A357186 -id:A357186 - cp's OEIS frontend

A357187 First differences A357186 = "Take the k-th composition in standard order for each part k of the n-th composition in standard order, then add up everything.".

1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, -1, 1, 0, 0, -1, 1, 0, 0, 0, 1, 0, 0, -1, 0, 1, 0, -1, 1, 0, 0, -2, 1, 1, 0, -1, 1, 0, 0, 0, 0, 1, 0, -1, 1, 0, 0, -2, 1, 0, 0, 0, 1, 0, 0, -1, 0, 1, 0, -1, 1, 0, 0, -3, 1, 1, 0, 0, 1, 0, 0, -1, 0, 1, 0, -1, 1, 0, 0, -1, 1, 0

Offset: 0

Gus Wiseman, Sep 28 2022

sign

Are there any terms > 1?

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.

			We have A357186(5) - A357186(4) = 3 - 2 = 1, so a(4) = 1.

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

See link for sequences related to standard compositions.
Positions of first appearances appear to all belong to A052955.
Differences of A357186 (row-sums of A357135).
The version for partitions is A357458, differences of A325033.
Cf. A000120, A029837, A029931, A048896, A058891, A070939, A133494, A333766, A357134, A357137.

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

a(n) = A357186(n + 1) - A357186(n).

A357135 Take the k-th composition in standard order for each part k of the n-th composition in standard order; then concatenate.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1

Offset: 0

Gus Wiseman, Sep 26 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.

			Triangle begins:
   0:
   1: 1
   2: 2
   3: 1 1
   4: 1 1
   5: 2 1
   6: 1 2
   7: 1 1 1
   8: 3
   9: 1 1 1
  10: 2 2
  11: 2 1 1
  12: 1 1 1
  13: 1 2 1
  14: 1 1 2
  15: 1 1 1 1

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

See link for sequences related to standard compositions.
Row n is the A357134(n)-th composition in standard order.
The version for Heinz numbers of partitions is A357139, cf. A003963.
Row sums are A357186, differences A357187.
Cf. A000120, A001511, A029931, A048896, A058891, A070939, A096111, A329395, A333766, A335404, A357137.

stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Join@@Table[Join@@stc/@stc[n],{n,0,30}]

Row n is the A357134(n)-th composition in standard order.

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[#]]&]

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.

cp's OEIS Frontend

A357187 First differences A357186 = "Take the k-th composition in standard order for each part k of the n-th composition in standard order, then add up everything.".

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A357135 Take the k-th composition in standard order for each part k of the n-th composition in standard order; then concatenate.

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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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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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