A367213 Number of integer partitions of n whose length (number of parts) is not equal to the sum of any submultiset.

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, 217, 274, 303, 379, 420, 523, 579, 709, 786, 960, 1061, 1285, 1423, 1714, 1885, 2265, 2498, 2966, 3280, 3881, 4268, 5049, 5548, 6507, 7170, 8391, 9194, 10744, 11778, 13677

Offset: 0

Author

Gus Wiseman, Nov 12 2023

Keywords

• nonn

Comments

These partitions are necessarily incomplete (A365924).

Are there any decreases after the initial terms?

Examples

			The a(3) = 1 through a(9) = 8 partitions:
  (3)  (4)    (5)    (6)      (7)      (8)        (9)
       (3,1)  (4,1)  (3,3)    (4,3)    (4,4)      (5,4)
                     (5,1)    (6,1)    (5,3)      (6,3)
                     (2,2,2)  (5,1,1)  (7,1)      (8,1)
                     (4,1,1)           (4,2,2)    (4,4,1)
                                       (6,1,1)    (5,2,2)
                                       (5,1,1,1)  (7,1,1)
                                                  (6,1,1,1)
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..65

Crossrefs

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.

sum-full sum-free comb-full comb-free

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partitions: A367212 A367213* A367218 A367219

strict: A367214 A367215 A367220 A367221

subsets: A367216 A367217 A367222 A367223

ranks: A367224 A367225 A367226 A367227

A000041 counts partitions, strict A000009.

A002865 counts partitions whose length is a part, complement A229816.

A007865/A085489/A151897 count certain types of sum-free subsets.

A108917 counts knapsack partitions, non-knapsack A366754.

A126796 counts complete partitions, incomplete A365924.

A237667 counts sum-free partitions, sum-full A237668.

A304792 counts subset-sums of partitions, strict A365925.

Triangles:

A008284 counts partitions by length, strict A008289.

A046663 counts partitions of n without a subset-sum k, strict A365663.

A365543 counts partitions of n with a subset-sum k, strict A365661.

A365658 counts partitions by number of subset-sums, strict A365832.

Cf. A000700, A124506, A238628, A240861, A364349, A364531, A365045, A365381, A365918, A366320.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n], FreeQ[Total/@Subsets[#], Length[#]]&]], {n,0,10}]

Extensions

a(41)-a(54) from Chai Wah Wu, Nov 13 2023