A367215 Number of strict integer partitions of n whose length (number of parts) is not equal to the sum of any subset.

0, 0, 1, 1, 2, 2, 2, 3, 3, 4, 5, 7, 8, 10, 12, 15, 18, 21, 25, 29, 34, 40, 46, 53, 62, 71, 82, 95, 109, 124, 143, 162, 185, 210, 240, 270, 308, 347, 393, 443, 500, 562, 634, 711, 798, 895, 1002, 1120, 1252, 1397, 1558, 1735, 1930, 2146, 2383, 2644, 2930, 3245

Offset

0,5

Comments

These partitions have Heinz numbers A367225 /\ A005117.

Links

Chai Wah Wu, Table of n, a(n) for n = 0..118

Example

The a(2) = 1 through a(11) = 7 strict partitions:
  (2)  (3)  (4)    (5)    (6)    (7)    (8)    (9)    (10)     (11)
            (3,1)  (4,1)  (5,1)  (4,3)  (5,3)  (5,4)  (6,4)    (6,5)
                                 (6,1)  (7,1)  (6,3)  (7,3)    (7,4)
                                               (8,1)  (9,1)    (8,3)
                                                      (5,4,1)  (10,1)
                                                               (5,4,2)
                                                               (6,4,1)
The a(2) = 1 through a(15) = 15 strict partitions (A..F = 10..15):
  2  3  4   5   6   7   8   9   A    B    C    D    E     F
        31  41  51  43  53  54  64   65   75   76   86    87
                    61  71  63  73   74   84   85   95    96
                            81  91   83   93   94   A4    A5
                                541  A1   B1   A3   B3    B4
                                     542  642  C1   D1    C3
                                     641  651  652  752   E1
                                          741  742  761   654
                                               751  842   762
                                               841  851   852
                                                    941   861
                                                    6521  942
                                                          951
                                                          A41
                                                          7521

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&FreeQ[Total/@Subsets[#], Length[#]]&]], {n, 0, 30}]

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

-------------------------------------------

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.

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

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

A188431 counts complete strict partitions, incomplete A365831.

A237667 counts sum-free partitions, ranks A364531.

A240861 counts strict partitions with length not a part, complement A240855.

A275972 counts strict knapsack partitions, non-strict A108917.

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

Triangles:

A008289 counts strict partitions by length, non-strict A008284.

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

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

A365832 counts strict partitions by subset-sums, non-strict A365658.

Cf. A002865, A229816, A238628, A364346, A364350, A365312, A365922, A366320.

Sequence in context: A074732 A089046 A054911 * A137267 A341451 A123576

Adjacent sequences: A367212 A367213 A367214 * A367216 A367217 A367218

Keyword

nonn

Author

Gus Wiseman, Nov 12 2023

Status

approved