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Number of integer partitions of n whose length cannot be written as a nonnegative linear combination of the distinct parts.
24

%I #15 Dec 30 2023 17:00:43

%S 0,0,1,1,1,1,3,2,4,4,7,6,11,9,16,16,23,22,35,33,48,50,69,70,99,99,136,

%T 142,187,194,261,267,346,367,468,489,626,650,824,870,1081,1135,1421,

%U 1485,1833,1942,2374,2501,3062,3220,3915,4145,4987,5274,6363,6709,8027

%N Number of integer partitions of n whose length cannot be written as a nonnegative linear combination of the distinct parts.

%e 3 cannot be written as a nonnegative linear combination of 2 and 5, so (5,2,2) is counted under a(9).

%e The a(2) = 1 through a(10) = 7 partitions:

%e (2) (3) (4) (5) (6) (7) (8) (9) (10)

%e (3,3) (4,3) (4,4) (5,4) (5,5)

%e (2,2,2) (5,3) (6,3) (6,4)

%e (4,2,2) (5,2,2) (7,3)

%e (4,4,2)

%e (6,2,2)

%e (2,2,2,2,2)

%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],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 A008284 counts partitions by length, strict A008289.

%Y A124506 appears to count combination-free subsets, differences of A326083.

%Y A365046 counts combination-full subsets, differences of A364914.

%Y Cf. A068911, A088314, A116861, A364345, A364350, A365073, A365312, A365380.

%K nonn

%O 0,7

%A _Gus Wiseman_, Nov 14 2023

%E a(31)-a(56) from _Chai Wah Wu_, Nov 15 2023