A365660 Number of integer partitions of 2n with exactly n distinct sums of nonempty submultisets.

1, 1, 1, 3, 2, 6, 6, 16, 12, 20, 26, 59, 45, 79, 94, 186, 142, 231, 244, 442, 470, 616, 746, 1340, 1053, 1548, 1852, 2780, 2826, 3874, 4320, 6617, 6286, 7924, 9178, 13180, 13634, 17494, 20356, 28220, 29176, 37188, 41932, 56037

Offset: 0

Author

Gus Wiseman, Sep 16 2023

Keywords

• nonn
• more

Comments

Are n = 1, 2, 4 the only n such that none of these partitions has 1?

Are n = 2, 4, 5, 8, 9 the only n such that none of these partitions is strict?

Examples

			The partition (433) has sums 3, 4, 6, 7, 10 so is counted under a(5).
The a(1) = 1 through a(7) = 16 partitions:
(2)  (2,2)  (4,2)    (4,2,2)    (4,3,3)      (6,4,2)        (6,5,3)
            (5,1)    (2,2,2,2)  (4,4,2)      (6,5,1)        (8,4,2)
            (2,2,2)             (6,2,2)      (4,4,2,2)      (8,5,1)
                                (8,1,1)      (6,2,2,2)      (9,3,2)
                                (4,2,2,2)    (4,2,2,2,2)    (9,4,1)
                                (2,2,2,2,2)  (2,2,2,2,2,2)  (10,3,1)
                                                            (11,2,1)
                                                            (4,4,4,2)
                                                            (5,3,3,3)
                                                            (6,4,2,2)
                                                            (8,2,2,2)
                                                            (11,1,1,1)
                                                            (4,4,2,2,2)
                                                            (6,2,2,2,2)
                                                            (4,2,2,2,2,2)
                                                            (2,2,2,2,2,2,2)
		

Crossrefs

For n instead of 2n we have A126796.

Central column n = 2k of A365658.

A000009 counts subsets summing to n.

A000124 counts distinct possible sums of subsets of {1..n}.

A002219 counts partitions of 2n with a submultiset summing to n.

A046663 counts partitions of n w/o a submultiset of sum k, strict A365663.

A122768 counts distinct nonempty submultisets of partitions.

A299701 counts sums of submultisets of prime indices, of partitions A304792.

A364272 counts sum-full strict partitions, sum-free A364349.

A365543 counts partitions of n w/ a submultiset of sum k, strict A365661.

Cf. A000041, A008967, A095944, A364911, A365376, A365377.

Programs

Mathematica

msubs[y_]:=primeMS/@Divisors[Times@@Prime/@y];
Table[Length[Select[IntegerPartitions[2n], Length[Union[Total/@Rest[msubs[#]]]]==n&]],{n,0,10}]

Python

from collections import Counter
from sympy.utilities.iterables import partitions, multiset_combinations
def A365660(n):
    c = 0
    for p in partitions(n<<1):
        q, s = list(Counter(p).elements()), set()
        for l in range(1,len(q)+1):
            for k in multiset_combinations(q,l):
                s.add(sum(k))
                if len(s) > n:
                    break
            else:
                continue
            break
        if len(s)==n:
            c += 1
    return c # Chai Wah Wu, Sep 20 2023

Extensions

a(21)-a(38) from Chai Wah Wu, Sep 20 2023

a(39)-a(43) from Chai Wah Wu, Sep 21 2023