A332295 -id:A332295 - cp's OEIS frontend

A332277 Number of widely totally normal integer partitions of n.

1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 4, 4, 2, 4, 4, 6, 3, 5, 7, 6, 8, 12, 9, 12, 13, 11, 12, 18, 17, 12, 32, 19, 25, 33, 30, 28, 44, 33, 43, 57, 51, 60, 83, 70, 83, 103, 96, 97, 125, 117, 134, 157, 157, 171, 226, 215, 238, 278, 302, 312, 359, 357, 396, 450, 444, 477, 580

Offset: 0

Gus Wiseman, Feb 12 2020

nonn

A sequence is widely totally normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has widely totally normal run-lengths.

Also the number of widely totally normal reversed integer partitions of n.

			The a(n) partitions for n = 1, 4, 10, 11, 16, 18:
  1  211   4321        33221        443221            543321
     1111  33211       322211       4432111           4333221
           322111      332111       1111111111111111  4432221
           1111111111  11111111111                    4433211
                                                      43322211
                                                      44322111
                                                      111111111111111111

Normal partitions are A000009.
Taking multiplicities instead of run-lengths gives A317245.
Constantly recursively normal partitions are A332272.
The Heinz numbers of these partitions are A332276.
The case of all compositions (not just partitions) is A332279.
The co-strong version is A332278.
The recursive version is A332295.
The narrow version is a(n) + 1 for n > 1.
Cf. A181819, A316496, A317081, A317256, A317491, A317588, A329746, A329747, A332289, A332290, A332291, A332296, A332297, A332336, A332337, A332340.

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

a(61)-a(66) from Jinyuan Wang, Jun 26 2020

A332289 Number of widely alternately co-strongly normal integer partitions of n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Feb 13 2020

nonn

An integer partition is widely alternately co-strongly normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly increasing run-lengths (co-strong) which, if reversed, are themselves a widely alternately co-strongly normal partition.

			The a(1) = 1, a(3) = 2, and a(10) = 3 partitions:
  (1)  (21)   (4321)
       (111)  (322111)
              (1111111111)
For example, starting with y = (4,3,2,2,1,1,1) and repeatedly taking run-lengths and reversing gives y -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1). These are all normal, have weakly increasing run-lengths, and the last is all 1's, so y is counted a(14).

Normal partitions are A000009.
Dominated by A317245.
The non-co-strong version is A332277.
The total (instead of alternate) version is A332278.
The Heinz numbers of these partitions are A332290.
The strong version is A332292.
The case of reversed partitions is (also) A332292.
The generalization to compositions is A332340.
Cf. A100883, A107429, A133808, A316496, A317081, A317256, A317491, A329746, A332291, A332295, A332297, A332337, A332338, A332339.

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

A332276 Heinz numbers of widely totally normal integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 30, 32, 60, 64, 90, 128, 150, 180, 210, 256, 300, 360, 450, 512, 540, 600, 630, 1024, 1050, 1350, 1500, 2048, 2100, 2250, 2310, 2520, 2940, 3150, 3780, 4096, 4200, 4410, 5880, 8192, 8820, 9450, 10500, 11550, 12600, 13230, 14700

Offset: 1

Gus Wiseman, Feb 12 2020

nonn

First differs from A317246 in having 630.
A sequence of positive integers is widely totally normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has widely totally normal run-lengths.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The sequence of terms together with their prime indices begins:
    1: {}
    2: {1}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   12: {1,1,2}
   16: {1,1,1,1}
   18: {1,2,2}
   30: {1,2,3}
   32: {1,1,1,1,1}
   60: {1,1,2,3}
   64: {1,1,1,1,1,1}
   90: {1,2,2,3}
  128: {1,1,1,1,1,1,1}
  150: {1,2,3,3}
  180: {1,1,2,2,3}
  210: {1,2,3,4}
  256: {1,1,1,1,1,1,1,1}
  300: {1,1,2,3,3}
  360: {1,1,1,2,2,3}
For example, starting with (4,3,2,2,1), the partition with Heinz number 630, and repeatedly taking run-lengths gives (4,3,2,2,1) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1). These are all normal and the last is all 1's, so 630 belongs to the sequence.

Contains all powers of two A000079 and the primorials A002110.
Heinz numbers of normal integer partitions are A055932.
The case of reversed integer partitions is A332276 (this sequence).
The enumeration of these partitions by sum is A332277.
The enumeration of the generalization to compositions is A332279.
The co-strong version is A332290.
The strong version is A332291.
Cf. A007097, A056239, A133808, A181819, A182850, A305732, A317081, A317089, A317090, A317246, A317492, A329747, A332295, A332296.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
gnaQ[y_]:=Or[y=={},Union[y]=={1},And[Union[y]==Range[Max[y]],gnaQ[Length/@Split[y]]]];
Select[Range[1000],gnaQ[primeMS[#]]&]

A332296 Number of narrowly totally normal compositions of n.

Original entry on oeis.org

1, 1, 2, 4, 5, 7, 13, 23, 30, 63, 120, 209, 369, 651, 1198, 2174, 3896, 7023, 12699, 22941, 41565

Offset: 0

Gus Wiseman, Feb 15 2020

A sequence is narrowly totally normal if either it is empty, a singleton (narrow), or it covers an initial interval of positive integers (normal) with narrowly totally normal run-lengths.

