A301854 - OEIS

%I #14 Sep 26 2023 14:58:54

%S 1,3,7,13,25,40,67,100,158,220,336,452,649,862,1228,1553,2155,2738,

%T 3674,4612,6124,7497,9857,12118,15524,18821,24152,28863,36549,44002,

%U 54576,65125,80943,95470,117991,139382,169389,199144,242925,283353,342139,400701,479001

%N Number of positive special sums of integer partitions of n.

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

%e The a(4) = 13 special positive subset-sums:

%e 1<=(1111), 2<=(1111), 3<=(1111), 4<=(1111),

%e 1<=(211),  3<=(211),  4<=(211),

%e 1<=(31),   3<=(31),   4<=(31),

%e 2<=(22),   4<=(22),

%e 4<=(4).

%t uqsubs[y_]:=Join@@Select[GatherBy[Union[Rest[Subsets[y]]],Total],Length[#]===1&];

%t Table[Total[Length/@uqsubs/@IntegerPartitions[n]],{n,25}]

%o (Python)

%o from collections import Counter

%o from sympy.utilities.iterables import partitions, multiset_combinations

%o 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

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Mar 27 2018

%E a(21)-a(35) from _Alois P. Heinz_, Apr 08 2018

%E a(36)-a(43) from _Chai Wah Wu_, Sep 26 2023