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%I #7 Nov 06 2019 12:43:02
%S 0,0,0,0,0,0,1,2,6,10,18,28,45,63,93,129,178,238,321,419,551,708,911,
%T 1158,1472,1845,2316,2883,3583,4421,5453,6680,8180,9964,12122,14687,
%U 17771,21418,25788,30949,37092,44324,52906,62980,74885,88832,105243,124429
%N Number of integer partitions of n whose number of nontrivial submultisets is greater than their number of distinct parts times their number of parts minus 1.
%C These partitions are conjectured to be precisely those that have a pair of multiset partitions such that no part of one is a submultiset of any part of the other (see A320632). For example, such a pair of partitions of {1,1,2,2} is ({{1,1},{2,2}}, {{1,2},{1,2}}).
%e The a(6) = 1 through a(10) = 18 partitions:
%e (2211) (3211) (3221) (3321) (3322)
%e (22111) (3311) (4221) (4321)
%e (4211) (4311) (4411)
%e (22211) (5211) (5221)
%e (32111) (32211) (5311)
%e (221111) (33111) (6211)
%e (42111) (32221)
%e (222111) (33211)
%e (321111) (42211)
%e (2211111) (43111)
%e (52111)
%e (222211)
%e (322111)
%e (331111)
%e (421111)
%e (2221111)
%e (3211111)
%e (22111111)
%e For example, the partition (4,2,2,1,1) has 16 nontrivial submultisets: {(1), (2), (4), (11), (21), ..., (2211), (4211), (4221)}, and 5 parts, 3 of which are distinct. Since 16 > (5 - 1) * 3 = 12, the partition (42211) is counted under a(10)
%t Table[Length[Select[IntegerPartitions[n],0<Times@@(1+Length/@Split[#])-2-(Length[#]-1)*Length[Union[#]]&]],{n,0,30}]
%Y The Heinz numbers of these partitions are conjectured to be A320632.
%Y A307409(n) is (omega(n) - 1) * nu(n).
%Y A328958(n) is sigma_0(n) - omega(n) * nu(n).
%Y A328959(n) is sigma_0(n) - 2 - (omega(n) - 1) * nu(n).
%Y Cf. A008284, A032741, A116608, A328956, A328961, A328963.
%K nonn
%O 0,8
%A _Gus Wiseman_, Nov 02 2019