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Number of integer partitions of n with frequency depth floor(sqrt(n)).
1

%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