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

1,3

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.

Links

Table of n, a(n) for n=1..45.

Formula

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

Example

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):

Mathematica

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

Prog

(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

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.

Sequence in context: A077638 A103015 A323859 * A098092 A221316 A106164

Adjacent sequences: A365915 A365916 A365917 * A365919 A365920 A365921

Keyword

nonn

Author

Gus Wiseman, Sep 23 2023

Extensions

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

Status

approved