A332278 -id:A332278 - cp's OEIS frontend

A316496 Number of totally strong integer partitions of n.

1, 1, 2, 3, 4, 5, 8, 8, 12, 13, 18, 20, 27, 27, 38, 41, 52, 56, 73, 77, 99, 105, 129, 145, 176, 186, 229, 253, 300, 329, 395, 427, 504, 555, 648, 716, 836, 905, 1065, 1173, 1340, 1475, 1703, 1860, 2140, 2349, 2671, 2944, 3365, 3666, 4167, 4582, 5160, 5668

Offset: 0

Gus Wiseman, Jul 29 2018

nonn

An integer partition is totally strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and are themselves a totally strong partition.

			The a(1) = 1 through a(8) = 12 totally strong partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (32)     (33)      (43)       (44)
             (111)  (31)    (41)     (42)      (52)       (53)
                    (1111)  (221)    (51)      (61)       (62)
                            (11111)  (222)     (331)      (71)
                                     (321)     (421)      (332)
                                     (2211)    (2221)     (431)
                                     (111111)  (1111111)  (521)
                                                          (2222)
                                                          (3311)
                                                          (22211)
                                                          (11111111)
For example, the partition (3,3,2,1) has run-lengths (2,1,1), which are weakly decreasing, but they have run-lengths (1,2), which are not weakly decreasing, so (3,3,2,1) is not totally strong.

Cf. A181819, A182850, A182857, A304660, A305563, A316597.
The Heinz numbers of these partitions are A316529.
The version for compositions is A332274.
The dual version is A332275.
The version for reversed partitions is (also) A332275.
The narrowly normal version is A332297.
The alternating version is A332339 (see also A317256).
Partitions with weakly decreasing run-lengths are A100882.
Cf. A025487, A100882, A133808, A317245, A317491, A332278, A332291, A332336.

totincQ[q_]:=Or[q=={},q=={1},And[GreaterEqual@@Length/@Split[q],totincQ[Length/@Split[q]]]];
Table[Length[Select[IntegerPartitions[n],totincQ]],{n,0,30}]

Updated with corrected terminology by Gus Wiseman, Mar 07 2020

A332292 Number of widely alternately strongly normal integer partitions of n.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Feb 16 2020

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

Also the number of widely alternately co-strongly normal reversed integer partitions of n.

			The a(1) = 1, a(3) = 2, and a(21) = 3 partitions:
  (1)  (21)   (654321)
       (111)  (4443321)
              (111111111111111111111)
For example, starting with the partition y = (4,4,4,3,3,2,1) and repeatedly taking run-lengths and reversing gives (4,4,4,3,3,2,1) -> (1,1,2,3) -> (1,1,2) -> (1,2) -> (1,1). All of these are normal with weakly decreasing run-lengths, and the last is all 1's, so y is counted under a(21).

Normal partitions are A000009.
The non-strong version is A332277.
The co-strong version is A332289.
The case of reversed partitions is (also) A332289.
The case of compositions is A332340.
Cf. A100883, A181819, A317081, A317245, A317256, A317491, A329746, A329747, A332278, A332290, A332291, A332297, A332337.

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

a(71)-a(77) from Jinyuan Wang, Jun 26 2020

A332340 Number of widely alternately co-strongly normal compositions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 9, 11, 13, 23, 53, 78, 120, 207, 357, 707, 1183, 2030, 3558, 6229, 10868

Offset: 0

Gus Wiseman, Feb 17 2020

An integer partition is widely alternately 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-length (co-strong) which, if reversed, are themselves a widely alternately co-strongly normal partition.

