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%I #5 Apr 23 2019 09:20:40
%S 1,1,1,1,2,1,3,1,3,8,11,11,19,17,25,29,83,113,124,171,190,242,289,368,
%T 399,796,981,1182,1442,1709,2096,2469,2990,3545,4276,5037,8417,10466,
%U 12824,15721,19104,23267,27981,33856,40515,48508,57826,68982,81493,446,738
%N Number of integer partitions of n with frequency depth floor(sqrt(n)).
%C The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).
%e The a(2) = 1 through a(12) = 19 partitions (A = 10, B = 11):
%e 2 3 22 11111 33 1111111 44 54 64 65 75
%e 1111 222 2222 63 73 74 84
%e 111111 11111111 72 82 83 93
%e 81 91 92 A2
%e 432 532 A1 B1
%e 531 541 542 543
%e 621 631 632 642
%e 222111 721 641 651
%e 3322 731 732
%e 4321 821 741
%e 4411 5321 831
%e 921
%e 4422
%e 5421
%e 5511
%e 6321
%e 332211
%e 333111
%e 22221111
%t fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#1]]&,ptn,Length[#1]>1&]]];
%t Table[Length[Select[IntegerPartitions[n],fdadj[#]==Floor[Sqrt[n]]&]],{n,0,30}]
%Y Cf. A117571, A181819, A225485, A323014, A323023, A325245, A325246, A325253, A325258, A325271, A325278, A325280, A325282.
%K nonn
%O 0,5
%A _Gus Wiseman_, Apr 22 2019
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