A332289 -id:A332289 - cp's OEIS frontend

A317256 Number of alternately co-strong integer partitions of n.

1, 1, 2, 3, 5, 6, 11, 13, 19, 25, 35, 42, 61, 74, 98, 122, 161, 194, 254, 304, 388, 472, 589, 700, 878, 1044, 1278, 1525, 1851, 2182, 2651, 3113, 3735, 4389, 5231, 6106, 7278, 8464, 9995, 11631, 13680, 15831, 18602, 21463, 25068, 28927, 33654, 38671, 44942, 51514

Offset: 0

Gus Wiseman, Jul 25 2018

nonn

A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.

Also the number of alternately strong reversed integer partitions of n.

			The a(1) = 1 through a(7) = 13 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (311)    (51)      (61)
                    (1111)  (2111)   (222)     (322)
                            (11111)  (321)     (421)
                                     (411)     (511)
                                     (2211)    (3211)
                                     (3111)    (4111)
                                     (21111)   (22111)
                                     (111111)  (31111)
                                               (211111)
                                               (1111111)
For example, starting with the partition y = (3,2,2,1,1) and repeatedly taking run-lengths and reversing gives (3,2,2,1,1) -> (2,2,1) -> (1,2), which is not weakly decreasing, so y is not  alternately co-strong. On the other hand, we have (3,3,2,2,1,1,1) -> (3,2,2) -> (2,1) -> (1,1) -> (2) -> (1), so (3,3,2,2,1,1,1) is counted under a(13).

Cf. A000041, A100883, A181819, A182850, A182857, A304660, A305563, A317081, A317086, A317245, A317258.
The Heinz numbers of these partitions are given by A317257.
The total (instead of alternating) version is A332275.
Dominates A332289 (the normal version).
The generalization to compositions is A332338.
The dual version is A332339.
The case of reversed partitions is (also) A332339.
Cf. A316496  A317491, A329746, A332292, A332297, A332340.

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

Updated with corrected terminology by Gus Wiseman, Mar 08 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

A332275 Number of totally co-strong integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 11, 12, 17, 22, 30, 32, 49, 53, 70, 82, 108, 119, 156, 171, 219, 250, 305, 336, 424, 468, 562, 637, 754, 835, 1011, 1108, 1304, 1461, 1692, 1873, 2212, 2417, 2787, 3109, 3562, 3911, 4536, 4947, 5653, 6265, 7076, 7758, 8883, 9669, 10945, 12040

Offset: 0

Gus Wiseman, Feb 12 2020

nonn

A sequence is totally co-strong if it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and are themselves a totally co-strong sequence.

Also the number of totally strong reversed integer partitions of n.

			The a(1) = 1 through a(7) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (21)   (22)    (32)     (33)      (43)
             (111)  (31)    (41)     (42)      (52)
                    (211)   (311)    (51)      (61)
                    (1111)  (2111)   (222)     (322)
                            (11111)  (321)     (421)
                                     (411)     (511)
                                     (2211)    (4111)
                                     (3111)    (22111)
                                     (21111)   (31111)
                                     (111111)  (211111)
                                               (1111111)
For example, the partition y = (5,4,4,4,3,3,3,2,2,2,2,2,2,1,1,1,1,1,1) has run-lengths (1,3,3,6,6), with run-lengths (1,2,2), with run-lengths (1,2), with run-lengths (1,1), with run-lengths (2), with run-lengths (1). All of these having weakly increasing run-lengths, and the last is (1), so y is counted under a(44).

The strong version is A316496.
The version for reversed partitions is (also) A316496.
The alternating version is A317256.
The generalization to compositions is A332274.
Cf. A001462, A100883, A181819, A182850, A317491, A329746, A332289, A332297, A332336, A332337, A332338, A332339, A332340.