			The a(1) = 1 through a(8) = 13 compositions:
  (1)  (11)  (12)   (121)   (122)    (123)     (1213)     (1232)
             (21)   (211)   (212)    (132)     (1231)     (1322)
             (111)  (1111)  (1211)   (213)     (1312)     (2123)
                            (11111)  (231)     (1321)     (2132)
                                     (312)     (2122)     (2312)
                                     (321)     (2131)     (2321)
                                     (1212)    (2311)     (3122)
                                     (2121)    (3121)     (3212)
                                     (111111)  (3211)     (12131)
                                               (12121)    (13121)
                                               (1111111)  (21212)
                                                          (122111)
                                                          (11111111)
For example, starting with the composition y = (122111) and repeatedly taking run-lengths and reversing gives (122111) -> (321) -> (111). All of these are normal with weakly increasing run-lengths and the last is all 1's, so y is counted under a(8).

Normal compositions are A107429.
Compositions with normal run-lengths are A329766.
The Heinz numbers of the case of partitions are A332290.
The case of partitions is A332289.
The total (instead of alternating) version is A332337.
Not requiring normality gives A332338.
The strong version is this same sequence.
The narrow version is a(n) + 1 for n > 1.
Cf. A181819, A317245, A317491, A329744, A329741, A329746, A332278, A332279, A332292.

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[Join@@Permutations/@IntegerPartitions[n],totnQ]],{n,0,10}]

A332277 Number of widely totally normal integer partitions of n.

Original entry on oeis.org

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

A332291 Heinz numbers of widely totally strongly normal integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 18, 30, 32, 64, 128, 210, 256, 450, 512, 1024, 2048, 2250, 2310, 4096, 8192, 16384, 30030, 32768, 65536, 131072, 262144, 510510, 524288

Offset: 1

Gus Wiseman, Feb 14 2020

An integer partition is widely totally strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) which are themselves a widely totally strongly normal partition.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
This sequence is closed under A304660, so there are infinitely many terms that are not powers of 2 or primorial numbers.

			The sequence of all widely totally strongly normal integer partitions together with their Heinz numbers begins:
      1: ()
      2: (1)
      4: (1,1)
      6: (2,1)
      8: (1,1,1)
     16: (1,1,1,1)
     18: (2,2,1)
     30: (3,2,1)
     32: (1,1,1,1,1)
     64: (1,1,1,1,1,1)
    128: (1,1,1,1,1,1,1)
    210: (4,3,2,1)
    256: (1,1,1,1,1,1,1,1)
    450: (3,3,2,2,1)
    512: (1,1,1,1,1,1,1,1,1)
   1024: (1,1,1,1,1,1,1,1,1,1)
   2048: (1,1,1,1,1,1,1,1,1,1,1)
   2250: (3,3,3,2,2,1)
   2310: (5,4,3,2,1)
   4096: (1,1,1,1,1,1,1,1,1,1,1,1)

Closed under A304660.
The non-strong version is A332276.
The co-strong version is A332293.
The case of reversed partitions is (also) A332293.
Heinz numbers of normal partitions with decreasing run-lengths are A025487.
Cf. A055932, A056239, A181819, A242031, A317089, A317246, A317257, A317492, A329747, A332277, A332278, A332290, A332292, A332297, A332337.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
totnQ[ptn_]:=Or[ptn=={},Union[ptn]=={1},And[Union[ptn]==Range[Max[ptn]],GreaterEqual@@Length/@Split[ptn],totnQ[Length/@Split[ptn]]]];
Select[Range[10000],totnQ[Reverse[primeMS[#]]]&]

A332297 Number of narrowly totally strongly normal integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2

Offset: 0

Gus Wiseman, Feb 15 2020

A partition 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 partition.

			The a(1) = 1, a(2) = 2, a(3) = 3, and a(55) = 4 partitions:
  (1)  (2)    (3)      (55)
       (1,1)  (2,1)    (10,9,8,7,6,5,4,3,2,1)
              (1,1,1)  (5,5,5,5,5,4,4,4,4,3,3,3,2,2,1)
                       (1)^55
For example, starting with the partition (3,3,2,2,1) and repeatedly taking run-lengths gives (3,3,2,2,1) -> (2,2,1) -> (2,1) -> (1,1) -> (2). The first four are normal and have weakly decreasing run-lengths, and the last is a singleton, so (3,3,2,2,1) is counted under a(11).

