A333766 -id:A333766 - cp's OEIS frontend

A357708 Numbers k such that the k-th composition in standard order has sum equal to twice its maximum part.

3, 10, 11, 13, 14, 36, 37, 38, 39, 41, 44, 50, 51, 52, 57, 60, 136, 137, 138, 139, 140, 141, 142, 143, 145, 152, 162, 163, 168, 177, 184, 196, 197, 198, 199, 200, 209, 216, 226, 227, 232, 241, 248, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539

Offset: 1

Gus Wiseman, Oct 14 2022

nonn

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 and corresponding compositions begin:
    3: (1,1)
   10: (2,2)
   11: (2,1,1)
   13: (1,2,1)
   14: (1,1,2)
   36: (3,3)
   37: (3,2,1)
   38: (3,1,2)
   39: (3,1,1,1)
   41: (2,3,1)
   44: (2,1,3)
   50: (1,3,2)
   51: (1,3,1,1)

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

See link for sequences related to standard compositions.
A066311 lists gapless numbers.
A124767 counts runs in standard compositions.
A333766 gives maximal part of standard compositions, minimal A333768.
A356844 ranks compositions with at least one 1.
Cf. A000120, A001511, A003754, A029931, A329395.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,1000],Max@@stc[#]==Total[stc[#]]/2&]

A358138 Difference between maximum and minimum part in the n-th composition in standard order.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 31 2022

nonn

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.

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

See link for sequences related to standard compositions.
The first and last parts are A065120 and A001511, difference A358135.
This is the maximum minus minimum part in row n of A066099.
The version for Heinz numbers of partitions is A243055.
The maximum and minimum parts are A333766 and A333768.
The partial sums of standard compositions are A358134, adjusted A242628.
A011782 counts compositions.
A351014 counts distinct runs in standard compositions.
Cf. A000120, A029837, A029931, A048896, A058891, A070939, A133494, A329395, A357187, A358133.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Table[Max[stc[n]]-Min[stc[n]],{n,1,100}]

a(n) = A333766(n) - A333768(n).

A358330 By concatenating the standard compositions of each part of the a(n)-th standard composition, we get a weakly increasing sequence.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 12, 14, 15, 18, 19, 24, 25, 26, 28, 30, 31, 32, 36, 38, 39, 40, 42, 50, 51, 56, 57, 58, 60, 62, 63, 64, 72, 73, 74, 76, 78, 79, 96, 100, 102, 103, 104, 106, 114, 115, 120, 121, 122, 124, 126, 127, 128, 129, 130, 136, 146, 147

Offset: 1

Gus Wiseman, Nov 10 2022

nonn

Note we shorten the language, "the k-th composition in standard order," to "the standard composition of k."

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 standard compositions begin:
   0: ()
   1: (1)
   2: (2)
   3: (1,1)
   4: (3)
   6: (1,2)
   7: (1,1,1)
   8: (4)
   9: (3,1)
  10: (2,2)
  12: (1,3)
  14: (1,1,2)
  15: (1,1,1,1)
  18: (3,2)
  19: (3,1,1)
  24: (1,4)
  25: (1,3,1)
  26: (1,2,2)
For example, the 532,488-th composition is (6,10,4), with standard compositions ((1,2),(2,2),(3)), with weakly increasing concatenation (1,2,2,2,3), so 532,488 is in the sequence.

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

See link for sequences related to standard compositions.
Standard compositions are listed by A066099.
Indices of rows of A357135 (ranked by A357134) that are weakly increasing.
Cf. A000120, A001511, A029931, A048896, A058891, A070939, A333766, A335404, A357137, A357186.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],OrderedQ[Join@@stc/@stc[#]]&]

A349152 Standard composition numbers of compositions into divisors. Numbers k such that all parts of the k-th composition in standard order are divisors of the sum of parts.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 10, 11, 13, 14, 15, 16, 31, 32, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 127, 128, 136, 138, 139, 141, 142, 143, 162, 163, 168, 170, 171, 173, 174, 175, 177, 181, 182, 183, 184

