A316651 -id:A316651 - cp's OEIS frontend

A316695 Number of series-reduced locally disjoint rooted trees whose leaves form the integer partition with Heinz number n.

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 8, 1, 1, 2, 3, 1, 4, 1, 10, 1, 1, 1, 12, 1, 1, 1, 8, 1, 4, 1, 3, 3, 1, 1, 23, 1, 3, 1, 3, 1, 8, 1, 8, 1, 1, 1, 16, 1, 1, 3, 24, 1, 4, 1, 3, 1, 4, 1, 37, 1, 1, 3, 3, 1, 4, 1, 23, 5, 1, 1, 16

Offset: 1

Gus Wiseman, Jul 10 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other (unequal) branch of the same root.

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

			The a(24) = 8 trees:
  (1(1(12)))
  (1(2(11)))
  (2(1(11)))
  (1(112))
  (2(111))
  (11(12))
  (12(11))
  (1112)

Cf. A000081, A000669, A001678, A056239, A141268, A292504, A296150, A316471, A316651, A316652, A316655.

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]]]];
disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
gro[m_]:=gro[m]=If[Length[m]==1,List/@m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
Table[Length[Select[gro[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],And@@Cases[#,q:{__List}:>disjointQ[q],{0,Infinity}]&]],{n,100}]

A316768 Number of series-reduced locally stable rooted trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 2, 4, 11, 29, 91, 284, 950, 3235, 11336, 40370, 146095, 534774, 1977891, 7377235, 27719883

Offset: 1

Gus Wiseman, Jul 12 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally stable if no branch is a submultiset of any other branch of the same root.

			The a(5) = 29 trees:
  5,
  (14),
  (23),
  (1(13)), (3(11)), (113),
  (1(22)), (2(12)), (122),
  (1(1(12))), (1(2(11))), (1(112)), (2(1(11))), (2(111)), ((11)(12)), (11(12)), (12(11)), (1112),
  (1(1(1(11)))), (1(1(111))), (1((11)(11))), (1(11(11))), (1(1111)), ((11)(1(11))), (11(1(11))), (11(111)), (1(11)(11)), (111(11)), (11111).
Missing from this list but counted by A141268 is ((11)(111)).

Cf. A000081, A000669, A001678, A141268, A292504, A316468, A316475, A316651, A316652, A316655.

submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
stableQ[u_]:=Apply[And,Outer[#1==#2||!submultisetQ[#1,#2]&&!submultisetQ[#2,#1]&,u,u,1],{0,1}];
nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],stableQ],{ptn,Rest[IntegerPartitions[n]]}],{n}];
Table[Length[nms[n]],{n,10}]

a(15)-a(16) from Robert Price, Sep 16 2018

A316767 Number of series-reduced locally stable rooted trees whose leaves form the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 8, 1, 1, 2, 3, 1, 4, 1, 10, 1, 1, 1, 12, 1, 1, 1, 8, 1, 4, 1, 3, 3, 1, 1, 24, 1, 3, 1, 3, 1, 8, 1, 8, 1, 1, 1, 17, 1, 1, 3, 24, 1, 4, 1, 3, 1, 4, 1, 39, 1, 1, 3, 3, 1, 4, 1, 24, 5, 1, 1, 17

Offset: 1

Gus Wiseman, Jul 12 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally stable if no branch is a submultiset of any other branch of the same root.

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

			The a(24) = 8 trees:
  (1(1(12)))
  (1(2(11)))
  (2(1(11)))
  (1(112))
  (2(111))
  (11(12))
  (12(11))
  (1112)

Cf. A000081, A000669, A001678, A056239, A141268, A292504, A296150, A316475, A316651, A316652, A316655.

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]]]];
stableQ[u_]:=Apply[And,Outer[#1==#2||Complement[#2,#1]=!={}&,u,u,1],{0,1}];
gro[m_]:=gro[m]=If[Length[m]==1,List/@m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
Table[Length[Select[gro[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],And@@Cases[#,q:{__List}:>stableQ[q],{0,Infinity}]&]],{n,100}]

A316769 Number of series-reduced locally stable rooted trees with n unlabeled leaves.

