cp's OEIS Frontend

A325331 Number of integer partitions of n whose multiplicities appear with distinct multiplicities that cover an initial interval of positive integers.

%I A325331 #8 May 02 2019 08:53:08
%S A325331 1,1,2,2,3,2,4,3,7,10,14,18,30,34,44,65,73,88,110,127,155,183,202,231,
%T A325331 277,301,339,382,430,461,551,579,681,762,896,1010,1255,1406,1752,2061,
%U A325331 2555,3001,3783,4437,5512,6611,8056,9539,11668,13692,16515,19435,23098
%N A325331 Number of integer partitions of n whose multiplicities appear with distinct multiplicities that cover an initial interval of positive integers.
%C A325331 Partitions with distinct multiplicities that cover an initial interval of positive integers are counted by A320348, with Heinz numbers A325337. Partitions whose multiplicities appear with distinct multiplicities are counted by A325329, with Heinz numbers A325369. Partitions whose multiplicities appear with multiplicities that cover an initial interval of positive integers of counted by A325330, with Heinz numbers A325370.
%C A325331 The Heinz numbers of these partitions are given by A325371.
%e A325331 The a(0) = 1 through a(8) = 7 partitions:
%e A325331   ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
%e A325331            (11)  (111)  (22)    (11111)  (33)      (3211)     (44)
%e A325331                         (1111)           (222)     (1111111)  (2222)
%e A325331                                          (111111)             (3221)
%e A325331                                                               (4211)
%e A325331                                                               (32111)
%e A325331                                                               (11111111)
%e A325331 For example, the partition p = (5,5,4,3,3,3,2,2) has multiplicities (2,3,1,2), which appear with multiplicities (1,2,1), which cover an initial interval but are not distinct, so p is not counted under a(27). The partition q = (5,5,5,4,4,4,3,3,2,2,1,1) has multiplicities (3,3,2,2,2), which appear with multiplicities (3,2), which are distinct but do not cover an initial interval, so q is not counted under a(39). The partition r = (3,3,2,1,1) has multiplicities (2,1,2), which appear with multiplicities (1,2), which are distinct and cover an initial interval, so r is counted under a(10).
%t A325331 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A325331 Table[Length[Select[IntegerPartitions[n],normQ[Length/@Split[Sort[Length/@Split[#]]]]&&UnsameQ@@Length/@Split[Sort[Length/@Split[#]]]&]],{n,0,30}]
%Y A325331 Cf. A098859, A130091, A317081, A317090, A320348, A325329, A325330, A325337, A325369, A325370, A325371.
%K A325331 nonn
%O A325331 0,3
%A A325331 _Gus Wiseman_, May 01 2019