A composition of n is a finite sequence of positive integers summing to n.

			The a(0) = 1 through a(6) = 13 compositions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)
           (11)  (12)   (112)   (122)    (123)
                 (21)   (121)   (212)    (132)
                 (111)  (211)   (221)    (213)
                        (1111)  (1121)   (231)
                                (1211)   (312)
                                (11111)  (321)
                                         (1212)
                                         (1221)
                                         (2112)
                                         (2121)
                                         (11211)
                                         (111111)
For example, starting with the composition (1,1,2,3,1,1) and repeatedly taking run-lengths gives (1,1,2,3,1,1) -> (2,1,1,2) -> (1,2,1) -> (1,1,1) -> (3). The first four are normal and the last is a singleton, so (1,1,2,3,1,1) is counted under a(9).

Normal compositions are A107429.
The wide version is A332279.
The wide recursive version (for partitions) is A332295.
The alternating version is A332296 (this sequence).
The strong version is A332336.
The co-strong version is (also) A332336.
Cf. A001462, A316496, A317081, A317245, A317491, A329744, A332276, A332277, A332278, A332297, A332337, A332340.

tinQ[q_]:=Or[Length[q]<=1,And[Union[q]==Range[Max[q]],tinQ[Length/@Split[q]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],tinQ]],{n,0,10}]

For n > 1, a(n) = A332279(n) + 1.

A332279 Number of widely totally normal compositions of n.

Original entry on oeis.org

1, 1, 1, 3, 4, 6, 12, 22, 29, 62, 119, 208, 368, 650, 1197, 2173, 3895, 7022, 12698, 22940, 41564

Offset: 0

Gus Wiseman, Feb 12 2020

A sequence is widely totally normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has widely totally normal run-lengths.

A composition of n is a finite sequence of positive integers with sum n.

			The a(1) = 1 through a(7) = 22 compositions:
  (1)  (11)  (12)   (112)   (122)    (123)     (1123)
             (21)   (121)   (212)    (132)     (1132)
             (111)  (211)   (221)    (213)     (1213)
                    (1111)  (1121)   (231)     (1231)
                            (1211)   (312)     (1312)
                            (11111)  (321)     (1321)
                                     (1212)    (2113)
                                     (1221)    (2122)
                                     (2112)    (2131)
                                     (2121)    (2212)
                                     (11211)   (2311)
                                     (111111)  (3112)
                                               (3121)
                                               (3211)
                                               (11221)
                                               (12112)
                                               (12121)
                                               (12211)
                                               (21121)
                                               (111211)
                                               (112111)
                                               (1111111)
For example, starting with y = (3,2,1,1,2,2,2,1,2,1,1,1,1) and repeatedly taking run-lengths gives y -> (1,1,2,3,1,1,4) -> (2,1,1,2,1) -> (1,2,1,1) -> (1,1,2) -> (2,1) -> (1,1). These are all normal and the last is all 1's, so y is counted under a(20).

Normal compositions are A107429.
Constantly recursively normal partitions are A332272.
The case of partitions is A332277.
The case of reversed partitions is (also) A332277.
The narrow version is A332296.
The strong version is A332337.
The co-strong version is (also) A332337.
Cf. A001462, A181819, A182850, A317081, A317245, A317491, A329744, A332276, A332289, A332292, A332295, A332297, A332336, A332340.

recnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],recnQ[Length/@Split[ptn]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],recnQ]],{n,0,10}]

For n > 1, a(n) = A332296(n) - 1.

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.

A332576 Number of integer partitions of n that are all 1's or whose run-lengths cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 6, 10, 12, 17, 21, 31, 35, 51, 59, 80, 97, 130, 153, 204, 244, 308, 376, 475, 564, 708, 851, 1043, 1247, 1533, 1816, 2216, 2633, 3174, 3766, 4526, 5324, 6376, 7520, 8917, 10479, 12415, 14524, 17134, 20035, 23489, 27423, 32091, 37286, 43512

Offset: 0

Gus Wiseman, Mar 05 2020

nonn

First differs from A317491 at a(11) = 31, A317491(11) = 30.

			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)

The narrow version is A317081.
Heinz numbers of these partitions first differ from A317492 in having 420.
Not counting constant-1 sequences gives A317081.
Dominated by A332295.
Cf. A000009, A001462, A181819, A182850, A317245, A317491, A329746, A329747, A332272, A332277.

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

a(n > 1) = A317081(n) + 1.

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

A332277 Number of widely totally normal integer partitions of n.

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A332289 Number of widely alternately co-strongly normal integer partitions of n.

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A332276 Heinz numbers of widely totally normal integer partitions.

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A332296 Number of narrowly totally normal compositions of n.

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A332279 Number of widely totally normal compositions of n.

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A332272 Number of narrowly recursively normal integer partitions of n.

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A332576 Number of integer partitions of n that are all 1's or whose run-lengths cover an initial interval of positive integers.

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A330937 Number of strictly recursively normal integer partitions of n.

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