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

Author

Gus Wiseman, Nov 25 2018

Keywords

• nonn
• more

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).

Examples

			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)
		

Crossrefs

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

Programs

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}]