A367219 Number of integer partitions of n whose length cannot be written as a nonnegative linear combination of the distinct parts.

0, 0, 1, 1, 1, 1, 3, 2, 4, 4, 7, 6, 11, 9, 16, 16, 23, 22, 35, 33, 48, 50, 69, 70, 99, 99, 136, 142, 187, 194, 261, 267, 346, 367, 468, 489, 626, 650, 824, 870, 1081, 1135, 1421, 1485, 1833, 1942, 2374, 2501, 3062, 3220, 3915, 4145, 4987, 5274, 6363, 6709, 8027

Offset

0,7

Links

Table of n, a(n) for n=0..56.

Example

3 cannot be written as a nonnegative linear combination of 2 and 5, so (5,2,2) is counted under a(9).
The a(2) = 1 through a(10) = 7 partitions:
  (2)  (3)  (4)  (5)  (6)      (7)    (8)      (9)      (10)
                      (3,3)    (4,3)  (4,4)    (5,4)    (5,5)
                      (2,2,2)         (5,3)    (6,3)    (6,4)
                                      (4,2,2)  (5,2,2)  (7,3)
                                                        (4,4,2)
                                                        (6,2,2)
                                                        (2,2,2,2,2)

Mathematica

combs[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n], combs[Length[#], Union[#]]=={}&]], {n, 0, 15}]

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 integer partitions, strict A000009.

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

A008284 counts partitions by length, strict A008289.

A124506 appears to count combination-free subsets, differences of A326083.

A365046 counts combination-full subsets, differences of A364914.

Cf. A068911, A088314, A116861, A364345, A364350, A365073, A365312, A365380.

Sequence in context: A175512 A240829 A284013 * A241412 A241445 A371180

Adjacent sequences: A367216 A367217 A367218 * A367220 A367221 A367222

Keyword

nonn

Author

Gus Wiseman, Nov 14 2023

Extensions

a(31)-a(56) from Chai Wah Wu, Nov 15 2023

Status

approved