A333769 -id:A333769 - cp's OEIS frontend

A337504 Number of compositions of 2*n with n maximal anti-runs.

1, 1, 3, 8, 13, 33, 112, 286, 769, 2288, 6695, 18745, 54654, 160888, 467402, 1362378, 4016517, 11807966, 34708018, 102451390, 302870005, 895207191, 2650590597, 7859253320, 23316653154, 69231883374, 205773157904, 612021943421, 1821435719846, 5424528040529, 16165017705176

Offset: 0

Gus Wiseman, Sep 04 2020

nonn

An anti-run is a sequence with no adjacent equal parts.

			The a(0) = 1 through a(4) = 13 compositions:
  ()  (2)  (2,2)    (2,2,2)      (2,2,2,2)
           (1,1,2)  (1,1,1,3)    (1,1,1,1,4)
           (2,1,1)  (1,1,2,2)    (1,1,2,2,2)
                    (2,2,1,1)    (2,2,2,1,1)
                    (3,1,1,1)    (4,1,1,1,1)
                    (1,1,1,2,1)  (1,1,1,1,3,1)
                    (1,1,2,1,1)  (1,1,1,2,2,1)
                    (1,2,1,1,1)  (1,1,1,3,1,1)
                                 (1,1,2,2,1,1)
                                 (1,1,3,1,1,1)
                                 (1,2,2,1,1,1)
                                 (1,3,1,1,1,1)
                                 (2,1,1,1,1,2)

Andrew Howroyd, Table of n, a(n) for n = 0..200

A106356 has this as main diagonal n = 2*k.
A336108 is the version for runs.
A337505 is the version for patterns.
A337564 is the version for runs in patterns.
A003242 counts anti-run compositions.
A011782 counts compositions.
A124767 counts runs in standard compositions.
A238343 counts compositions by descents.
A333213 counts compositions by weak ascents.
A333381 counts anti-runs in standard compositions.
A333382 counts adjacent unequal pairs in standard compositions.
A333489 ranks anti-runs.
A333755 counts compositions by number of runs.
A333769 gives run-lengths in standard compositions.
A337565 gives anti-run lengths in standard compositions.
Cf. A106351, A124762, A233564, A235998, A238130, A238279, A333214, A333216.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[2*n],Length[Split[#,UnsameQ]]==n&]],{n,0,10}]

a(n)={polcoef(polcoef(1 - y + y*(y-1)/(y - 1 - sum(d=1, 2*n, (y-1)^d*x^d/(1 - x^d) + O(x^(2*n+1)))), 2*n, x), n, y)} \\ Andrew Howroyd, Feb 02 2021

a(n) = [x^(2*n)*y^n] 1 - y + y*(y-1)/(y - 1 - Sum_{d>=1} (y-1)^d*x^d/(1 - x^d)). - Andrew Howroyd, Feb 02 2021

Terms a(11) and beyond from Andrew Howroyd, Feb 02 2021

A357180 First run-length of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 2, 3, 5, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 6, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 2

Offset: 0

Gus Wiseman, Sep 24 2022

nonn

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

			Composition 87 in standard order is (2,2,1,1,1), so a(87) = 2.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
For parts instead of run-lengths we have A065120, last A001511.
The version for Heinz numbers of partitions is A067029, last A071178.
This is the first part of row n of A333769.
For minimal instead of first we have A357138, maximal A357137.
The last instead of first run-length is A357181.
A051903 gives maximal part in prime signature.
A061395 gives maximal prime index.
A124767 counts runs in standard compositions.
A286470 gives maximal difference of prime indices.
A333766 gives maximal part of standard compositions, minimal A333768.
A353847 ranks run-sums of standard compositions.
Cf. A000120, A003754, A029931, A070939, A329395, A356841, A356844, A357134, A357135, A357136.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[If[n==0,0,First[Length/@Split[stc[n]]]],{n,0,100}]

A337506 Triangle read by rows where T(n,k) is the number of length-n sequences covering an initial interval of positive integers with k maximal anti-runs.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 8, 4, 1, 0, 44, 24, 6, 1, 0, 308, 176, 48, 8, 1, 0, 2612, 1540, 440, 80, 10, 1, 0, 25988, 15672, 4620, 880, 120, 12, 1, 0, 296564, 181916, 54852, 10780, 1540, 168, 14, 1, 0, 3816548, 2372512, 727664, 146272, 21560, 2464, 224, 16, 1

Offset: 0

Gus Wiseman, Sep 06 2020

An anti-run is a sequence with no adjacent equal parts. The number of maximal anti-runs is one more than the number of adjacent equal parts.