Offset: 1

Gus Wiseman, Nov 15 2021

nonn

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 and corresponding compositions begin:
      0: ()              36: (3,3)           54: (1,2,1,2)
      1: (1)             37: (3,2,1)         55: (1,2,1,1,1)
      2: (2)             38: (3,1,2)         57: (1,1,3,1)
      3: (1,1)           39: (3,1,1,1)       58: (1,1,2,2)
      4: (3)             41: (2,3,1)         59: (1,1,2,1,1)
      7: (1,1,1)         42: (2,2,2)         60: (1,1,1,3)
      8: (4)             43: (2,2,1,1)       61: (1,1,1,2,1)
     10: (2,2)           44: (2,1,3)         62: (1,1,1,1,2)
     11: (2,1,1)         45: (2,1,2,1)       63: (1,1,1,1,1,1)
     13: (1,2,1)         46: (2,1,1,2)       64: (7)
     14: (1,1,2)         47: (2,1,1,1,1)    127: (1,1,1,1,1,1,1)
     15: (1,1,1,1)       50: (1,3,2)        128: (8)
     16: (5)             51: (1,3,1,1)      136: (4,4)
     31: (1,1,1,1,1)     52: (1,2,3)        138: (4,2,2)
     32: (6)             53: (1,2,2,1)      139: (4,2,1,1)

Looking at length instead of parts gives A096199.
These composition are counted by A100346.
A version counting subsets instead of compositions is A125297.
An unordered version is A326841, counted by A018818.
A011782 counts compositions.
A316413 ranks partitions with sum divisible by length, counted by A067538.
A319333 ranks partitions with sum equal to lcm, counted by A074761.
Cf. A003242, A056239, A238279, A326836, A326842.
Statistics of standard compositions:
- The compositions themselves are the rows of A066099.
- Number of parts is given by A000120, distinct A334028.
- Sum and product of parts are given by A070939 and A124758.
- Maximum and minimum parts are given by A333766 and A333768.
Classes of standard compositions:
- Partitions and strict partitions are ranked by A114994 and A333256.
- Multisets and sets are ranked by A225620 and A333255.
- Strict and constant compositions are ranked by A233564 and A272919.
- Permutations are ranked by A333218.
- Relatively prime compositions are ranked by A291166*, complement A291165.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],#==0||Divisible[Total[stc[#]],LCM@@stc[#]]&]

A358333 By concatenating the standard compositions for each part of the n-th standard composition, we get a sequence of length a(n). Row-lengths of A357135.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Nov 10 2022

nonn

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 92 in standard order is (2,1,1,3), with compositions ((2),(1),(1),(1,1)) so a(92) = 5.

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

See link for sequences related to standard compositions (A066099).
Dominates A000120.
Row-lengths of A357135, which is ranked by A357134.
A related sequence is A358330.
Cf. A001511, A029931, A048896, A058891, A070939, A096111, A333766, A357137, A357139, A357186, A357187.

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

Sum of A000120 over row n of A066099.

A358525 Number of distinct permutations of the n-th composition in standard order.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 1, 3, 2, 3, 3, 1, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 3, 4, 3, 4, 4, 1, 1, 2, 2, 3, 1, 6, 6, 4, 2, 6, 1, 6, 6, 6, 6, 5, 2, 3, 6, 4, 6, 6, 6, 5, 3, 4, 6, 5, 4, 5, 5, 1, 1, 2, 2, 3, 2, 6, 6, 4, 2, 3, 3, 12, 3, 12, 12, 5, 2, 6, 3, 12, 3, 4

Offset: 0

Gus Wiseman, Nov 21 2022

nonn

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 a(45) = 6 permutations are: (2121), (2112), (2211), (1221), (1212), (1122).

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

See link for sequences related to standard compositions.
Positions of 1's are A272919.
Cf. A000120, A048896, A070939, A329395, A333766, A358371, A358372, A358379, A358505, A358507, A358508.

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

cp's OEIS Frontend

A357708 Numbers k such that the k-th composition in standard order has sum equal to twice its maximum part.

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A358138 Difference between maximum and minimum part in the n-th composition in standard order.

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A358330 By concatenating the standard compositions of each part of the a(n)-th standard composition, we get a weakly increasing sequence.

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A349152 Standard composition numbers of compositions into divisors. Numbers k such that all parts of the k-th composition in standard order are divisors of the sum of parts.

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A358333 By concatenating the standard compositions for each part of the n-th standard composition, we get a sequence of length a(n). Row-lengths of A357135.

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A358525 Number of distinct permutations of the n-th composition in standard order.

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