cp's OEIS Frontend

A364349 Number of strict integer partitions of n containing the sum of no subset of the parts.

%I A364349 #7 Jul 30 2023 09:20:19
%S A364349 1,1,1,2,2,3,3,5,5,8,7,11,11,15,14,21,21,28,29,38,38,51,50,65,68,82,
%T A364349 83,108,106,130,136,163,168,206,210,248,266,307,322,381,391,457,490,
%U A364349 553,582,675,703,797,854,952,1000,1147,1187,1331,1437,1564,1656,1869
%N A364349 Number of strict integer partitions of n containing the sum of no subset of the parts.
%C A364349 First differs from A275972 in counting (7,5,3,1), which is not knapsack.
%e A364349 The partition y = (7,5,3,1) has no subset with sum in y, so is counted under a(16).
%e A364349 The partition y = (15,8,4,2,1) has subset {1,2,4,8} with sum in y, so is not counted under a(31).
%e A364349 The a(1) = 1 through a(9) = 8 partitions:
%e A364349   (1)  (2)  (3)    (4)    (5)    (6)    (7)      (8)      (9)
%e A364349             (2,1)  (3,1)  (3,2)  (4,2)  (4,3)    (5,3)    (5,4)
%e A364349                           (4,1)  (5,1)  (5,2)    (6,2)    (6,3)
%e A364349                                         (6,1)    (7,1)    (7,2)
%e A364349                                         (4,2,1)  (5,2,1)  (8,1)
%e A364349                                                           (4,3,2)
%e A364349                                                           (5,3,1)
%e A364349                                                           (6,2,1)
%t A364349 Table[Length[Select[IntegerPartitions[n],Function[ptn,UnsameQ@@ptn&&Select[Subsets[ptn,{2,Length[ptn]}],MemberQ[ptn,Total[#]]&]=={}]]],{n,0,30}]
%Y A364349 For subsets of {1..n} we have A151897, complement A364534.
%Y A364349 The non-strict version is A237667, ranked by A364531.
%Y A364349 The complement in strict partitions is counted by A364272.
%Y A364349 The linear combination-free version is A364350.
%Y A364349 The binary version is A364533, allowing re-used parts A364346.
%Y A364349 A000041 counts partitions, strict A000009.
%Y A364349 A008284 counts partitions by length, strict A008289.
%Y A364349 A108917 counts knapsack partitions, strict A275972.
%Y A364349 A236912 counts sum-free partitions (not re-using parts), complement A237113.
%Y A364349 A323092 counts double-free partitions, ranks A320340.
%Y A364349 Cf. A007865, A025065, A085489, A093971, A111133, A240861, A320347, A325862, A363226, A364345.
%K A364349 nonn
%O A364349 0,4
%A A364349 _Gus Wiseman_, Jul 29 2023