A365831 Number of incomplete strict integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.

0, 0, 1, 1, 2, 3, 3, 4, 6, 8, 9, 11, 13, 16, 21, 25, 31, 36, 43, 50, 59, 69, 82, 96, 113, 131, 155, 179, 208, 239, 276, 315, 362, 414, 472, 539, 614, 698, 795, 902, 1023, 1158, 1311, 1479, 1672, 1881, 2118, 2377, 2671, 2991, 3354, 3748, 4194, 4679, 5223, 5815

Offset: 0

Author

Gus Wiseman, Sep 28 2023

Keywords

• nonn

Examples

			The strict partition (14,5,4,2,1) has no subset summing to 13 so is counted under a(26).
The a(2) = 1 through a(10) = 9 strict partitions:
  (2)  (3)  (4)    (5)    (6)    (7)    (8)      (9)      (10)
            (3,1)  (3,2)  (4,2)  (4,3)  (5,3)    (5,4)    (6,4)
                   (4,1)  (5,1)  (5,2)  (6,2)    (6,3)    (7,3)
                                 (6,1)  (7,1)    (7,2)    (8,2)
                                        (4,3,1)  (8,1)    (9,1)
                                        (5,2,1)  (4,3,2)  (5,3,2)
                                                 (5,3,1)  (5,4,1)
                                                 (6,2,1)  (6,3,1)
                                                          (7,2,1)
		

Crossrefs

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

The strict complement is A188431, non-strict A126796 (ranks A325781).

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

The non-strict version is A365924, ranks A365830.

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.

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

Cf. A006827, A047967, A299701, A304792, A364350, A365658, A365918, A365921.

Programs

Mathematica

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