A332295 Number of widely recursively normal integer partitions of n.

1, 1, 2, 3, 4, 6, 6, 10, 12, 17, 21, 30, 34, 48, 54, 74, 86, 113, 132, 169, 200, 246, 293, 360, 422, 512, 599, 726, 840, 1009, 1181, 1401, 1631, 1940, 2240, 2636, 3069, 3567, 4141, 4846, 5556, 6470, 7505, 8627, 9936, 11523, 13176, 15151, 17430, 19935, 22846

Offset: 0

Author

Gus Wiseman, Feb 16 2020

Keywords

• nonn

Comments

A sequence is widely recursively normal if either it is all 1's (wide) or its run-lengths cover an initial interval of positive integers (normal) and are themselves a widely recursively normal sequence.

Examples

			The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (31)    (32)     (42)      (43)       (53)
             (111)  (211)   (41)     (51)      (52)       (62)
                    (1111)  (221)    (321)     (61)       (71)
                            (311)    (411)     (322)      (332)
                            (11111)  (111111)  (331)      (422)
                                               (421)      (431)
                                               (511)      (521)
                                               (3211)     (611)
                                               (1111111)  (3221)
                                                          (4211)
                                                          (11111111)
For example, starting with y = (4,3,2,2,1) and repeatedly taking run-lengths gives (4,3,2,2,1) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1), all of which have normal run-lengths, so y is widely recursively normal. On the other hand, starting with y and repeatedly taking multiplicities gives (4,3,2,2,1) -> (2,1,1,1) -> (3,1), so y is not fully normal (A317491).
Starting with y = (5,4,3,3,2,2,2,1,1) and repeatedly taking run-lengths gives (5,4,3,3,2,2,2,1,1) -> (1,1,2,3,2) -> (2,1,1,1) -> (1,3), so y is not widely recursively normal. On the other hand, starting with y and repeatedly taking multiplicities gives (5,4,3,3,2,2,2,1,1) -> (3,2,2,1,1) -> (2,2,1) -> (2,1) -> (1,1), so y is fully normal (A317491).
		

Crossrefs

The narrow version is A000012.

Partitions with normal multiplicities are A317081.

The Heinz numbers of these partitions are a proper superset of A317492.

Accepting any constant sequence instead of just 1's gives A332272.

The total (instead of recursive) version is A332277.

The case of reversed partitions is this same sequence.

The alternating (instead of recursive) version is this same sequence.

Dominated by A332576.

Cf. A000009, A001462, A181819, A182850, A317245, A317491, A329746, A329747, A332276.

Programs

Mathematica

recnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[Length/@Split[ptn]]==Range[Max[Length/@Split[ptn]]],recnQ[Length/@Split[ptn]]]];
Table[Length[Select[IntegerPartitions[n],recnQ]],{n,0,30}]