OFFSET
1,2
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.
EXAMPLE
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).
MATHEMATICA
uqsubs[y_]:=Join@@Select[GatherBy[Union[Rest[Subsets[y]]], Total], Length[#]===1&];
Table[Total[Length/@uqsubs/@IntegerPartitions[n]], {n, 25}]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 27 2018
EXTENSIONS
a(21)-a(35) from Alois P. Heinz, Apr 08 2018
a(36)-a(43) from Chai Wah Wu, Sep 26 2023
STATUS
approved