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.

0, 0, 0, 0, 0, 0, 1, 2, 6, 10, 18, 28, 45, 63, 93, 129, 178, 238, 321, 419, 551, 708, 911, 1158, 1472, 1845, 2316, 2883, 3583, 4421, 5453, 6680, 8180, 9964, 12122, 14687, 17771, 21418, 25788, 30949, 37092, 44324, 52906, 62980, 74885, 88832, 105243, 124429

Offset: 0

Author

Gus Wiseman, Nov 02 2019

Keywords

• nonn

Comments

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}}).

Examples

			The a(6) = 1 through a(10) = 18 partitions:
  (2211)  (3211)   (3221)    (3321)     (3322)
          (22111)  (3311)    (4221)     (4321)
                   (4211)    (4311)     (4411)
                   (22211)   (5211)     (5221)
                   (32111)   (32211)    (5311)
                   (221111)  (33111)    (6211)
                             (42111)    (32221)
                             (222111)   (33211)
                             (321111)   (42211)
                             (2211111)  (43111)
                                        (52111)
                                        (222211)
                                        (322111)
                                        (331111)
                                        (421111)
                                        (2221111)
                                        (3211111)
                                        (22111111)
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)
		

Crossrefs

The Heinz numbers of these partitions are conjectured to be A320632.

A307409(n) is (omega(n) - 1) * nu(n).

A328958(n) is sigma_0(n) - omega(n) * nu(n).

A328959(n) is sigma_0(n) - 2 - (omega(n) - 1) * nu(n).

Cf. A008284, A032741, A116608, A328956, A328961, A328963.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],0