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A367219 Number of integer partitions of n whose length cannot be written as a nonnegative linear combination of the distinct parts.

%I A367219 #15 Dec 30 2023 17:00:43
%S A367219 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 A367219 142,187,194,261,267,346,367,468,489,626,650,824,870,1081,1135,1421,
%U A367219 1485,1833,1942,2374,2501,3062,3220,3915,4145,4987,5274,6363,6709,8027
%N A367219 Number of integer partitions of n whose length cannot be written as a nonnegative linear combination of the distinct parts.
%e A367219 3 cannot be written as a nonnegative linear combination of 2 and 5, so (5,2,2) is counted under a(9).
%e A367219 The a(2) = 1 through a(10) = 7 partitions:
%e A367219   (2)  (3)  (4)  (5)  (6)      (7)    (8)      (9)      (10)
%e A367219                       (3,3)    (4,3)  (4,4)    (5,4)    (5,5)
%e A367219                       (2,2,2)         (5,3)    (6,3)    (6,4)
%e A367219                                       (4,2,2)  (5,2,2)  (7,3)
%e A367219                                                         (4,4,2)
%e A367219                                                         (6,2,2)
%e A367219                                                         (2,2,2,2,2)
%t A367219 combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A367219 Table[Length[Select[IntegerPartitions[n],combs[Length[#],Union[#]]=={}&]],{n,0,15}]
%Y A367219 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 A367219                sum-full   sum-free   comb-full  comb-free
%Y A367219               -------------------------------------------
%Y A367219   partitions:  A367212    A367213    A367218    A367219*
%Y A367219   strict:      A367214    A367215    A367220    A367221
%Y A367219   subsets:     A367216    A367217    A367222    A367223
%Y A367219   ranks:       A367224    A367225    A367226    A367227
%Y A367219 A000041 counts integer partitions, strict A000009.
%Y A367219 A002865 counts partitions whose length is a part, complement A229816.
%Y A367219 A008284 counts partitions by length, strict A008289.
%Y A367219 A124506 appears to count combination-free subsets, differences of A326083.
%Y A367219 A365046 counts combination-full subsets, differences of A364914.
%Y A367219 Cf. A068911, A088314, A116861, A364345, A364350, A365073, A365312, A365380.
%K A367219 nonn
%O A367219 0,7
%A A367219 _Gus Wiseman_, Nov 14 2023
%E A367219 a(31)-a(56) from _Chai Wah Wu_, Nov 15 2023