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.

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).

Links

Table of n, a(n) for n=1..40.

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}]

Crossrefs

Cf. A063834, A066723, A213427, A265947, A299201, A300383, A317141, A321468, A321472.

Sequence in context: A096443 A126442 A129306 * A114094 A179255 A332285

Adjacent sequences: A322074 A322075 A322076 * A322078 A322079 A322080

Keyword

nonn,more

Author

Gus Wiseman, Nov 25 2018

Status

approved