Original entry on oeis.org

1, 1, 2, 5, 11, 29, 74, 205, 578, 1683, 4978, 15000, 45672, 140600, 436421, 1364876, 4295403, 13594685, 43238514

Offset: 1

Gus Wiseman, Jul 12 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally stable if no branch is a proper submultiset of any other branch of the same root.

			The a(5) = 11 trees:
  (o(o(o(oo))))
  (o(o(ooo)))
  (o((oo)(oo)))
  (o(oo(oo)))
  (o(oooo))
  ((oo)(o(oo)))
  (oo(o(oo)))
  (oo(ooo))
  (o(oo)(oo))
  (ooo(oo))
  (ooooo)
Missing from this list but counted by A000669 is ((oo)(ooo)).

Cf. A000081, A000669, A001678, A141268, A316475, A316651, A316652, A316655.

submultisetQ[M_,N_]:=Or[Length[M]==0,MatchQ[{Sort[List@@M],Sort[List@@N]},{{x_,Z___},{_,x_,W___}}/;submultisetQ[{Z},{W}]]];
stableQ[u_]:=Apply[And,Outer[#1==#2||!submultisetQ[#1,#2]&&!submultisetQ[#2,#1]&,u,u,1],{0,1}];
nms[n_]:=nms[n]=If[n==1,{{1}},Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],stableQ],{ptn,Rest[IntegerPartitions[n]]}]];
Table[Length[nms[n]],{n,12}]

a(17)-a(19) from Robert Price, Sep 14 2018

A316770 Number of series-reduced locally nonintersecting rooted identity trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 1, 2, 3, 6, 13, 28, 64, 153, 379, 939, 2385, 6121, 15871, 41529, 109509, 290607, 775842, 2081874, 5612176, 15191329, 41274052

Offset: 1

Gus Wiseman, Jul 12 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally nonintersecting if the intersection of all branches directly under any given root is empty. It is an identity tree if no branch appears multiple times under the same root.

			The a(6) = 13 trees:
  (1(1(1(12))))
  (1(1(13)))
  (1(2(12)))
  (2(1(12)))
  (12(12))
  (1(14))
  (1(23))
  (2(13))
  (3(12))
  (123)
  (15)
  (24)
  6
Examples of series-reduced rooted identity trees that are not locally nonintersecting are ((12)(13)) and ((12)(1(12))).

Cf. A000081, A000669, A001678, A141268, A292504, A316500, A316651, A316652, A316655, A316696, A316768.

nonintQ[u_]:=Intersection@@u=={};
nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],And[UnsameQ@@#,nonintQ[#]]&],{ptn,Rest[IntegerPartitions[n]]}],{n}];
Table[Length[nms[n]],{n,15}]

a(21)-a(22) from Robert Price, Sep 14 2018

A316771 Number of series-reduced locally nonintersecting rooted trees whose leaves form the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 2, 1, 1, 1, 0, 1, 2, 1, 2, 1, 1, 1, 4, 0, 1, 0, 2, 1, 4, 1, 0, 1, 1, 1, 6, 1, 1, 1, 4, 1, 4, 1, 2, 2, 1, 1, 8, 0, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 12, 1, 1, 2, 0, 1, 4, 1, 2, 1, 4, 1, 17, 1, 1, 2, 2, 1, 4, 1, 8, 0, 1, 1, 12, 1, 1

Offset: 1

Gus Wiseman, Jul 12 2018

nonn

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally nonintersecting if the intersection of all branches directly under any given root is empty.

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

			The a(36) = 6 trees:
  (1(2(12)))
  (2(1(12)))
  (1(122))
  (2(112))
  (12(12))
  (1122)

Cf. A000081, A000669, A001678, A056239, A141268, A292504, A296150, A316503, A316651, A316652, A316655.

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]]]];
gro[m_]:=gro[m]=If[Length[m]==1,List/@m,Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])]];
Table[Length[Select[gro[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],And@@Cases[#,q:{__List}:>Intersection@@q=={},{0,Infinity}]&]],{n,100}]

A316772 Number of series-reduced locally nonintersecting rooted trees whose leaves form an integer partition of n.

