A301854 Number of positive special sums of integer partitions of n.

1, 3, 7, 13, 25, 40, 67, 100, 158, 220, 336, 452, 649, 862, 1228, 1553, 2155, 2738, 3674, 4612, 6124, 7497, 9857, 12118, 15524, 18821, 24152, 28863, 36549, 44002, 54576, 65125, 80943, 95470, 117991, 139382, 169389, 199144, 242925, 283353, 342139, 400701, 479001

Offset: 1

Author

Gus Wiseman, Mar 27 2018

Keywords

• nonn

Comments

A positive special sum of an integer partition y is a number n > 0 such that exactly one submultiset of y sums to n.

Examples

			The a(4) = 13 special positive subset-sums:
1<=(1111), 2<=(1111), 3<=(1111), 4<=(1111),
1<=(211),  3<=(211),  4<=(211),
1<=(31),   3<=(31),   4<=(31),
2<=(22),   4<=(22),
4<=(4).
		

Crossrefs

Cf. A000712, A108917, A122768, A275972, A276024, A284640, A299701, A299702, A299729, A301829, A301830, A301854, A301855, A301856.

Programs

Mathematica

uqsubs[y_]:=Join@@Select[GatherBy[Union[Rest[Subsets[y]]],Total],Length[#]===1&];
Table[Total[Length/@uqsubs/@IntegerPartitions[n]],{n,25}]

Python

from collections import Counter
from sympy.utilities.iterables import partitions, multiset_combinations
def A301854(n): return sum(sum(1 for r in Counter(sum(q) for l in range(1,len(p)+1) for q in multiset_combinations(p,l)).values() if r==1) for p in (tuple(Counter(x).elements()) for x in partitions(n))) # Chai Wah Wu, Sep 26 2023

Extensions

a(21)-a(35) from Alois P. Heinz, Apr 08 2018

a(36)-a(43) from Chai Wah Wu, Sep 26 2023