A325253 -id:A325253 - cp's OEIS frontend

A325271 Number of integer partitions of n with frequency depth round(sqrt(n)).

1, 1, 1, 1, 2, 1, 3, 4, 6, 8, 11, 11, 19, 44, 53, 63, 83, 113, 124, 171, 190, 344, 429, 525, 662, 796, 981, 1182, 1442, 1709, 2096, 2663, 3406, 4315, 5426, 6784, 8417, 10466, 12824, 15721, 19104, 23267, 27981, 5, 14, 36, 76, 143, 269, 446, 738, 1143, 1754

Offset: 0

Gus Wiseman, Apr 23 2019

nonn

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 a(2) = 1 through a(10) = 11 partitions:
  (2)  (111)  (22)    (11111)  (33)      (43)   (53)    (54)      (64)
              (1111)           (222)     (52)   (62)    (63)      (73)
                               (111111)  (61)   (71)    (72)      (82)
                                         (421)  (431)   (81)      (91)
                                                (521)   (432)     (532)
                                                (3311)  (531)     (541)
                                                        (621)     (631)
                                                        (222111)  (721)
                                                                  (3322)
                                                                  (4321)
                                                                  (4411)

Cf. A117571, A181819, A225485, A323014, A323023, A325245, A325246, A325272, A325253, A325258, A325278, A325280, A325282.

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

A325252 Number of integer partitions of n with frequency depth floor(sqrt(n)).

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Apr 22 2019

nonn

			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

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

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}]

cp's OEIS Frontend

A325271 Number of integer partitions of n with frequency depth round(sqrt(n)).

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