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%I #4 May 02 2019 08:52:47
%S 1,1,2,3,4,4,8,7,13,18,25,30,52,57,81,109,140,167,230,267,354,428,532,
%T 630,815,942,1166,1385,1695,1966,2440,2810,3422,4008,4828,5630,6847,
%U 7905,9527,11135,13340,15498,18636,21591,25769,30086,35630,41379,49150,56880
%N Number of integer partitions of n whose multiplicities appear with distinct multiplicities.
%C The Heinz numbers of these partitions are given by A325369.
%C Partitions whose parts appear with distinct multiplicities are counted by A098859, with Heinz numbers A130091.
%e The a(0) = 1 through a(8) = 13 partitions:
%e () (1) (2) (3) (4) (5) (6) (7) (8)
%e (11) (21) (22) (32) (33) (43) (44)
%e (111) (31) (41) (42) (52) (53)
%e (1111) (11111) (51) (61) (62)
%e (222) (421) (71)
%e (321) (3211) (431)
%e (2211) (1111111) (521)
%e (111111) (2222)
%e (3221)
%e (3311)
%e (4211)
%e (32111)
%e (11111111)
%e For example, in (4,2,1,1), the multiplicities are 1 and 2, and 2 appears 1 time while 1 appears 2 times, so (4,2,1,1) is counted under a(8).
%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@Length/@Split[Sort[Length/@Split[#]]]&]],{n,0,30}]
%Y Cf. A098859, A130091, A317081, A320348, A325326, A325330, A325331, A325333, A325337, A325369.
%K nonn
%O 0,3
%A _Gus Wiseman_, May 01 2019