%I #15 Mar 10 2019 21:00:47
%S 1,1,0,0,2,0,1,0,1,1,2,0,3,0,2,0,4,1,2,1,4,1,3,1,5,3,5,1,6,2,6,1,7,2,
%T 7,2,11,4,8,3,11,5,10,4,13,5,11,5,16,8,14,5,19,8,18,6,22,8,22,7,26,10,
%U 25,8,33,12,29,11,36,13,34,12,40,16,41,14,47,17,45,16,55
%N Number of integer partitions of n whose multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are equal to the distinct parts in decreasing order.
%C These are a kind of self-describing partitions (cf. A001462, A304679).
%C The Heinz numbers of these partitions are given by A324571.
%C The case where the distinct parts are taken in increasing order is counted by A033461, with Heinz numbers given by A109298.
%e The first 19 terms count the following integer partitions:
%e 1: (1)
%e 4: (22)
%e 4: (211)
%e 6: (3111)
%e 8: (41111)
%e 9: (333)
%e 10: (511111)
%e 10: (322111)
%e 12: (6111111)
%e 12: (4221111)
%e 12: (33222)
%e 14: (71111111)
%e 14: (52211111)
%e 16: (811111111)
%e 16: (622111111)
%e 16: (4444)
%e 16: (442222)
%e 17: (43331111)
%e 18: (9111111111)
%e 18: (7221111111)
%e 19: (533311111)
%t Table[Length[Select[IntegerPartitions[n],Union[#]==Length/@Split[#]&]],{n,0,30}]
%Y Cf. A001156, A033461, A109298, A117144, A276078, A324524, A324571.
%Y Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.
%K nonn
%O 0,5
%A _Gus Wiseman_, Mar 08 2019
%E More terms from _Alois P. Heinz_, Mar 08 2019