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%I #8 May 02 2019 08:53:08
%S 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 277,301,339,382,430,461,551,579,681,762,896,1010,1255,1406,1752,2061,
%U 2555,3001,3783,4437,5512,6611,8056,9539,11668,13692,16515,19435,23098
%N Number of integer partitions of n whose multiplicities appear with distinct multiplicities that cover an initial interval of positive integers.
%C 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 The Heinz numbers of these partitions are given by A325371.
%e The a(0) = 1 through a(8) = 7 partitions:
%e () (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (111) (22) (11111) (33) (3211) (44)
%e (1111) (222) (1111111) (2222)
%e (111111) (3221)
%e (4211)
%e (32111)
%e (11111111)
%e 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 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t Table[Length[Select[IntegerPartitions[n],normQ[Length/@Split[Sort[Length/@Split[#]]]]&&UnsameQ@@Length/@Split[Sort[Length/@Split[#]]]&]],{n,0,30}]
%Y Cf. A098859, A130091, A317081, A317090, A320348, A325329, A325330, A325337, A325369, A325370, A325371.
%K nonn
%O 0,3
%A _Gus Wiseman_, May 01 2019