Original entry on oeis.org

1, 1, 2, 4, 11, 27, 75, 202, 565, 1602, 4617, 13472, 39781, 118604, 356605, 1080178, 3293109, 10097356, 31118507, 96341035

Offset: 1

Gus Wiseman, Jul 12 2018

A rooted tree is series-reduced if all non-leaf nodes have at least two branches. It is locally nonintersecting if the intersection of all branches directly under any given root is empty.

			The a(6) = 27 trees:
6,
(15),
(24),
(1(14)), (114),
(1(23)), (2(13)), (3(12)), (123),
(1(1(13))), (1(113)), (11(13)), (1113),
(1(2(12))), (1(122)), (2(1(12))), (2(112)), (12(12)), (1122),
(1(1(1(12)))), (1(1(112))), (1(11(12))), (1(1112)), (11(1(12))), (11(112)), (111(12)), (11112).

Cf. A000081, A000669, A001678, A141268, A292504, A316500, A316501, A316503, A316651, A316652, A316655.

nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]],Intersection@@#=={}&],{ptn,Rest[IntegerPartitions[n]]}],{n}];
Table[Length[nms[n]],{n,10}]

a(17)-a(20) from Robert Price, Sep 14 2018

A319291 Number of series-reduced locally disjoint rooted trees with n leaves spanning an initial interval of positive integers.

Original entry on oeis.org

1, 2, 12, 107, 1299, 20764, 412957, 9817743

Offset: 1

Gus Wiseman, Sep 16 2018

			The a(3) = 12 series-reduced locally disjoint rooted trees:
  (1(11))
   (111)
  (1(22))
  (2(12))
   (122)
  (1(12))
  (2(11))
   (112)
  (1(23))
  (2(13))
  (3(12))
   (123)
The trees counted by A316651(4) but not by a(4):
  ((11)(12))
  ((12)(13))
  ((12)(22))
  ((12)(23))
  ((13)(23))

Cf. A000081, A007562, A301700, A316473, A316475, A316495, A316651, A316694, A316695, A316696, A316697, A319286.

disjointQ[u_]:=Apply[And,Outer[#1==#2||Intersection[#1,#2]=={}&,u,u,1],{0,1}];
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]]]];
gro[m_]:=gro[m]=If[Length[m]==1,{m},Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],disjointQ]];
allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Sum[Length[gro[m]],{m,allnorm[n]}],{n,5}]

A319292 Number of series-reduced locally nonintersecting rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

Original entry on oeis.org

1, 1, 6, 48, 455, 5700, 83138, 1454870

Offset: 1

Gus Wiseman, Sep 16 2018

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is locally nonintersecting if the intersection of all branches directly under any given node with at least two branches is empty.

			The a(3) = 6 trees are: (1(12)), (112), (1(23)), (2(13)), (3(12)), (123). Missing from this list but counted by A316651 are: (1(11)), (2(11)), (111).

Cf. A000081, A007562, A273873, A289509, A301700, A316470, A316473, A316475, A316651, A319271.

nonintQ[u_]:=Intersection@@u=={};
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]]]];
gro[m_]:=gro[m]=If[Length[m]==1,{m},Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],nonintQ]];
Table[Sum[Length[gro[m]],{m,Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n]}],{n,5}]

cp's OEIS Frontend

A316695 Number of series-reduced locally disjoint rooted trees whose leaves form the integer partition with Heinz number n.

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A316768 Number of series-reduced locally stable rooted trees whose leaves form an integer partition of n.

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A316767 Number of series-reduced locally stable rooted trees whose leaves form the integer partition with Heinz number n.

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A316769 Number of series-reduced locally stable rooted trees with n unlabeled leaves.

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A316770 Number of series-reduced locally nonintersecting rooted identity trees whose leaves form an integer partition of n.

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A316771 Number of series-reduced locally nonintersecting rooted trees whose leaves form the integer partition with Heinz number n.

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A316772 Number of series-reduced locally nonintersecting rooted trees whose leaves form an integer partition of n.

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A319291 Number of series-reduced locally disjoint rooted trees with n leaves spanning an initial interval of positive integers.

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A319292 Number of series-reduced locally nonintersecting rooted trees whose leaves span an initial interval of positive integers with multiplicities an integer partition of n.

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