:=Select[mps[m],And[And@@UnsameQ@@@#,Min@@Length/@#>1,stableQ[#]]&];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[n],multanti[#]!={}&]],{n,8}]

A322012 Number of s-positive simple labeled graphs with n vertices.

Original entry on oeis.org

1, 2, 8, 60, 1009

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). A graph is s-positive if, in the expansion of its chromatic symmetric function in terms of Schur functions, all coefficients are nonnegative.

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.

a(n) >= A321979(n).

Cf. A000569, A006125, A229048, A240936, A277203, A321895, A321924, A321925, A321931, A321994.

cp's OEIS Frontend

A029890 Number of odd graphical partitions.

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A029891 Number of even graphical partitions.

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A174439 Partial sums of A001523.

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

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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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A029893 Number of graphical partitions with up to n parts (?).

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A069831 Number of graphical partitions of simple Eulerian graphs (partitions given by the degrees of vertices of simple (no loops or multiple edges) graphs having only vertices of even degrees) having n edges.

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A321177 Heinz numbers of integer partitions that are the vertex-degrees of some set system with no singletons.

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Mathematica

A321184 Number of integer partitions of n that are the vertex-degrees of some multiset of nonempty sets, none of which is a proper subset of any other, with no singletons.

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Mathematica

A322012 Number of s-positive simple labeled graphs with n vertices.

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