A382078 Number of integer partitions of n that cannot be partitioned into a set of sets.

0, 0, 1, 1, 2, 2, 5, 6, 9, 13, 17, 23, 33, 42, 58, 76, 97, 126, 168, 207, 266, 343, 428, 534, 675, 832, 1039, 1279, 1575, 1933, 2381, 2881, 3524, 4269, 5179, 6237, 7525, 9033, 10860, 12969, 15512, 18475, 22005, 26105, 30973, 36642, 43325, 51078, 60184, 70769, 83152

Offset: 0

Author

Gus Wiseman, Mar 18 2025

Keywords

• nonn

Comments

First differs from A240309 at a(11) = 23, A240309(11) = 25.

First differs from A381990 at a(17) = 126, A381990(17) = 127.

Examples

			The partition y = (2,2,1,1,1) can be partitioned into sets in the following ways:
  {{1},{1,2},{1,2}}
  {{1},{1},{2},{1,2}}
  {{1},{1},{1},{2},{2}}
But none of these is itself a set, so y is counted under a(7).
The a(2) = 1 through a(8) = 9 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)
               (1111)  (11111)  (222)     (4111)     (2222)
                                (3111)    (22111)    (5111)
                                (21111)   (31111)    (22211)
                                (111111)  (211111)   (41111)
                                          (1111111)  (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
		

Crossrefs

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.

For normal multisets see A292432, A292444, A116539.

These partitions are ranked by A293243, complement A382200.

The MM-numbers of these multiset partitions (set of sets) are A302494.

Twice-partitions of this type are counted by A358914.

For distinct sums we have A381990 (ranks A381806), complement A381992 (ranks A382075).

The complement is counted by A382077, unique A382079.

A000041 counts integer partitions, strict A000009.

A050320 counts multiset partitions of prime indices into sets.

A050326 counts multiset partitions into distinct sets, complement A050345.

A265947 counts refinement-ordered pairs of integer partitions.

Cf. A293511, A299202, A317142, A381441, A381454, A381717, A381718, A381870.

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]]]];
Table[Length[Select[IntegerPartitions[n],Length[Select[mps[#],UnsameQ@@#&&And@@UnsameQ@@@#&]]==0&]],{n,0,9}]

Extensions

a(19)-a(50) from Bert Dobbelaere, Mar 29 2025