A325253 Number of integer partitions of n with adjusted frequency depth ceiling(sqrt(n)).

1, 1, 1, 1, 2, 2, 4, 4, 6, 8, 17, 26, 25, 44, 53, 63, 83, 128, 168, 212, 273, 344, 429, 525, 662, 796, 684, 910, 1211, 1595, 2060, 2663, 3406, 4315, 5426, 6784, 8417, 0, 0, 0, 0, 0, 1, 5, 14, 36, 76, 143, 269, 446, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,5

Comments

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is one 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..60.

Example

The a(2) = 1 through a(11) = 26 partitions:
    11  111  22    32  42    43   53    54      433        443
             1111  41  51    52   62    63      442        533
                       321   61   71    72      622        551
                       2211  421  431   81      811        722
                                  521   432     3331       911
                                  3311  531     4222       3332
                                        621     7111       5222
                                        222111  61111      8111
                                                222211     32222
                                                322111     33311
                                                331111     44111
                                                511111     71111
                                                2221111    222221
                                                4111111    322211
                                                22111111   332111
                                                31111111   422111
                                                211111111  611111
                                                           2222111
                                                           3221111
                                                           3311111
                                                           5111111
                                                           22211111
                                                           41111111
                                                           221111111
                                                           311111111
                                                           2111111111

Mathematica

fdadj[ptn_List]:=If[ptn=={}, 0, Length[NestWhileList[Sort[Length/@Split[#1]]&, ptn, Length[#1]>1&]]];
Table[Length[Select[IntegerPartitions[n], fdadj[#]==Ceiling[Sqrt[n]]&]], {n, 0, 30}]

Crossrefs

Cf. A117571, A181819, A225485, A323014, A323023, A325245, A325246, A325252, A325258, A325271, A325278, A325280, A325282.

Sequence in context: A029940 A045674 A276065 * A143483 A323093 A264788

Adjacent sequences: A325250 A325251 A325252 * A325254 A325255 A325256

Keyword

nonn

Author

Gus Wiseman, Apr 22 2019

Status

approved