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A325252 Number of integer partitions of n with frequency depth floor(sqrt(n)). 1
1, 1, 1, 1, 2, 1, 3, 1, 3, 8, 11, 11, 19, 17, 25, 29, 83, 113, 124, 171, 190, 242, 289, 368, 399, 796, 981, 1182, 1442, 1709, 2096, 2469, 2990, 3545, 4276, 5037, 8417, 10466, 12824, 15721, 19104, 23267, 27981, 33856, 40515, 48508, 57826, 68982, 81493, 446, 738 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

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).

LINKS

Table of n, a(n) for n=0..50.

EXAMPLE

The a(2) = 1 through a(12) = 19 partitions (A = 10, B = 11):

  2  3  22    11111  33      1111111  44        54      64    65    75

        1111         222              2222      63      73    74    84

                     111111           11111111  72      82    83    93

                                                81      91    92    A2

                                                432     532   A1    B1

                                                531     541   542   543

                                                621     631   632   642

                                                222111  721   641   651

                                                        3322  731   732

                                                        4321  821   741

                                                        4411  5321  831

                                                                    921

                                                                    4422

                                                                    5421

                                                                    5511

                                                                    6321

                                                                    332211

                                                                    333111

                                                                    22221111

MATHEMATICA

fdadj[ptn_List]:=If[ptn=={}, 0, Length[NestWhileList[Sort[Length/@Split[#1]]&, ptn, Length[#1]>1&]]];

Table[Length[Select[IntegerPartitions[n], fdadj[#]==Floor[Sqrt[n]]&]], {n, 0, 30}]

CROSSREFS

Cf. A117571, A181819, A225485, A323014, A323023, A325245, A325246, A325253, A325258, A325271, A325278, A325280, A325282.

Sequence in context: A308673 A324287 A213594 * A243334 A234809 A135591

Adjacent sequences:  A325249 A325250 A325251 * A325253 A325254 A325255

KEYWORD

nonn

AUTHOR

Gus Wiseman, Apr 22 2019

STATUS

approved

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Last modified January 19 15:37 EST 2020. Contains 331049 sequences. (Running on oeis4.)