cp's OEIS Frontend

A325254 Number of integer partitions of n with the maximum adjusted frequency depth for partitions of n.

%I A325254 #15 Sep 13 2023 22:48:02
%S A325254 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 A325254 499,684,910,1211,1595,2060,2663,3406,4315,5426,6784,8417,10466,12824,
%U A325254 15721,19104,23267,1,5,14,36,76,143,269,446,738,1143,1754,2570,3742,5269
%N A325254 Number of integer partitions of n with the maximum adjusted frequency depth for partitions of n.
%C A325254 The Heinz numbers of these partitions are given by A325283.
%C A325254 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 A325254 Essentially, the last numbers of rows of the array in A225485. - _Clark Kimberling_, Sep 13 2022
%e A325254 The a(1) = 1 through a(11) = 17 partitions:
%e A325254   1  11  21  211  221   411    3211  3221   3321    5221     4322
%e A325254                   311   3111         4211   4221    5311     4331
%e A325254                   2111  21111        32111  4311    6211     4421
%e A325254                                             5211    32221    5411
%e A325254                                             32211   33211    6221
%e A325254                                             42111   42211    6311
%e A325254                                             321111  43111    7211
%e A325254                                                     52111    33221
%e A325254                                                     421111   42221
%e A325254                                                     3211111  43211
%e A325254                                                              52211
%e A325254                                                              53111
%e A325254                                                              62111
%e A325254                                                              431111
%e A325254                                                              521111
%e A325254                                                              4211111
%e A325254                                                              32111111
%t A325254 nn=30;
%t A325254 fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
%t A325254 mfds=Table[Max@@fdadj/@IntegerPartitions[n],{n,nn}];
%t A325254 Table[Length[Select[IntegerPartitions[n],fdadj[#]==mfds[[n]]&]],{n,0,nn}]
%Y A325254 Cf. A011784, A181819, A182850, A182857, A225486, A323014, A323023, A325246, A325258, A325278, A325281, A325282, A325283.
%Y A325254 Integer partition triangles: A008284 (first omega), A116608 (second omega), A325242 (third omega), A325268 (second-to-last omega), A225485 or A325280 (length/frequency depth).
%K A325254 nonn
%O A325254 0,6
%A A325254 _Gus Wiseman_, Apr 16 2019