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A322077
In the ranked poset of integer partitions ordered by refinement, number of integer partitions coarser (greater) than or equal to the integer partition whose multiplicities are the prime indices of n in weakly decreasing order.
0
1, 1, 2, 2, 3, 4, 5, 5, 8, 6, 7, 9, 11, 10, 12, 13, 15, 18, 22, 15, 19, 14, 30, 24, 22, 21, 40, 23, 42, 29, 56, 36, 27, 29, 34, 47, 77, 41, 39, 40
OFFSET
1,3
COMMENTS
This partition (reversed row n of A305936) is generally not the same as the integer partition with Heinz number n. For example, 12 is the Heinz number of (2,1,1), while the integer partition whose multiplicities are (2,1,1) is (3,2,1,1).
EXAMPLE
The list of a(1) = 1 through a(18) = 18 coarser partitions:
() (1) (2) (3) (3) (4) (4) (6) (6) (5) (5)
(11) (21) (21) (22) (22) (33) (33) (32) (32)
(111) (31) (31) (42) (42) (41) (41)
(211) (211) (51) (51) (221) (221)
(1111) (321) (222) (311) (311)
(321) (2111) (2111)
(411) (11111)
(2211)
.
(7) (6) (6) (7) (10) (7) (9)
(43) (33) (33) (43) (55) (43) (54)
(52) (42) (42) (52) (64) (52) (63)
(61) (51) (51) (61) (73) (61) (72)
(322) (222) (222) (322) (82) (322) (81)
(331) (321) (321) (331) (91) (331) (333)
(421) (411) (411) (421) (433) (421) (432)
(511) (2211) (2211) (511) (442) (511) (441)
(3211) (3111) (3111) (2221) (532) (2221) (522)
(21111) (21111) (3211) (541) (3211) (531)
(111111) (4111) (631) (4111) (621)
(22111) (721) (22111) (711)
(4321) (31111) (3222)
(211111) (3321)
(1111111) (4221)
(4311)
(5211)
(32211)
MATHEMATICA
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]]]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]], {#1}]&, If[n==1, {}, Flatten[Cases[FactorInteger[n]//Reverse, {p_, k_}:>Table[PrimePi[p], {k}]]]]];
Table[Length[Union[Sort/@Apply[Plus, mps[nrmptn[n]], {2}]]], {n, 20}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 25 2018
STATUS
approved