			Triangle begins:
  1
  0      1
  0      2      1
  0      8      4      1
  0     44     24      6      1
  0    308    176     48      8      1
  0   2612   1540    440     80     10      1
  0  25988  15672   4620    880    120     12      1
  0 296564 181916  54852  10780   1540    168     14      1
Row n = 3 counts the following sequences (empty column indicated by dot):
  .  (1,2,1)  (1,1,2)  (1,1,1)
     (1,2,3)  (1,2,2)
     (1,3,2)  (2,1,1)
     (2,1,2)  (2,2,1)
     (2,1,3)
     (2,3,1)
     (3,1,2)
     (3,2,1)

Andrew Howroyd, Table of n, a(n) for n = 0..1325

A000670 gives row sums.
A005649 gives column k = 1.
A337507 gives column k = 2.
A337505 gives the diagonal n = 2*k.
A106356 is the version for compositions.
A238130/A238279/A333755 is the version for runs in compositions.
A335461 has the reversed rows (except zeros).
A003242 counts anti-run compositions.
A124762 counts adjacent equal terms in standard compositions.
A124767 counts maximal runs in standard compositions.
A333381 counts maximal anti-runs in standard compositions.
A333382 counts adjacent unequal terms in standard compositions.
A333489 ranks anti-run compositions.
A333769 gives maximal run-lengths in standard compositions.
A337565 gives maximal anti-run lengths in standard compositions.
Cf. A019472, A052841, A060223, A106351, A269134, A325535, A337564.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],Length[Split[#,UnsameQ]]==k&]],{n,0,5},{k,0,n}]

\\ here b(n) is A005649.
b(n) = {sum(k=0, n, stirling(n,k,2)*(k + 1)!)}
T(n,k)=if(n==0, k==0, b(n-k)*binomial(n-1,k-1)) \\ Andrew Howroyd, Dec 31 2020

T(n,k) = A005649(n-k) * binomial(n-1,k-1) for k > 0. - Andrew Howroyd, Dec 31 2020

Terms a(45) and beyond from Andrew Howroyd, Dec 31 2020

A357181 Last run-length of the n-th composition in standard order.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 5, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1

Offset: 0

Gus Wiseman, Sep 24 2022

nonn

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

			Composition 87 in standard order is (2,2,1,1,1), so a(87) = 3.

Gus Wiseman, Statistics, classes, and transformations of standard compositions

See link for sequences related to standard compositions.
For parts instead of run-lengths we have A001511, first A065120.
For Heinz numbers of partitions we have A071178, first A067029.
This is the last part of row n of A333769.
For maximal instead of last we have A357137, minimal A357138.
The first instead of last run-length is A357180.
A051903 gives maximal part of prime signature.
A061395 gives maximal prime index.
A124767 counts runs in standard compositions.
A286470 gives maximal difference of prime indices.
A333766 gives maximal part of standard composition, minimal A333768.
A353847 ranks run-sums of standard compositions.
Cf. A000120, A003754, A029931, A070939, A329395, A356841, A356844, A357134, A357135, A357136.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[If[n==0,0,Last[Length/@Split[stc[n]]]],{n,0,100}]

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[#]]&]

A337507 Number of length-n sequences covering an initial interval of positive integers with exactly two maximal anti-runs, or with one pair of adjacent equal parts.

Original entry on oeis.org

0, 0, 1, 4, 24, 176, 1540, 15672, 181916, 2372512, 34348932, 546674120, 9486840748, 178285201008, 3607174453844, 78177409231768, 1806934004612220, 44367502983673664, 1153334584544496676, 31643148872573831016

Offset: 0

Gus Wiseman, Sep 06 2020

nonn

An anti-run is a sequence with no adjacent equal parts. For example, the maximal anti-runs in (3,1,1,2,2,2,1) are ((3,1),(1,2),(2),(2,1)). In general, there is one more maximal anti-run than the number of pairs of adjacent equal parts.

			The a(4) = 24 sequences:
  (2,1,2,2)  (2,1,3,3)  (3,1,2,2)
  (2,2,1,2)  (2,3,3,1)  (3,2,2,1)
  (1,2,2,1)  (3,3,1,2)  (1,1,2,3)
  (2,1,1,2)  (3,3,2,1)  (1,1,3,2)
  (1,1,2,1)  (1,2,2,3)  (2,1,1,3)
  (1,2,1,1)  (1,3,2,2)  (2,3,1,1)
  (1,2,3,3)  (2,2,1,3)  (3,1,1,2)
  (1,3,3,2)  (2,2,3,1)  (3,2,1,1)

A002133 is the version for runs in partitions.
A106357 is the version for compositions.
A337506 has this as column k = 2.
A000670 counts patterns.
A005649 counts anti-run patterns.
A003242 counts anti-run compositions.
A106356 counts compositions by number of maximal anti-runs.
A124762 counts adjacent equal terms in standard compositions.
A124767 counts maximal runs in standard compositions.
A238130/A238279/A333755 count maximal runs in compositions.
A333381 counts maximal anti-runs in standard compositions.
A333382 counts adjacent unequal terms in standard compositions.
A333489 ranks anti-run compositions.
A333769 gives maximal run lengths in standard compositions.
A337565 gives maximal anti-run lengths in standard compositions.
Cf. A019472, A052841, A060223, A106351, A269134, A335461, A337505, A337564.

kv=2;
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],Length[Split[#,UnsameQ]]==kv&]],{n,0,6}]

a(n > 0) = (n - 1)*A005649(n - 2).

cp's OEIS Frontend

A337504 Number of compositions of 2*n with n maximal anti-runs.

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A357180 First run-length of the n-th composition in standard order.

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A337506 Triangle read by rows where T(n,k) is the number of length-n sequences covering an initial interval of positive integers with k maximal anti-runs.

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A357181 Last run-length of the n-th composition in standard order.

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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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A337507 Number of length-n sequences covering an initial interval of positive integers with exactly two maximal anti-runs, or with one pair of adjacent equal parts.

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