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%I #6 Nov 15 2023 08:23:08
%S 1,1,0,1,1,2,3,3,4,5,7,7,10,11,15,17,22,25,32,37,46,53,65,75,90,105,
%T 124,143,168,193,224,258,297,340,390,446,509,580,660,751,852,967,1095,
%U 1240,1401,1584,1786,2015,2269,2554,2869,3226,3617,4056,4541,5084
%N Number of strict integer partitions of n whose length (number of parts) can be written as a nonnegative linear combination of the parts.
%C The non-strict version is A367218.
%e The a(3) = 1 through a(10) = 7 strict partitions:
%e (2,1) (3,1) (3,2) (4,2) (5,2) (6,2) (7,2) (8,2)
%e (4,1) (5,1) (6,1) (7,1) (8,1) (9,1)
%e (3,2,1) (4,2,1) (4,3,1) (4,3,2) (5,3,2)
%e (5,2,1) (5,3,1) (5,4,1)
%e (6,2,1) (6,3,1)
%e (7,2,1)
%e (4,3,2,1)
%t combs[n_,y_]:=With[{s=Table[{k,i},{k,y}, {i,0,Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
%t Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&combs[Length[#], Union[#]]!={}&]], {n,0,15}]
%Y The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum or linear combination of the parts. The current sequence is starred.
%Y sum-full sum-free comb-full comb-free
%Y -------------------------------------------
%Y partitions: A367212 A367213 A367218 A367219
%Y strict: A367214 A367215 A367220* A367221
%Y subsets: A367216 A367217 A367222 A367223
%Y ranks: A367224 A367225 A367226 A367227
%Y A000041 counts integer partitions, strict A000009.
%Y A002865 counts partitions whose length is a part, complement A229816.
%Y A188431 counts complete strict partitions, incomplete A365831.
%Y A240855 counts strict partitions whose length is a part, complement A240861.
%Y A364272 counts sum-full strict partitions, sum-free A364349.
%Y A365046 counts combination-full subsets, differences of A364914.
%Y Cf. A008289, A088314, A116861, A124506, A363225, A364346, A364350, A364916, A365073, A365311, A365312.
%K nonn
%O 0,6
%A _Gus Wiseman_, Nov 14 2023