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%I #5 May 18 2019 22:46:44
%S 1,1,0,1,1,1,2,1,2,2,4,4,5,6,7,8,10,11,13,16,18,23,26,32,36,43,48,57,
%T 64,74,84,98,110,127,145,165,189,215,244,277,316,356,403,455,513,577,
%U 650,727,817,913,1024,1143,1279,1425,1592,1773,1977,2198,2448,2717
%N Number of integer partitions of n covering an initial interval of positive integers and containing all of their distinct multiplicities.
%C The Heinz numbers of these partitions are given by A325708.
%e The initial terms count the following partitions:
%e 1: (1)
%e 3: (21)
%e 4: (211)
%e 5: (221)
%e 6: (321)
%e 6: (2211)
%e 7: (3211)
%e 8: (3221)
%e 8: (32111)
%e 9: (3321)
%e 9: (32211)
%e 10: (4321)
%e 10: (33211)
%e 10: (32221)
%e 10: (322111)
%e 11: (43211)
%e 11: (33221)
%e 11: (332111)
%e 11: (322211)
%e 12: (43221)
%e 12: (432111)
%e 12: (33321)
%e 12: (332211)
%e 12: (3222111)
%t Table[Length[Select[IntegerPartitions[n],Range[Length[Union[#]]]==Union[#]&&SubsetQ[Sort[#],Sort[Length/@Split[#]]]&]],{n,0,30}]
%Y Cf. A000009 (partitions covering an initial interval), A055932, A109297, A114639, A114640, A290689, A324753, A325702, A325706, A325708.
%K nonn
%O 0,7
%A _Gus Wiseman_, May 18 2019
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