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Number of integer partitions of n covering an initial interval of positive integers and containing all of their distinct multiplicities.
4

%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