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

%I A365660 #17 Sep 21 2023 11:11:01
%S A365660 1,1,1,3,2,6,6,16,12,20,26,59,45,79,94,186,142,231,244,442,470,616,
%T A365660 746,1340,1053,1548,1852,2780,2826,3874,4320,6617,6286,7924,9178,
%U A365660 13180,13634,17494,20356,28220,29176,37188,41932,56037
%N A365660 Number of integer partitions of 2n with exactly n distinct sums of nonempty submultisets.
%C A365660 Are n = 1, 2, 4 the only n such that none of these partitions has 1?
%C A365660 Are n = 2, 4, 5, 8, 9 the only n such that none of these partitions is strict?
%e A365660 The partition (433) has sums 3, 4, 6, 7, 10 so is counted under a(5).
%e A365660 The a(1) = 1 through a(7) = 16 partitions:
%e A365660 (2)  (2,2)  (4,2)    (4,2,2)    (4,3,3)      (6,4,2)        (6,5,3)
%e A365660             (5,1)    (2,2,2,2)  (4,4,2)      (6,5,1)        (8,4,2)
%e A365660             (2,2,2)             (6,2,2)      (4,4,2,2)      (8,5,1)
%e A365660                                 (8,1,1)      (6,2,2,2)      (9,3,2)
%e A365660                                 (4,2,2,2)    (4,2,2,2,2)    (9,4,1)
%e A365660                                 (2,2,2,2,2)  (2,2,2,2,2,2)  (10,3,1)
%e A365660                                                             (11,2,1)
%e A365660                                                             (4,4,4,2)
%e A365660                                                             (5,3,3,3)
%e A365660                                                             (6,4,2,2)
%e A365660                                                             (8,2,2,2)
%e A365660                                                             (11,1,1,1)
%e A365660                                                             (4,4,2,2,2)
%e A365660                                                             (6,2,2,2,2)
%e A365660                                                             (4,2,2,2,2,2)
%e A365660                                                             (2,2,2,2,2,2,2)
%t A365660 msubs[y_]:=primeMS/@Divisors[Times@@Prime/@y];
%t A365660 Table[Length[Select[IntegerPartitions[2n], Length[Union[Total/@Rest[msubs[#]]]]==n&]],{n,0,10}]
%o A365660 (Python)
%o A365660 from collections import Counter
%o A365660 from sympy.utilities.iterables import partitions, multiset_combinations
%o A365660 def A365660(n):
%o A365660     c = 0
%o A365660     for p in partitions(n<<1):
%o A365660         q, s = list(Counter(p).elements()), set()
%o A365660         for l in range(1,len(q)+1):
%o A365660             for k in multiset_combinations(q,l):
%o A365660                 s.add(sum(k))
%o A365660                 if len(s) > n:
%o A365660                     break
%o A365660             else:
%o A365660                 continue
%o A365660             break
%o A365660         if len(s)==n:
%o A365660             c += 1
%o A365660     return c # _Chai Wah Wu_, Sep 20 2023
%Y A365660 For n instead of 2n we have A126796.
%Y A365660 Central column n = 2k of A365658.
%Y A365660 A000009 counts subsets summing to n.
%Y A365660 A000124 counts distinct possible sums of subsets of {1..n}.
%Y A365660 A002219 counts partitions of 2n with a submultiset summing to n.
%Y A365660 A046663 counts partitions of n w/o a submultiset of sum k, strict A365663.
%Y A365660 A122768 counts distinct nonempty submultisets of partitions.
%Y A365660 A299701 counts sums of submultisets of prime indices, of partitions A304792.
%Y A365660 A364272 counts sum-full strict partitions, sum-free A364349.
%Y A365660 A365543 counts partitions of n w/ a submultiset of sum k, strict A365661.
%Y A365660 Cf. A000041, A008967, A095944, A364911, A365376, A365377.
%K A365660 nonn,more
%O A365660 0,4
%A A365660 _Gus Wiseman_, Sep 16 2023
%E A365660 a(21)-a(38) from _Chai Wah Wu_, Sep 20 2023
%E A365660 a(39)-a(43) from _Chai Wah Wu_, Sep 21 2023