A353390 -id:A353390 - cp's OEIS frontend

A353428 Number of integer compositions of n with all parts and all run-lengths > 2.

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 1, 0, 2, 0, 2, 4, 0, 0, 8, 3, 0, 10, 4, 4, 15, 4, 8, 24, 7, 8, 42, 16, 10, 59, 31, 27, 87, 37, 52, 149, 62, 66, 233, 121, 111, 342, 207, 204, 531, 308, 351, 864, 487, 536, 1373, 864, 865, 2057, 1440, 1509, 3232

Offset: 0

Gus Wiseman, May 16 2022

nonn

			The a(n) compositions for selected n:
  n=16:   n=18:     n=20:    n=21:      n=24:
----------------------------------------------------
  (4444)  (666)     (5555)   (777)      (888)
          (333333)  (44444)  (333444)   (6666)
                             (444333)   (333555)
                             (3333333)  (444444)
                                        (555333)
                                        (3333444)
                                        (4443333)
                                        (33333333)

Alois P. Heinz, Table of n, a(n) for n = 0..2000

Allowing any multiplicities gives A078012, partitions A008483.
The version for no (instead of all) parts or run-lengths > 2 is A137200.
Allowing any parts gives A353400, partitions A100405.
The version for partitions is A353501, ranked by A353502.
The version for > 1 instead of > 2 is A353508, partitions A339222.
A003242 counts anti-run compositions, ranked by A333489.
A008466 counts compositions with some part > 2.
A011782 counts compositions.
A114901 counts compositions with no runs of length 1, ranked by A353427.
A128695 counts compositions with no run-lengths > 2.
A261983 counts non-anti-run compositions.
A335464 counts compositions with a run-length > 2.
Cf. A032020, A044813, A098859, A165413, A318928, A329738, A329739, A333755, A353390, A353429.

b:= proc(n, h) option remember; `if`(n=0, 1, add(
     `if`(i=h, 0, add(b(n-i*j, i), j=3..n/i)), i=3..n/3))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..80);  # Alois P. Heinz, May 18 2022

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[#,1|2]&&!MemberQ[Length/@Split[#],1|2]&]],{n,0,15}]

a(26)-a(66) from Alois P. Heinz, May 17 2022

A353508 Number of integer compositions of n with no ones or runs of length 1.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 2, 0, 2, 1, 4, 0, 8, 2, 11, 4, 21, 5, 37, 12, 57, 25, 104, 38, 177, 79, 292, 149, 513, 251, 876, 482, 1478, 871, 2562, 1533, 4387, 2815, 7473, 5036, 12908, 8935, 22135, 16085, 37940, 28611, 65422, 50731, 112459, 90408, 193386, 160119, 333513

Offset: 0

Gus Wiseman, May 17 2022

nonn

			The a(0) = 1 through a(14) = 11 compositions (empty columns indicated by dots, 0 is the empty composition):
  0  .  .  .  22  .  33   .  44    333  55     .  66      22333  77
                     222     2222       2233      444     33322  2255
                                        3322      2244           3344
                                        22222     3333           4433
                                                  4422           5522
                                                  22233          22244
                                                  33222          44222
                                                  222222         222233
                                                                 223322
                                                                 332222
                                                                 2222222

Alois P. Heinz, Table of n, a(n) for n = 0..1000

The version for partitions is A339222.
Compositions counted by their run-lengths:
- For run-lengths <= 1 we have A003242, ranked by A333489.
- For run-lengths = 2 we have A003242 aerated.
- For run-lengths > 1 we have A114901, ranked by A353427.
- For run-lengths <= 2 we have A128695 matching A335464.
- For run-lengths > 2 we have A353400, partitions A100405.
- For run-lengths all prime we have A353401.
- For run-lengths and parts > 2 we have A353428.
A008466 counts compositions with some part > 2.
A011782 counts compositions.
A106356 counts compositions by number of adjacent equal parts.
A261983 counts non-anti-run compositions.
A274174 counts compositions with equal parts contiguous.
Cf. A005811, A078012, A238279, A329739, A333755, A351013, A353390.

b:= proc(n,h) option remember; `if`(n=0, 1, add(
     `if`(i<>h, add(b(n-i*j, i), j=2..n/i), 0), i=2..n/2))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, May 17 2022

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[#,1]&&!MemberQ[Length/@Split[#],1]&]],{n,0,15}]

a(41)-a(52) from Alois P. Heinz, May 17 2022

A353696 Numbers k such that the k-th composition in standard order (A066099) is empty, a singleton, or has run-lengths that are a consecutive subsequence that is already counted.

Original entry on oeis.org

0, 1, 2, 4, 8, 10, 16, 32, 43, 58, 64, 128, 256, 292, 349, 442, 512, 586, 676, 697, 826, 1024, 1210, 1338, 1393, 1394, 1396, 1594, 2048, 2186, 2234, 2618, 2696, 2785, 2786, 2792, 3130, 4096, 4282, 4410, 4666, 5178, 5569, 5570, 5572, 5576, 5584, 6202, 8192

Offset: 1

Gus Wiseman, May 22 2022

nonn

First differs from the non-consecutive version A353431 in lacking 22318, corresponding to the binary word 101011100101110 and standard composition (2,2,1,1,3,2,1,1,2), whose run-lengths (2,2,1,1,2,1) are a subsequence but not a consecutive subsequence.

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The terms together with their corresponding compositions begin:
    0: ()
    1: (1)
    2: (2)
    4: (3)
    8: (4)
   10: (2,2)
   16: (5)
   32: (6)
   43: (2,2,1,1)
   58: (1,1,2,2)
   64: (7)
  128: (8)
  256: (9)
  292: (3,3,3)
  349: (2,2,1,1,2,1)
  442: (1,2,1,1,2,2)
  512: (10)
  586: (3,3,2,2)
  676: (2,2,3,3)
  697: (2,2,1,1,3,1)
  826: (1,3,1,1,2,2)

Non-recursive non-consecutive for partitions: A325755, counted by A325702.
Non-consecutive: A353431, counted by A353391.
Non-consecutive for partitions: A353393, counted by A353426.
Non-recursive non-consecutive: A353402, counted by A353390.
Counted by: A353430.
Non-recursive: A353432, counted by A353392.
A005811 counts runs in binary expansion.
A011782 counts compositions.
A066099 lists compositions in standard order, run-lengths A333769.
Statistics of standard compositions:
- Length is A000120, sum A070939.
- Runs are counted by A124767, distinct A351014.
- Subsequences are counted by A334299, contiguous A124770/A124771.
- Runs-resistance is A333628.
Classes of standard compositions:
- Partitions are A114994, strict A333255, multisets A225620, sets A333256.
- Runs are A272919, counted by A000005.
- Golomb rulers are A333222, counted by A169942.
- Anti-runs are A333489, counted by A003242.
Cf. A032020, A114640, A181819, A228351, A329739, A318928, A325705, A329738, A333224, A353427, A353403.

stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
yoyQ[y_]:=Length[y]<=1||MemberQ[Join@@Table[Take[y,{i,j}],{i,Length[y]},{j,i,Length[y]}],Length/@Split[y]]&&yoyQ[Length/@Split[y]];
Select[Range[0,1000],yoyQ[stc[#]]&]

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A353428 Number of integer compositions of n with all parts and all run-lengths > 2.

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A353508 Number of integer compositions of n with no ones or runs of length 1.

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Maple

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A353696 Numbers k such that the k-th composition in standard order (A066099) is empty, a singleton, or has run-lengths that are a consecutive subsequence that is already counted.

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