A317245 -id:A317245 - cp's OEIS frontend

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

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}]

A317492 Heinz numbers of fully normal integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78

Offset: 1

Gus Wiseman, Jul 30 2018

nonn

An integer partition is fully normal if either it is of the form (1,1,...,1) or its multiplicities span an initial interval of positive integers and, sorted in weakly decreasing order, are themselves fully normal.

Cf. A055932, A056239, A181819, A182850, A296150, A305732, A317089, A317090, A317245, A317246, A317491, A317493.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
fulnrmQ[ptn_]:=With[{qtn=Sort[Length/@Split[ptn],Greater]},Or[ptn=={}||Union[ptn]=={1},And[Union[qtn]==Range[Max[qtn]],fulnrmQ[qtn]]]];
Select[Range[100],fulnrmQ[Reverse[primeMS[#]]]&]

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

Original entry on oeis.org

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

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.

A332336 Number of narrowly totally strongly normal compositions of n.

Original entry on oeis.org

1, 1, 2, 4, 4, 4, 10, 10, 13, 24, 55, 78, 117, 206, 353, 698, 1175, 2014, 3539, 6210, 10831

Offset: 0

Gus Wiseman, Feb 15 2020

A sequence is narrowly totally strongly normal if either it is empty, a singleton (narrow), or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) that are themselves a narrowly totally strongly normal sequence.

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

			The a(1) = 1 through a(8) = 13 compositions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (112)   (212)    (123)     (1213)     (1232)
             (21)   (121)   (221)    (132)     (1231)     (2123)
             (111)  (1111)  (11111)  (213)     (1312)     (2132)
                                     (231)     (1321)     (2312)
                                     (312)     (2131)     (2321)
                                     (321)     (3121)     (3212)
                                     (1212)    (11221)    (12131)
                                     (2121)    (12121)    (13121)
                                     (111111)  (1111111)  (21212)
                                                          (22112)
                                                          (111221)
                                                          (11111111)
For example, starting with (22112) and repeated taking run-lengths gives (22112) -> (221) -> (21) -> (11) -> (2). The first four are normal with weakly decreasing run-lengths, and the last is a singleton, so (22112) is counted under a(8).

Normal compositions are A107429.
The non-strong version is A332296.
The case of partitions is A332297.
The co-strong version is A332336 (this sequence).
The wide version is A332337.
Cf. A025487, A316496, A317081, A317245, A317256, A317491, A329744, A332279, A332291, A332292, A332338, A332340.

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

For n > 1, a(n) = A332337(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.

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}]

A317589 Heinz numbers of uniformly normal integer partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 23, 25, 27, 29, 30, 31, 32, 36, 37, 41, 43, 47, 49, 53, 59, 60, 61, 64, 67, 71, 73, 79, 81, 83, 89, 90, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 150, 151, 157, 163, 167, 169

Offset: 1

Gus Wiseman, Aug 01 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

An integer partition is uniformly normal if either (1) it is of the form (x, x, ..., x) for some x > 0, or (2a) it spans an initial interval of positive integers, and (2b) its multiplicities, sorted in weakly decreasing order, are themselves a uniformly normal integer partition.

Cf. A055932, A056239, A181819, A182850, A296150, A304687, A304818, A317089, A317090, A317245, A317246, A317492, A317588, A317590.

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

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.

A325330 Number of integer partitions of n whose multiplicities have multiplicities that cover an initial interval of positive integers.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 11, 16, 22, 31, 44, 55, 77, 96, 127, 158, 208, 251, 329, 400, 501, 610, 766, 915, 1141, 1368, 1677, 2005, 2454, 2913, 3553, 4219, 5110, 6053, 7300, 8644, 10376, 12238, 14645, 17216, 20504, 24047, 28501, 33336, 39373, 45871, 53926, 62745

Offset: 0

Gus Wiseman, May 01 2019

nonn

Partitions whose parts cover an initial interval of positive integers are counted by A000009, with Heinz numbers A055932. Partitions whose multiplicities cover an initial interval of positive integers are counted by A317081, with Heinz numbers A317090. Partitions whose parts and multiplicities both cover an initial interval of positive integers are counted by A317088, with Heinz numbers A317089. Partitions whose multiplicities at every depth cover an initial interval of positive integers are counted by A317245, with Heinz numbers A317246.

The Heinz numbers of these partitions are given by A325370.

			The a(0) = 1 through a(8) = 16 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
           (11)  (111)  (22)    (221)    (33)      (322)      (44)
                        (211)   (311)    (222)     (331)      (332)
                        (1111)  (2111)   (411)     (511)      (422)
                                (11111)  (3111)    (2221)     (611)
                                         (21111)   (3211)     (2222)
                                         (111111)  (4111)     (3221)
                                                   (22111)    (4211)
                                                   (31111)    (5111)
                                                   (211111)   (22211)
                                                   (1111111)  (32111)
                                                              (41111)
                                                              (221111)
                                                              (311111)
                                                              (2111111)
                                                              (11111111)
For example, the partition (5,5,4,3,3,3,2,2) has multiplicities (2,1,3,2) with multiplicities (1,2,1) which cover the initial interval {1,2}, so (5,5,4,3,3,3,2,2) is counted under a(27).

Cf. A000837, A055932, A317081, A317088, A317089, A317090, A317245, A320348, A325331, A325333, A325337, A325370.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[IntegerPartitions[n],normQ[Length/@Split[Sort[Length/@Split[#]]]]&]],{n,0,30}]

cp's OEIS Frontend

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

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A317492 Heinz numbers of fully normal integer partitions.

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

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A332296 Number of narrowly totally normal compositions of n.

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A332336 Number of narrowly totally strongly normal compositions of n.

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A332279 Number of widely totally normal compositions of n.

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A332295 Number of widely recursively normal integer partitions of n.

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A317589 Heinz numbers of uniformly normal integer partitions.

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A332272 Number of narrowly recursively normal integer partitions of n.

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