A317077 -id:A317077 - cp's OEIS frontend

A317752 Number of multiset partitions of normal multisets of size n such that the blocks have empty intersection.

0, 1, 8, 49, 305, 1984, 13686, 100124, 776885, 6386677, 55532358, 509549386, 4921352952, 49899820572, 529807799836, 5876162077537, 67928460444139, 816764249684450, 10195486840926032, 131896905499007474, 1765587483656124106, 24419774819813602870

Offset: 1

Gus Wiseman, Aug 06 2018

nonn

A multiset is normal if it spans an initial interval of positive integers.

			The a(3) = 8 multiset partitions with empty intersection:
  {{2},{1,1}}
  {{1},{2,2}}
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
  {{1},{1},{2}}
  {{1},{2},{2}}
  {{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 1..100

Cf. A007716, A255903, A255906, A317073, A281116, A317077, A317532, A317533.

Cf. A317748, A317751, A317755, A317757, A317776.

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]]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Length[Join@@Table[Select[mps[m],Intersection@@#=={}&],{m,allnorm[n]}]],{n,6}]

P(n,k)={1/prod(i=1, n, (1 - x^i*y + O(x*x^n))^binomial(k+i-1, k-1))}
R(n,k)={my(p=P(n,k), q=p/(1-y+O(y*y^n))); Vec(sum(i=2, n, polcoef(p,i,y) + polcoef(q,i,y)*sum(j=1, n\i, (-1)^j*binomial(k,j)*x^(i*j))), -n)}
seq(n)={sum(k=2, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Feb 05 2021

Terms a(9) and beyond from Andrew Howroyd, Feb 05 2021

A317755 Number of multiset partitions of strongly normal multisets of size n such that the blocks have empty intersection.

Original entry on oeis.org

0, 1, 6, 30, 130, 629, 2930, 15019, 78224, 438626, 2548481

Offset: 1

Gus Wiseman, Aug 06 2018

A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

			The a(3) = 6 strongly normal multiset partitions with empty intersection:
  {{2},{1,1}}
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
  {{1},{1},{2}}
  {{1},{2},{3}}

Cf. A007716, A035310, A255906, A317073, A281116, A317077, A317078, A317532, A317533.

Cf. A317748, A317751, A317752, A317757, A317775.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Join@@Table[Select[mps[m],Intersection@@#=={}&],{m,strnorm[n]}]],{n,6}]

a(10)-a(11) from Robert Price, May 08 2021

A317073 Number of antichains of multisets with multiset-join a normal multiset of size n.

Original entry on oeis.org

1, 1, 3, 16, 198, 9890, 8592538

Offset: 0

Gus Wiseman, Jul 20 2018

An antichain of multisets is a finite set of finite nonempty multisets, none of which is a submultiset of any other. A multiset is normal if it spans an initial interval of positive integers. The multiset-join of a set of multisets has the same vertices with multiplicities equal to the maxima of the multiplicities in the edges.

			The a(3) = 16 antichains of multisets:
  (111),
  (122), (12)(22), (1)(22),
  (112), (11)(12), (2)(11),
  (123), (13)(23), (12)(23), (12)(13), (12)(13)(23), (3)(12), (2)(13), (1)(23), (1)(2)(3).

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A048143, A255906, A285572, A303837, A303838, A304998, A305001.

Cf. A317074, A317075, A317077, A317079.

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]]]];
multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}]
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
auu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]],submultisetQ],multijoin@@#==m&];
Table[Length[Join@@Table[auu[m],{m,allnorm[n]}]],{n,5}]

a(6) from Robert Price, Jun 21 2021

A317078 Number of connected multiset partitions of strongly normal multisets of size n.

Original entry on oeis.org

1, 1, 3, 6, 18, 46, 172, 563, 2347

Offset: 0

Gus Wiseman, Jul 20 2018

A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

			The a(3) = 6 connected multiset partitions are (111), (1)(11), (1)(1)(1), (112), (1)(12), (123).

Cf. A007716, A007718, A048143, A293994, A303837, A303838, A304716, A305078.

Cf. A317074, A317076, A317077, A317080.

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]]]];
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]]],multijoin@@s[[c[[1]]]]]]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Length/@Table[Join@@Table[Select[mps[m],Length[csm[#]]==1&],{m,strnorm[n]}],{n,8}]

A317075 Number of connected antichains of multisets with multiset-join a normal multiset of size n.

Original entry on oeis.org

1, 1, 2, 10, 147, 8998

Offset: 0

Gus Wiseman, Jul 20 2018

An antichain of multisets is a finite set of finite nonempty multisets, none of which is a submultiset of any other. A multiset is normal if it spans an initial interval of positive integers. The multiset-join of a multiset system has the same vertices with multiplicities equal to the maxima of the multiplicities in the edges.

			The a(3) = 10 connected antichains of multisets:
  (111),
  (122), (12)(22),
  (112), (11)(12),
  (123), (13)(23), (12)(23), (12)(13), (12)(13)(23).

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A048143, A007718, A255906, A286520, A303837, A303838, A304716, A305001, A305078.

Cf. A317073, A317076, A317077, A317080.

