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

Author

Gus Wiseman, Sep 26 2023

Keywords

• nonn

Comments

The complement (complete partitions) is A126796.

Examples

			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)
		

Links

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

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.

Programs

Mathematica

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

Formula

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