A329746 -id:A329746 - cp's OEIS frontend

A332338 Number of alternately co-strong compositions of n.

1, 1, 2, 4, 7, 12, 24, 39, 72, 125, 224, 387, 697, 1205, 2141, 3736, 6598, 11516, 20331, 35526, 62507, 109436, 192200, 336533, 590582, 1034187

Offset: 0

Gus Wiseman, Feb 17 2020

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.

			The a(1) = 1 through a(5) = 12 compositions:
  (1)  (2)   (3)    (4)     (5)
       (11)  (12)   (13)    (14)
             (21)   (22)    (23)
             (111)  (31)    (32)
                    (112)   (41)
                    (121)   (113)
                    (1111)  (131)
                            (212)
                            (221)
                            (1112)
                            (1121)
                            (11111)
For example, starting with the composition y = (1,6,2,2,1,1,1,1) and repeatedly taking run-lengths and reversing gives (1,6,2,2,1,1,1,1) -> (4,2,1,1) -> (2,1,1) -> (2,1) -> (1,1) -> (2). All of these have weakly increasing run-lengths and the last is a singleton, so y is counted under a(15).

The case of partitions is A317256.
The recursive (rather than alternating) version is A332274.
The total (rather than alternating) version is (also) A332274.
The strong version is this same sequence.
The case of reversed partitions is A332339.
The normal version is A332340(n) + 1 for n > 1.
Cf. A001462, A100883, A181819, A182850, A316496, A317257, A329744, A329746, A332275, A332292, A332296.

tniQ[q_]:=Or[q=={},q=={1},And[LessEqual@@Length/@Split[q],tniQ[Reverse[Length/@Split[q]]]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],tniQ]],{n,0,10}]

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}]

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}]

A329750 Triangle read by rows where T(n,k) is the number of compositions of n >= 1 with runs-resistance n - k, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 2, 6, 6, 1, 1, 0, 4, 9, 15, 3, 1, 0, 2, 16, 22, 22, 1, 1, 0, 0, 8, 37, 38, 41, 3, 1, 0, 0, 0, 26, 86, 69, 72, 2, 1, 0, 0, 0, 2, 78, 175, 124, 129, 3, 1, 0, 0, 0, 0, 14, 202, 367, 226, 213, 1, 1, 0, 0, 0, 0, 0, 52, 469, 750, 376, 395, 5, 1

Offset: 1

Gus Wiseman, Nov 21 2019

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

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

			Triangle begins:
   1
   1   1
   2   1   1
   2   3   2   1
   2   6   6   1   1
   0   4   9  15   3   1
   0   2  16  22  22   1   1
   0   0   8  37  38  41   3   1
   0   0   0  26  86  69  72   2   1
   0   0   0   2  78 175 124 129   3   1
   0   0   0   0  14 202 367 226 213   1   1
   0   0   0   0   0  52 469 750 376 395   5   1
Row n = 6 counts the following compositions:
  (1,1,3,1)    (1,1,4)      (1,5)      (3,3)          (6)
  (1,3,1,1)    (4,1,1)      (2,4)      (2,2,2)
  (1,1,1,2,1)  (1,1,1,3)    (4,2)      (1,1,1,1,1,1)
  (1,2,1,1,1)  (1,2,2,1)    (5,1)
               (2,1,1,2)    (1,2,3)
               (3,1,1,1)    (1,3,2)
               (1,1,1,1,2)  (1,4,1)
               (1,1,2,1,1)  (2,1,3)
               (2,1,1,1,1)  (2,3,1)
                            (3,1,2)
                            (3,2,1)
                            (1,1,2,2)
                            (1,2,1,2)
                            (2,1,2,1)
                            (2,2,1,1)

Row sums are A000079.
Column sums are A329768.
The version with rows reversed is A329744.
Cf. A000740, A008965, A098504, A242882, A318928, A329745, A329746, A329747, A329767.

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==n-k&]],{n,10},{k,n}]

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.

A329743 Number of compositions of n with runs-resistance n - 3.

Original entry on oeis.org

0, 0, 0, 1, 2, 6, 9, 16, 8

Offset: 0

Gus Wiseman, Nov 21 2019

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

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

			The a(3) = 1 through a(8) = 8 compositions:
  (3)  (22)    (14)   (114)    (1123)    (12113)
       (1111)  (23)   (411)    (1132)    (12212)
               (32)   (1113)   (1141)    (13112)
               (41)   (1221)   (1411)    (21131)
               (131)  (2112)   (2122)    (21221)
               (212)  (3111)   (2212)    (31121)
                      (11112)  (2311)    (121112)
                      (11211)  (3211)    (211121)
                      (21111)  (11131)
                               (11212)
                               (11221)
                               (12211)
                               (13111)
                               (21211)
                               (111121)
                               (121111)
For example, repeatedly taking run-lengths starting with (1,2,1,1,3) gives (1,2,1,1,3) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1) -> (2), which is 5 steps, and 5 = 8 - 3, so (1,2,1,1,3) is counted under a(8).

Column k = n - 3 of A329744.
Column k = 3 of A329750.
Compositions with runs-resistance 2 are A329745.
Cf. A000740, A008965, A098504, A242882, A318928, A329746, A329747, A329767.

runsres[q_]:=If[Length[q]==1,0,Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==n-3&]],{n,10}]

A329768 Number of finite sequences of positive integers whose sum minus runs-resistance is n.

Original entry on oeis.org

8, 17, 42, 104, 242, 541, 1212, 2664, 5731, 12314

Offset: 1

Gus Wiseman, Nov 21 2019

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

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

			The a(1) = 8 and a(2) = 17 compositions whose sum minus runs-resistance is n:
  (1)        (2)
  (1,1)      (1,3)
  (1,2)      (3,1)
  (2,1)      (1,1,1)
  (1,1,2)    (1,1,3)
  (2,1,1)    (1,2,1)
  (1,1,2,1)  (1,2,2)
  (1,2,1,1)  (2,2,1)
             (3,1,1)
             (1,1,1,2)
             (1,1,3,1)
             (1,3,1,1)
             (2,1,1,1)
             (1,1,1,2,1)
             (1,2,1,1,1)
             (1,2,1,1,2)
             (2,1,1,2,1)

Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003.

Column sums of A329750.

Cf. A000740, A008965, A098504, A242882, A318928, A329744, A329745, A329746, A329747, A329767.

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

A332338 Number of alternately co-strong compositions of n.

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

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A332295 Number of widely recursively normal integer partitions of n.

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A329750 Triangle read by rows where T(n,k) is the number of compositions of n >= 1 with runs-resistance n - k, 1 <= k <= 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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A329743 Number of compositions of n with runs-resistance n - 3.

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A329768 Number of finite sequences of positive integers whose sum minus runs-resistance is n.

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A330937 Number of strictly recursively normal integer partitions of n.

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