proposed

approved

editing

proposed

Cf. A000081, A000311, A000669, A001678, A004111, A005804, A034691, A056239, A141268, A181821, A292504, A296150, A300660, A304660.

Cf. A316651, A316652, A316653. A316654, A316655.

allocated for Gus WisemanNumber of series-reduced rooted identity trees whose leaves span an initial interval of positive integers with multiplicities the integer partition with Heinz number n.

0, 1, 0, 1, 0, 1, 0, 4, 3, 1, 0, 9, 0, 1, 6, 26, 0, 36, 0, 16, 10, 1, 0, 92, 21, 1, 197, 25, 0, 100, 0, 236, 15, 1, 53, 474

1,8

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is an identity tree if no branch appears multiple times under the same root.

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

a(prime(n>1)) = 0.

a(2^n) = A000311(n).

Sequence of sets of trees begins:

1:

2: 1

3:

4: (12)

5:

6: (1(12))

7:

8: (1(23)), (2(13)), (3(12)), (123)

9: (1(2(12))), (2(1(12))), (12(12))

10: (1(1(12)))

11:

12: (1(1(23))), (1(2(13))), (1(3(12))), (1(123)), (2(1(13))), (3(1(12))), ((12)(13)), (12(13)), (13(12))

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_]:=If[Length[m]==1, m, Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m], Length[#]>1&])], UnsameQ@@#&]];

Table[Length[gro[Flatten[MapIndexed[Table[#2, {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]]]]]], {n, 30}]

Cf. A000081, A000311, A000669, A001678, A004111, A005804, A034691, A056239, A141268, A181821, A292504, A296150, A300660, A304660.

allocated

nonn,more

Gus Wiseman, Jul 09 2018

approved

editing

allocated for Gus Wiseman

allocated

approved