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

0,8

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}}).

Links

Table of n, a(n) for n=0..47.

Example

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)

Mathematica

Table[Length[Select[IntegerPartitions[n], 0<Times@@(1+Length/@Split[#])-2-(Length[#]-1)*Length[Union[#]]&]], {n, 0, 30}]

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.

Sequence in context: A290220 A320678 A050844 * A282428 A174316 A095357

Adjacent sequences: A328957 A328958 A328959 * A328961 A328962 A328963

Keyword

nonn

Author

Gus Wiseman, Nov 02 2019

Status

approved