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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

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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.

LINKS

Table of n, a(n) for n=1..43.

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

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

Sequence in context: A360783 A026103 A328652 * A092463 A259343 A210613

Adjacent sequences: A301851 A301852 A301853 * A301855 A301856 A301857

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