|
%I #15 Sep 13 2023 22:48:02
%S 0,1,1,1,1,3,3,1,3,7,10,17,27,38,1,4,8,17,31,52,83,122,181,257,361,
%T 499,684,910,1211,1595,2060,2663,3406,4315,5426,6784,8417,10466,12824,
%U 15721,19104,23267,1,5,14,36,76,143,269,446,738,1143,1754,2570,3742,5269
%N Number of integer partitions of n with the maximum adjusted frequency depth for partitions of n.
%C The Heinz numbers of these partitions are given by A325283.
%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). The enumeration of integer partitions by adjusted frequency depth is given by A325280. The adjusted frequency depth of the integer partition with Heinz number n is given by A323014. The maximum adjusted frequency depth for integer partitions of n is given by A325282.
%C Essentially, the last numbers of rows of the array in A225485. - _Clark Kimberling_, Sep 13 2022
%e The a(1) = 1 through a(11) = 17 partitions:
%e 1 11 21 211 221 411 3211 3221 3321 5221 4322
%e 311 3111 4211 4221 5311 4331
%e 2111 21111 32111 4311 6211 4421
%e 5211 32221 5411
%e 32211 33211 6221
%e 42111 42211 6311
%e 321111 43111 7211
%e 52111 33221
%e 421111 42221
%e 3211111 43211
%e 52211
%e 53111
%e 62111
%e 431111
%e 521111
%e 4211111
%e 32111111
%t nn=30;
%t fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
%t mfds=Table[Max@@fdadj/@IntegerPartitions[n],{n,nn}];
%t Table[Length[Select[IntegerPartitions[n],fdadj[#]==mfds[[n]]&]],{n,0,nn}]
%Y Cf. A011784, A181819, A182850, A182857, A225486, A323014, A323023, A325246, A325258, A325278, A325281, A325282, A325283.
%Y Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).
%K nonn
%O 0,6
%A _Gus Wiseman_, Apr 16 2019
|
