A365924 Number of incomplete integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

0, 0, 1, 1, 3, 3, 6, 7, 12, 14, 22, 25, 38, 46, 64, 76, 106, 124, 167, 199, 261, 309, 402, 471, 604, 714, 898, 1053, 1323, 1542, 1911, 2237, 2745, 3201, 3913, 4536, 5506, 6402, 7706, 8918, 10719, 12364, 14760, 17045, 20234, 23296, 27600, 31678, 37365, 42910, 50371, 57695, 67628, 77300, 90242, 103131, 119997

Offset

0,5

Comments

The complement (complete partitions) is A126796.

Links

Joerg Arndt, Table of n, a(n) for n = 0..10000

Formula

a(n) = A000041(n) - A126796(n).

Example

The a(0) = 0 through a(8) = 12 partitions:
  .  .  (2)  (3)  (4)    (5)    (6)      (7)      (8)
                  (2,2)  (3,2)  (3,3)    (4,3)    (4,4)
                  (3,1)  (4,1)  (4,2)    (5,2)    (5,3)
                                (5,1)    (6,1)    (6,2)
                                (2,2,2)  (3,2,2)  (7,1)
                                (4,1,1)  (3,3,1)  (3,3,2)
                                         (5,1,1)  (4,2,2)
                                                  (4,3,1)
                                                  (5,2,1)
                                                  (6,1,1)
                                                  (2,2,2,2)
                                                  (5,1,1,1)

Mathematica

nmz[y_]:=Complement[Range[Total[y]], Total/@Subsets[y]];
Table[Length[Select[IntegerPartitions[n], Length[nmz[#]]>0&]], {n, 0, 15}]

Crossrefs

For parts instead of sums we have A047967/A365919, ranks A080259/A055932.

The complement is A126796, ranks A325781, strict A188431.

These partitions have ranks A365830.

The strict case is A365831.

Row sums of A365923 without the first column, strict A365545.

A000041 counts integer partitions, strict A000009.

A046663 counts partitions w/o a submultiset summing to k, strict A365663.

A276024 counts positive subset-sums of partitions, strict A284640.

A325799 counts non-subset-sums of prime indices.

A364350 counts combination-free strict partitions.

A365543 counts partitions with a submultiset summing to k, strict A365661.

Cf. A002865, A006827, A018818, A264401, A299701, A304792, A364272, A365658, A365918, A365921.

Sequence in context: A240449 A088571 A325834 * A241832 A027187 A056508

Adjacent sequences: A365921 A365922 A365923 * A365925 A365926 A365927

Keyword

nonn

Author

Gus Wiseman, Sep 26 2023

Status

approved