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

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

A332339 Number of alternately co-strong reversed integer partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 8, 12, 14, 18, 20, 29, 28, 40, 45, 54, 59, 82, 81, 108, 118, 141, 154, 204, 204, 255, 285, 339, 363, 458, 471, 580, 632, 741, 806, 983, 1015, 1225, 1341, 1562, 1667, 2003, 2107, 2491, 2712, 3101, 3344, 3962, 4182, 4860, 5270, 6022, 6482

Offset: 0

Gus Wiseman, Feb 17 2020

nonn

A sequence is alternately co-strong if either it is empty, equal to (1), or its run-lengths are weakly increasing (co-strong) and, when reversed, are themselves an alternately co-strong sequence.

Also the number of alternately strong integer partitions of n.

			The a(1) = 1 through a(8) = 12 reversed partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (12)   (13)    (14)     (15)      (16)       (17)
             (111)  (22)    (23)     (24)      (25)       (26)
                    (1111)  (122)    (33)      (34)       (35)
                            (11111)  (123)     (124)      (44)
                                     (222)     (133)      (125)
                                     (1122)    (1222)     (134)
                                     (111111)  (1111111)  (233)
                                                          (1133)
                                                          (2222)
                                                          (11222)
                                                          (11111111)
For example, starting with the composition y = (1,2,3,3,4,4,4) and repeatedly taking run-lengths and reversing gives (1,2,3,3,4,4,4) -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1) -> (2) -> (1). All of these have weakly increasing run-lengths and the last is equal to (1), so y is counted under a(21).

The total (instead of alternating) version is A316496.
Alternately strong partitions are A317256.
The case of ordinary (not reversed) partitions is (also) A317256.
The generalization to compositions is A332338.
Cf. A100883, A181819, A182850, A317257, A329744, A329746, A332275, A332289, A332292, A332340.

tniQ[q_]:=Or[q=={},q=={1},And[LessEqual@@Length/@Split[q],tniQ[Reverse[Length/@Split[q]]]]];
Table[Length[Select[Sort/@IntegerPartitions[n],tniQ]],{n,0,30}]

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.

A332290 Heinz numbers of widely alternately co-strongly normal integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 30, 32, 60, 64, 128, 210, 256, 360, 512, 1024, 2048, 2310, 2520, 4096, 8192, 16384, 30030, 32768, 65536, 75600, 131072, 262144, 510510, 524288

Offset: 1

Gus Wiseman, Feb 14 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) and has weakly increasing run-lengths (co-strong) which, if reversed, are themselves a widely alternately co-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 A181821, so there are infinitely many terms that are not powers of 2 or primorial numbers.

			The sequence of all widely alternately co-strongly normal integer partitions together with their Heinz numbers begins:
      1: ()
      2: (1)
      4: (1,1)
      6: (2,1)
      8: (1,1,1)
     12: (2,1,1)
     16: (1,1,1,1)
     30: (3,2,1)
     32: (1,1,1,1,1)
     60: (3,2,1,1)
     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)
    360: (3,2,2,1,1,1)
    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)
   2310: (5,4,3,2,1)
   2520: (4,3,2,2,1,1,1)
For example, starting with y = (4,3,2,2,1,1,1), which has Heinz number 2520, and repeatedly taking run-lengths and reversing gives (4,3,2,2,1,1,1) -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1). These are all normal with weakly increasing run-lengths and the last is all 1's, so 2520 belongs to the sequence.

Closed under A181821.
The non-co-strong version is A332276.
The enumeration of these partitions by sum is A332289.
The total (rather than alternating) version is A332293.
Cf. A055932, A056239, A100883, A133808, A181819, A317089, A317090, A317246, A317257, A317492, A329747, A332292, A332340.

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]],LessEqual@@Length/@Split[ptn],totnQ[Reverse[Length/@Split[ptn]]]]];
Select[Range[10000],totnQ[Reverse[primeMS[#]]]&]

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

A317256 Number of alternately co-strong integer partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A332277 Number of widely totally normal integer partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A332275 Number of totally co-strong integer partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A332339 Number of alternately co-strong reversed integer partitions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A332279 Number of widely totally normal compositions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A332290 Heinz numbers of widely alternately co-strongly normal integer partitions.

Views