A365918 Number of distinct non-subset-sums of integer partitions of n.

0, 1, 2, 6, 8, 19, 24, 46, 60, 101, 124, 206, 250, 378, 462, 684, 812, 1165, 1380, 1927, 2268, 3108, 3606, 4862, 5648, 7474, 8576, 11307, 12886, 16652, 19050, 24420, 27584, 35225, 39604, 49920, 56370, 70540, 78608, 98419, 109666, 135212, 151176, 185875, 205308

Offset: 1

Author

Gus Wiseman, Sep 23 2023

Keywords

• nonn

Comments

For an integer partition y of n, we call a positive integer k <= n a non-subset-sum iff there is no submultiset of y summing to k.

Examples

			The a(6) = 19 ways, showing each partition and its non-subset-sums:
       (6): 1,2,3,4,5
      (51): 2,3,4
      (42): 1,3,5
     (411): 3
      (33): 1,2,4,5
     (321):
    (3111):
     (222): 1,3,5
    (2211):
   (21111):
  (111111):
		

Crossrefs

Row sums of A046663, strict A365663.

The zero-full complement (subset-sums) is A304792.

The strict case is A365922.

Weighted row-sums of A365923, rank statistic A325799, complement A365658.

A000041 counts integer partitions, strict A000009.

A126796 counts complete partitions, ranks A325781, strict A188431.

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

A365924 counts incomplete partitions, ranks A365830, strict A365831.

Cf. A006827, A122768, A364272, A364350, A364839, A365919, A365921.

Programs

Mathematica

Table[Total[Length[Complement[Range[n],Total/@Subsets[#]]]&/@IntegerPartitions[n]],{n,10}]

Python

# uses A304792_T
from sympy import npartitions
def A365918(n): return (n+1)*npartitions(n)-A304792_T(n,n,(0,),1) # Chai Wah Wu, Sep 25 2023

Formula

a(n) = (n+1)*A000041(n) - A304792(n).

Extensions

a(21)-a(45) from Chai Wah Wu, Sep 25 2023