A367399 Number of strict integer partitions of n whose length is not the sum of any two distinct parts.

1, 1, 1, 2, 2, 3, 3, 4, 5, 7, 8, 10, 13, 15, 19, 22, 27, 31, 38, 43, 51, 59, 70, 79, 94, 107, 124, 143, 165, 188, 218, 248, 283, 324, 369, 419, 476, 540, 610, 691, 778, 878, 987, 1111, 1244, 1399, 1563, 1750, 1954, 2184, 2432, 2714, 3016, 3358, 3730, 4143

Offset

0,4

Links

Table of n, a(n) for n=0..55.

Example

The strict partition y = (6,4,2,1) has semi-sums {3,5,6,7,8,10}, which do not include 4, so y is counted under a(13).
The a(6) = 3 through a(13) = 15 strict partitions:
  (6)    (7)    (8)      (9)      (10)     (11)     (12)       (13)
  (4,2)  (4,3)  (5,3)    (5,4)    (6,4)    (6,5)    (7,5)      (7,6)
  (5,1)  (5,2)  (6,2)    (6,3)    (7,3)    (7,4)    (8,4)      (8,5)
         (6,1)  (7,1)    (7,2)    (8,2)    (8,3)    (9,3)      (9,4)
                (4,3,1)  (8,1)    (9,1)    (9,2)    (10,2)     (10,3)
                         (4,3,2)  (5,3,2)  (10,1)   (11,1)     (11,2)
                         (5,3,1)  (5,4,1)  (5,4,2)  (5,4,3)    (12,1)
                                  (6,3,1)  (6,3,2)  (6,4,2)    (6,4,3)
                                           (6,4,1)  (6,5,1)    (6,5,2)
                                           (7,3,1)  (7,3,2)    (7,4,2)
                                                    (7,4,1)    (7,5,1)
                                                    (8,3,1)    (8,3,2)
                                                    (5,4,2,1)  (8,4,1)
                                                               (9,3,1)
                                                               (6,4,2,1)

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#, {2}], Length[#]]&]], {n, 0, 15}]

Crossrefs

The following sequences count and rank integer partitions and finite sets according to whether their length is a subset-sum, linear combination, or semi-sum of the parts. The current sequence is starred.

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

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partitions: A367212 A367213 A367218 A367219 A367394 A367398

strict: A367214 A367215 A367220 A367221 A367395 A367399*

subsets: A367216 A367217 A367222 A367223 A367396 A367400

ranks: A367224 A367225 A367226 A367227 A367397 A367401

A000041 counts partitions, strict A000009.

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

A365924 counts incomplete partitions, strict A365831.

A236912 counts partitions with no semi-sum of the parts, ranks A364461.

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

A366738 counts semi-sums of partitions, strict A366741.

A367403 counts partitions without covering semi-sums, strict A367411.

Triangles:

A008284 counts partitions by length, strict A008289.

A365541 counts subsets with a semi-sum k.

A367404 counts partitions with a semi-sum k, strict A367405.

Cf. A000700, A126796, A188431, A238628, A237113, A363225, A364272, A365658, A365918, A365925, A367410.

Sequence in context: A036823 A035575 A036816 * A113788 A018243 A127207

Adjacent sequences: A367396 A367397 A367398 * A367400 A367401 A367402

Keyword

nonn

Author

Gus Wiseman, Nov 19 2023

Status

approved