Normal partitions are A000009.
The non-totally normal version is A316496.
The widely alternating version is A332292.
The non-strong case of compositions is A332296.
The case of compositions is A332336.
The wide version is a(n) - 1 for n > 1.
Cf. A001462, A025487, A100883, A181819, A182850, A317081, A317245, A317256, A317491, A332275, A332277, A332278, A332291, A332337.

tinQ[q_]:=Or[q=={},Length[q]==1,And[Union[q]==Range[Max[q]],GreaterEqual@@Length/@Split[q],tinQ[Length/@Split[q]]]];
Table[Length[Select[IntegerPartitions[n],tinQ]],{n,0,30}]

a(60)-a(80) from Jinyuan Wang, Jun 26 2020

A332337 Number of widely totally strongly normal compositions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 3, 9, 9, 12, 23, 54, 77, 116, 205, 352, 697, 1174, 2013, 3538, 6209, 10830

Offset: 0

Gus Wiseman, Feb 15 2020

A sequence is widely totally strongly normal if either it is all 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) that are themselves a widely totally strongly normal sequence.

			The a(1) = 1 through a(8) = 12 compositions:
  (1)  (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). These are all normal with weakly decreasing run-lengths, and the last is all 1's, so (22112) is counted under a(8).

Normal compositions are A107429.
The case of partitions is A332278.
The non-strong version is A332279.
Heinz numbers in the case of partitions are A332291.
The narrow version is A332336.
The alternating version is A332340.
The co-strong version is this same sequence.
Cf. A025487, A100883, A181819, A317245, A317491, A329744, A332274, A332276, A332277, A332292, A332293, A332296.

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

For n > 1, a(n) = A332336(n) - 1.

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}]

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.

A332293 Heinz numbers of widely totally co-strongly normal integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 30, 32, 64, 128, 180, 210, 256, 360, 512, 1024, 2048, 2310, 4096, 8192, 16384, 30030, 32768, 65536, 75600, 131072, 262144, 510510, 524288

Offset: 1

Gus Wiseman, Feb 16 2020

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

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}
    30: {1,2,3}
    32: {1,1,1,1,1}
    64: {1,1,1,1,1,1}
   128: {1,1,1,1,1,1,1}
   180: {1,1,2,2,3}
   210: {1,2,3,4}
   256: {1,1,1,1,1,1,1,1}
   360: {1,1,1,2,2,3}
   512: {1,1,1,1,1,1,1,1,1}
  1024: {1,1,1,1,1,1,1,1,1,1}
  2048: {1,1,1,1,1,1,1,1,1,1,1}
  2310: {1,2,3,4,5}
  4096: {1,1,1,1,1,1,1,1,1,1,1,1}
  8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
For example, 180 is the Heinz number of (3,2,2,1,1), with run-lengths (3,2,2,1,1) -> (1,2,2) -> (1,2) -> (1,1). These are all normal with weakly increasing multiplicities and the last is all 1's, so 180 belongs to the sequence.

A subset of A055932.
Closed under A181819.
The non-co-strong version is A332276.
The enumeration of these partitions by sum is A332278.
The alternating version is A332290.
The strong version is A332291.
The case of reversed partitions is (also) A332291.
Cf. A000009, A056239, A133808, A182850, A304660, A317089, A317246, A317257, A317492, A329747, A332277, A332289.

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

cp's OEIS Frontend

A316496 Number of totally strong integer partitions of n.

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A332292 Number of widely alternately strongly normal integer partitions of n.

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A332340 Number of widely alternately co-strongly normal compositions of n.

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A332277 Number of widely totally normal integer partitions of n.

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A332291 Heinz numbers of widely totally strongly normal integer partitions.

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A332297 Number of narrowly totally strongly normal integer partitions of n.

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A332337 Number of widely totally strongly normal compositions of n.

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

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A332296 Number of narrowly totally normal compositions of n.

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