A367220 Number of strict integer partitions of n whose length (number of parts) can be written as a nonnegative linear combination of the parts.

1, 1, 0, 1, 1, 2, 3, 3, 4, 5, 7, 7, 10, 11, 15, 17, 22, 25, 32, 37, 46, 53, 65, 75, 90, 105, 124, 143, 168, 193, 224, 258, 297, 340, 390, 446, 509, 580, 660, 751, 852, 967, 1095, 1240, 1401, 1584, 1786, 2015, 2269, 2554, 2869, 3226, 3617, 4056, 4541, 5084

Offset: 0

Author

Gus Wiseman, Nov 14 2023

Keywords

• nonn

Comments

The non-strict version is A367218.

Examples

			The a(3) = 1 through a(10) = 7 strict partitions:
  (2,1)  (3,1)  (3,2)  (4,2)    (5,2)    (6,2)    (7,2)    (8,2)
                (4,1)  (5,1)    (6,1)    (7,1)    (8,1)    (9,1)
                       (3,2,1)  (4,2,1)  (4,3,1)  (4,3,2)  (5,3,2)
                                         (5,2,1)  (5,3,1)  (5,4,1)
                                                  (6,2,1)  (6,3,1)
                                                           (7,2,1)
                                                           (4,3,2,1)
		

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.

A188431 counts complete strict partitions, incomplete A365831.

A240855 counts strict partitions whose length is a part, complement A240861.

A364272 counts sum-full strict partitions, sum-free A364349.

A365046 counts combination-full subsets, differences of A364914.

Cf. A008289, A088314, A116861, A124506, A363225, A364346, A364350, A364916, A365073, A365311, A365312.

Programs

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], UnsameQ@@#&&combs[Length[#], Union[#]]!={}&]], {n,0,15}]