cp's OEIS Frontend

A328960 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.

%I A328960 #7 Nov 06 2019 12:43:02
%S A328960 0,0,0,0,0,0,1,2,6,10,18,28,45,63,93,129,178,238,321,419,551,708,911,
%T A328960 1158,1472,1845,2316,2883,3583,4421,5453,6680,8180,9964,12122,14687,
%U A328960 17771,21418,25788,30949,37092,44324,52906,62980,74885,88832,105243,124429
%N A328960 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 A328960 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 A328960 The a(6) = 1 through a(10) = 18 partitions:
%e A328960   (2211)  (3211)   (3221)    (3321)     (3322)
%e A328960           (22111)  (3311)    (4221)     (4321)
%e A328960                    (4211)    (4311)     (4411)
%e A328960                    (22211)   (5211)     (5221)
%e A328960                    (32111)   (32211)    (5311)
%e A328960                    (221111)  (33111)    (6211)
%e A328960                              (42111)    (32221)
%e A328960                              (222111)   (33211)
%e A328960                              (321111)   (42211)
%e A328960                              (2211111)  (43111)
%e A328960                                         (52111)
%e A328960                                         (222211)
%e A328960                                         (322111)
%e A328960                                         (331111)
%e A328960                                         (421111)
%e A328960                                         (2221111)
%e A328960                                         (3211111)
%e A328960                                         (22111111)
%e A328960 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 A328960 Table[Length[Select[IntegerPartitions[n],0<Times@@(1+Length/@Split[#])-2-(Length[#]-1)*Length[Union[#]]&]],{n,0,30}]
%Y A328960 The Heinz numbers of these partitions are conjectured to be A320632.
%Y A328960 A307409(n) is (omega(n) - 1) * nu(n).
%Y A328960 A328958(n) is sigma_0(n) - omega(n) * nu(n).
%Y A328960 A328959(n) is sigma_0(n) - 2 - (omega(n) - 1) * nu(n).
%Y A328960 Cf. A008284, A032741, A116608, A328956, A328961, A328963.
%K A328960 nonn
%O A328960 0,8
%A A328960 _Gus Wiseman_, Nov 02 2019