A317141 In the ranked poset of integer partitions ordered by refinement, number of integer partitions coarser (greater) than or equal to the integer partition with Heinz number n.

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 8, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 10, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 9, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 12, 1, 2, 4, 4, 2, 5, 1, 11, 5, 2, 1, 10, 2

Offset

1,4

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 1..65536

Example

The a(24) = 6 partitions coarser than or equal to (2111) are (2111), (311), (221), (32), (41), (5), with Heinz numbers 24, 20, 18, 15, 14, 11.

Maple

g:= l-> `if`(l=[], {[]}, (t-> map(sort, map(x->
        [seq(subsop(i=x[i]+t, x), i=1..nops(x)),
        [x[], t]][], g(subsop(-1=[][], l)))))(l[-1])):
a:= n-> nops(g(map(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]))):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 22 2018

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
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]]]];
ptncaps[ptn_]:=Union[Sort/@Apply[Plus, mps[ptn], {2}]];
Table[Length[ptncaps[primeMS[n]]], {n, 100}]

Crossrefs

Cf. A002846, A056239, A213427, A215366, A265947, A296150, A296150, A299201, A300383, A317142, A317143.

Sequence in context: A292886 A317508 A323438 * A317791 A318559 A326334

Adjacent sequences: A317138 A317139 A317140 * A317142 A317143 A317144

Keyword

nonn

Author

Gus Wiseman, Jul 22 2018

Status

approved