cp's OEIS Frontend

A382079 Number of integer partitions of n that can be partitioned into a set of sets in exactly one way.

%I A382079 #14 Mar 29 2025 17:25:18
%S A382079 1,1,1,1,2,3,3,4,6,5,10,9,13,14,21,20,32,31,42,47,63,62,90,94,117,138,
%T A382079 170,186,235,260,315,363,429,493,588,674,795,901,1060,1209,1431,1608,
%U A382079 1896,2152,2515,2854,3310,3734,4368,4905,5686
%N A382079 Number of integer partitions of n that can be partitioned into a set of sets in exactly one way.
%e A382079 The unique multiset partition for (3222111) is {{1},{2},{1,2},{1,2,3}}.
%e A382079 The a(1) = 1 through a(12) = 13 partitions:
%e A382079   1  2  3  4    5    6     7    8      9      A      B      C
%e A382079            211  221  411   322  332    441    433    443    552
%e A382079                 311  2211  331  422    522    442    533    633
%e A382079                            511  611    711    622    551    822
%e A382079                                 3311   42111  811    722    A11
%e A382079                                 32111         3322   911    4422
%e A382079                                               4411   42221  5511
%e A382079                                               32221  53111  33321
%e A382079                                               43111  62111  52221
%e A382079                                               52111         54111
%e A382079                                                             63111
%e A382079                                                             72111
%e A382079                                                             3222111
%t A382079 ssfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[ssfacs[n/d],Min@@#>d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
%t A382079 Table[Length[Select[IntegerPartitions[n],Length[ssfacs[Times@@Prime/@#]]==1&]],{n,0,15}]
%Y A382079 Normal multiset partitions of this type are counted by A116539, see A381718.
%Y A382079 These partitions are ranked by A293511.
%Y A382079 MM-numbers of these multiset partitions (sets of sets) are A302494, see A302478, A382201.
%Y A382079 Twice-partitions of this type (sets of sets) are counted by A358914, see A279785.
%Y A382079 For at least one choice we have A382077 (ranks A382200), see A381992 (ranks A382075).
%Y A382079 For no choices we have A382078 (ranks A293243), see A381990 (ranks A381806).
%Y A382079 For distinct block-sums instead of blocks we have A382460, ranked by A381870.
%Y A382079 Set multipartitions: A089259, A116540, A270995, A296119, A318360.
%Y A382079 A000041 counts integer partitions, strict A000009.
%Y A382079 A050320 counts multiset partitions of prime indices into sets.
%Y A382079 A050326 counts multiset partitions of prime indices into distinct sets, see A381633.
%Y A382079 A265947 counts refinement-ordered pairs of integer partitions.
%Y A382079 Cf. A002846, A213427, A299202, A317142, A381078, A381441, A381454, A381636.
%K A382079 nonn,more
%O A382079 0,5
%A A382079 _Gus Wiseman_, Mar 20 2025
%E A382079 a(21)-a(50) from _Bert Dobbelaere_, Mar 29 2025