A325244 - OEIS

%I #5 Apr 16 2019 15:26:49

%S 0,0,0,1,1,2,3,4,7,12,16,21,33,38,50,75,87,111,150,185,244,307,373,

%T 461,585,702,856,1043,1255,1498,1822,2143,2565,3064,3607,4251,5064,

%U 5920,6953,8174,9503,11064,12927,14921,17320,19986,23067,26485,30499,34894

%N Number of integer partitions of n with one fewer distinct multiplicities than distinct parts.

%C For example, (32211) has two distinct multiplicities (1, 2) and three distinct parts (1, 2, 3) so is counted under a(9).

%C The Heinz numbers of these partitions are given by A325259.

%e The a(3) = 1 through a(10) = 16 partitions:

%e   (21)  (31)  (32)  (42)    (43)    (53)     (54)      (64)

%e               (41)  (51)    (52)    (62)     (63)      (73)

%e                     (2211)  (61)    (71)     (72)      (82)

%e                             (3211)  (3221)   (81)      (91)

%e                                     (3311)   (3321)    (3322)

%e                                     (4211)   (4221)    (4411)

%e                                     (32111)  (4311)    (5221)

%e                                              (5211)    (5311)

%e                                              (32211)   (6211)

%e                                              (42111)   (32221)

%e                                              (222111)  (33211)

%e                                              (321111)  (42211)

%e                                                        (43111)

%e                                                        (52111)

%e                                                        (421111)

%e                                                        (3211111)

%t Table[Length[Select[IntegerPartitions[n],Length[Union[#]]==Length[Union[Length/@Split[#]]]+1&]],{n,0,30}]

%Y Cf. A008284, A071625, A090858, A098859, A116608, A117571, A127002, A325242, A325259, A325270.

%K nonn

%O 0,6

%A _Gus Wiseman_, Apr 15 2019