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]]]];
multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}];
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
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]]],multijoin@@s[[c[[1]]]]]]]]];
allnorm[n_]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
cuu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]],submultisetQ],And[multijoin@@#==m,Length[csm[#]]==1]&];
Table[Length[Join@@Table[cuu[m],{m,allnorm[n]}]],{n,5}]

A317080 Number of unlabeled connected antichains of multisets with multiset-join a multiset of size n.

Original entry on oeis.org

1, 1, 2, 6, 34, 392

Offset: 0

Gus Wiseman, Jul 20 2018

An antichain of multisets is a finite set of finite nonempty multisets, none of which is a submultiset of any other. The multiset-join of a multiset system has the same vertices with multiplicities equal to the maxima of the multiplicities in the edges.

			Non-isomorphic representatives of the a(3) = 6 connected antichains of multisets:
  (111),
  (122), (12)(22),
  (123), (13)(23), (12)(13)(23).

Goran Kilibarda and Vladeta Jovovic, Antichains of Multisets, Journal of Integer Sequences, Vol. 7 (2004).

Cf. A007716, A007718, A048143, A261006, A286520, A293993, A293994, A304716, A304998.

Cf. A317073, A317074, A317075, A317076, A317077, A317078, A317079.

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]]]];
multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}]
submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
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]]],multijoin@@s[[c[[1]]]]]]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
sysnorm[m_]:=First[Sort[sysnorm[m,1]]];
sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
cuu[m_]:=Select[stableSets[Union[Rest[Subsets[m]]],submultisetQ],And[multijoin@@#==m,Length[csm[#]]==1]&];
Table[Length[Union[sysnorm/@Join@@Table[cuu[m],{m,strnorm[n]}]]],{n,5}]

A317785 Number of locally connected rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 21, 24, 34, 42, 55, 67, 91, 109, 144, 177, 228, 281, 366, 448, 579, 720, 916, 1142

Offset: 1

Gus Wiseman, Aug 06 2018

An unlabeled rooted tree is locally connected if the branches directly under any given node are connected as a hypergraph.

			The a(11) = 12 locally connected rooted trees:
  ((((((((((o))))))))))
  ((((((((o)(o))))))))
  (((((((o))((o)))))))
  ((((((o)))(((o))))))
  (((((o))))((((o)))))
  ((((((o)(o)(o))))))
  (((((o))((o)(o)))))
  ((((o))((o))((o))))
  ((((o)(o)(o)(o))))
  (((o))((o)(o)(o)))
  (((o)(o))((o)(o)))
  ((o)(o)(o)(o)(o))

Gus Wiseman, All 42 locally connected rooted trees with 16 nodes.

Cf. A000081, A276625, A286518, A286520, A301700, A304714, A316473, A316475, A317077, A317078, A317708, A317787.

multijoin[mss__]:=Join@@Table[Table[x, {Max[Count[#, x]&/@{mss}]}], {x, Union[mss]}];
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]]],multijoin@@s[[c[[1]]]]]]]]];
rurt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]],Or[Length[#]==1,Length[csm[#]]==1]&],{ptn,IntegerPartitions[n-1]}]];
Table[Length[rurt[n]],{n,10}]

A317786 Matula-Goebel numbers of locally connected rooted trees.

Original entry on oeis.org

1, 2, 3, 5, 9, 11, 23, 25, 27, 31, 81, 83, 97, 103, 115, 121, 125, 127, 243, 419, 431, 509, 515, 529, 563, 575, 625, 631, 661, 691, 709, 729, 961, 1067, 1331, 1543, 2095, 2187, 2369, 2575, 2645, 2875, 2897, 3001, 3125, 3637, 3691, 3803, 4091, 4201, 4637, 4663

Offset: 1

Gus Wiseman, Aug 07 2018

nonn

An unlabeled rooted tree is locally connected if the branches directly under any given node are connected as a hypergraph.

			The sequence of locally connected trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   5: (((o)))
   9: ((o)(o))
  11: ((((o))))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  31: (((((o)))))
  81: ((o)(o)(o)(o))
  83: ((((o)(o))))
  97: ((((o))((o))))

A subset of A184155.

Cf. A000081, A276625, A286518, A286520, A304714, A316470, A316495, A316502, A317077, A317078, A317785.

multijoin[mss__]:=Join@@Table[Table[x,{Max[Count[#,x]&/@{mss}]}],{x,Union[mss]}];
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]]], multijoin@@s[[c[[1]]]]]]]]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
rupQ[n_]:=Or[n==1,If[PrimeQ[n],rupQ[PrimePi[n]],And[Length[csm[primeMS/@primeMS[n]]]==1,And@@rupQ/@PrimePi/@FactorInteger[n][[All,1]]]]];
Select[Range[1000],rupQ[#]&]

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A317752 Number of multiset partitions of normal multisets of size n such that the blocks have empty intersection.

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A317755 Number of multiset partitions of strongly normal multisets of size n such that the blocks have empty intersection.

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A317073 Number of antichains of multisets with multiset-join a normal multiset of size n.

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A317078 Number of connected multiset partitions of strongly normal multisets of size n.

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A317075 Number of connected antichains of multisets with multiset-join a normal multiset of size n.

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A317080 Number of unlabeled connected antichains of multisets with multiset-join a multiset of size n.

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A317785 Number of locally connected rooted trees with n nodes.

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A317786 Matula-Goebel numbers of locally connected rooted trees.

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