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%I #19 Dec 30 2023 17:01:59
%S 0,0,1,1,2,2,5,4,7,8,12,13,19,21,29,33,45,49,67,73,97,108,139,152,196,
%T 217,274,303,379,420,523,579,709,786,960,1061,1285,1423,1714,1885,
%U 2265,2498,2966,3280,3881,4268,5049,5548,6507,7170,8391,9194,10744,11778,13677
%N Number of integer partitions of n whose length (number of parts) is not equal to the sum of any submultiset.
%C These partitions are necessarily incomplete (A365924).
%C Are there any decreases after the initial terms?
%H Chai Wah Wu, <a href="/A367213/b367213.txt">Table of n, a(n) for n = 0..65</a>
%e The a(3) = 1 through a(9) = 8 partitions:
%e (3) (4) (5) (6) (7) (8) (9)
%e (3,1) (4,1) (3,3) (4,3) (4,4) (5,4)
%e (5,1) (6,1) (5,3) (6,3)
%e (2,2,2) (5,1,1) (7,1) (8,1)
%e (4,1,1) (4,2,2) (4,4,1)
%e (6,1,1) (5,2,2)
%e (5,1,1,1) (7,1,1)
%e (6,1,1,1)
%t Table[Length[Select[IntegerPartitions[n], FreeQ[Total/@Subsets[#], Length[#]]&]], {n,0,10}]
%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 partitions, strict A000009.
%Y A002865 counts partitions whose length is a part, complement A229816.
%Y A007865/A085489/A151897 count certain types of sum-free subsets.
%Y A108917 counts knapsack partitions, non-knapsack A366754.
%Y A126796 counts complete partitions, incomplete A365924.
%Y A237667 counts sum-free partitions, sum-full A237668.
%Y A304792 counts subset-sums of partitions, strict A365925.
%Y Triangles:
%Y A008284 counts partitions by length, strict A008289.
%Y A046663 counts partitions of n without a subset-sum k, strict A365663.
%Y A365543 counts partitions of n with a subset-sum k, strict A365661.
%Y A365658 counts partitions by number of subset-sums, strict A365832.
%Y Cf. A000700, A124506, A238628, A240861, A364349, A364531, A365045, A365381, A365918, A366320.
%K nonn
%O 0,5
%A _Gus Wiseman_, Nov 12 2023
%E a(41)-a(54) from _Chai Wah Wu_, Nov 13 2023