A332576 -id:A332576 - cp's OEIS frontend

A332278 Number of widely totally co-strongly normal integer partitions of n.

1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2

Offset: 0

Gus Wiseman, Mar 05 2020

A sequence of integers is widely totally co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) with weakly increasing run-lengths (co-strong) which are themselves a widely totally co-strongly normal sequence.

Is this sequence bounded?

			The a(1) = 1 through a(20) = 2 partitions:
   1: (1)
   2: (11)
   3: (21),(111)
   4: (211),(1111)
   5: (11111)
   6: (321),(111111)
   7: (1111111)
   8: (11111111)
   9: (32211),(111111111)
  10: (4321),(322111),(1111111111)
  11: (11111111111)
  12: (111111111111)
  13: (1111111111111)
  14: (11111111111111)
  15: (54321),(111111111111111)
  16: (1111111111111111)
  17: (11111111111111111)
  18: (111111111111111111)
  19: (1111111111111111111)
  20: (4332221111),(11111111111111111111)

Not requiring co-strength gives A332277.
The strong version is A332297(n) - 1 for n > 1.
The narrow version is a(n) - 1 for n > 1.
The alternating version is A332289.
The Heinz numbers of these partitions are A332293.
The case of compositions is A332337.
Cf. A000009, A100883, A107429, A133808, A181819, A316496, A317245, A317491, A329746, A332279, A332290, A332291, A332292, A332296, A332576.

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

a(71)-a(78) from Jinyuan Wang, Jun 26 2020

A332295 Number of widely recursively normal integer partitions of n.

Original entry on oeis.org

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

Gus Wiseman, Feb 16 2020

nonn

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.

			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).

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.

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}]

A332272 Number of narrowly recursively normal integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 8, 10, 14, 18, 23, 30, 37, 46, 52, 70, 80, 100, 116, 146, 171, 203, 236, 290, 332, 401, 458, 547, 626, 744, 851, 1004, 1157, 1353, 1553, 1821, 2110, 2434, 2810, 3250, 3741, 4304, 4949, 5661, 6510, 7450, 8501, 9657, 11078, 12506, 14329, 16185

Offset: 0

Gus Wiseman, Mar 08 2020

nonn

A sequence is narrowly recursively normal if either it is constant (narrow) or its run-lengths are a narrowly recursively normal sequence covering an initial interval of positive integers (normal).

			The a(6) = 8 partitions are (6), (51), (42), (411), (33), (321), (222), (111111). Missing from this list are (3111), (2211), (21111).
The a(1) = 1 through a(8) = 14 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (211)   (221)    (51)      (61)       (62)
                    (1111)  (311)    (222)     (322)      (71)
                            (11111)  (321)     (331)      (332)
                                     (411)     (421)      (422)
                                     (111111)  (511)      (431)
                                               (3211)     (521)
                                               (1111111)  (611)
                                                          (2222)
                                                          (3221)
                                                          (4211)
                                                          (11111111)

The strict instead of narrow version is A330937.
The normal case is A332277.
The widely normal case is A332277(n) - 1 for n > 1.
The wide version is A332295(n) - 1.
Cf. A000009, A107429, A181819, A316496, A317081, A317245, A317491, A329744, A329746, A329766, A332576.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
recnQ[ptn_]:=With[{qtn=Length/@Split[ptn]},Or[Length[qtn]<=1,And[normQ[qtn],recnQ[qtn]]]];
Table[Length[Select[IntegerPartitions[n],recnQ]],{n,0,30}]

For n > 1, a(n) = A317491(n) + A000005(n) - 2.

A330937 Number of strictly recursively normal integer partitions of n.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 15, 20, 27, 35, 49, 58, 81, 100, 126, 160, 206, 246, 316, 374, 462, 564, 696, 813, 1006, 1195, 1441, 1701, 2058, 2394, 2896, 3367, 4007, 4670, 5542, 6368, 7540, 8702, 10199, 11734, 13760, 15734, 18384, 21008, 24441, 27893, 32380, 36841

Offset: 0

Gus Wiseman, Mar 09 2020

nonn

A sequence is strictly recursively normal if either it empty, its run-lengths are distinct (strict), or its run-lengths cover an initial interval of positive integers (normal) and are themselves a strictly recursively normal sequence.

			The a(1) = 1 through a(9) = 15 partitions:
  (1)  (2)  (3)   (4)    (5)    (6)    (7)     (8)     (9)
            (21)  (31)   (32)   (42)   (43)    (53)    (54)
                  (211)  (41)   (51)   (52)    (62)    (63)
                         (221)  (321)  (61)    (71)    (72)
                         (311)  (411)  (322)   (332)   (81)
                                       (331)   (422)   (432)
                                       (421)   (431)   (441)
                                       (511)   (521)   (522)
                                       (3211)  (611)   (531)
                                               (3221)  (621)
                                               (4211)  (711)
                                                       (3321)
                                                       (4221)
                                                       (4311)
                                                       (5211)
                                                       (32211)

The narrow instead of strict version is A332272.
A wide instead of strict version is A332295(n) - 1 for n > 1.
Cf. A107429, A181819, A316496, A317081, A317245, A317491, A329744, A329746, A329766, A332277, A332576.

normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
recnQ[ptn_]:=With[{qtn=Length/@Split[ptn]},Or[ptn=={},UnsameQ@@qtn,And[normQ[qtn],recnQ[qtn]]]];
Table[Length[Select[IntegerPartitions[n],recnQ]],{n,0,30}]

cp's OEIS Frontend

A332278 Number of widely totally co-strongly normal integer partitions of n.

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A332295 Number of widely recursively normal integer partitions of n.

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A332272 Number of narrowly recursively normal integer partitions of n.

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A330937 Number of strictly recursively normal integer partitions of n.

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