A316652 -id:A316652 - cp's OEIS frontend

A316767 Number of series-reduced locally stable 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, 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

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions