cp's OEIS Frontend

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

%I A382078 #11 Mar 29 2025 13:40:24
%S A382078 0,0,1,1,2,2,5,6,9,13,17,23,33,42,58,76,97,126,168,207,266,343,428,
%T A382078 534,675,832,1039,1279,1575,1933,2381,2881,3524,4269,5179,6237,7525,
%U A382078 9033,10860,12969,15512,18475,22005,26105,30973,36642,43325,51078,60184,70769,83152
%N A382078 Number of integer partitions of n that cannot be partitioned into a set of sets.
%C A382078 First differs from A240309 at a(11) = 23, A240309(11) = 25.
%C A382078 First differs from A381990 at a(17) = 126, A381990(17) = 127.
%e A382078 The partition y = (2,2,1,1,1) can be partitioned into sets in the following ways:
%e A382078   {{1},{1,2},{1,2}}
%e A382078   {{1},{1},{2},{1,2}}
%e A382078   {{1},{1},{1},{2},{2}}
%e A382078 But none of these is itself a set, so y is counted under a(7).
%e A382078 The a(2) = 1 through a(8) = 9 partitions:
%e A382078   (11)  (111)  (22)    (2111)   (33)      (2221)     (44)
%e A382078                (1111)  (11111)  (222)     (4111)     (2222)
%e A382078                                 (3111)    (22111)    (5111)
%e A382078                                 (21111)   (31111)    (22211)
%e A382078                                 (111111)  (211111)   (41111)
%e A382078                                           (1111111)  (221111)
%e A382078                                                      (311111)
%e A382078                                                      (2111111)
%e A382078                                                      (11111111)
%t A382078 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A382078 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A382078 Table[Length[Select[IntegerPartitions[n],Length[Select[mps[#],UnsameQ@@#&&And@@UnsameQ@@@#&]]==0&]],{n,0,9}]
%Y A382078 More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
%Y A382078 For normal multisets see A292432, A292444, A116539.
%Y A382078 These partitions are ranked by A293243, complement A382200.
%Y A382078 The MM-numbers of these multiset partitions (set of sets) are A302494.
%Y A382078 Twice-partitions of this type are counted by A358914.
%Y A382078 For distinct sums we have A381990 (ranks A381806), complement A381992 (ranks A382075).
%Y A382078 The complement is counted by A382077, unique A382079.
%Y A382078 A000041 counts integer partitions, strict A000009.
%Y A382078 A050320 counts multiset partitions of prime indices into sets.
%Y A382078 A050326 counts multiset partitions into distinct sets, complement A050345.
%Y A382078 A265947 counts refinement-ordered pairs of integer partitions.
%Y A382078 Cf. A293511, A299202, A317142, A381441, A381454, A381717, A381718, A381870.
%K A382078 nonn
%O A382078 0,5
%A A382078 _Gus Wiseman_, Mar 18 2025
%E A382078 a(19)-a(50) from _Bert Dobbelaere_, Mar 29 2025