A367213 - OEIS

%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