A354904 Numbers k such that the k-th composition in standard order is not the sequence of run-sums of any other composition.

3, 7, 11, 14, 15, 19, 23, 27, 28, 29, 30, 31, 35, 39, 43, 46, 47, 51, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 67, 71, 75, 78, 79, 83, 87, 91, 92, 93, 94, 95, 99, 103, 107, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127

Offset

1,1

Comments

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.

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).

The first term k such that the k-th composition in standard order does not have ones sandwiching the same prime number an even number of times is k = 3221, corresponding to the composition (1,3,3,2,2,1).

Links

Table of n, a(n) for n=1..61.

Example

The terms and their corresponding compositions begin:
   3: (1,1)
   7: (1,1,1)
  11: (2,1,1)
  14: (1,1,2)
  15: (1,1,1,1)
  19: (3,1,1)
  23: (2,1,1,1)
  27: (1,2,1,1)
  28: (1,1,3)
  29: (1,1,2,1)
  30: (1,1,1,2)
  31: (1,1,1,1,1)

Mathematica

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

Crossrefs

The standard compositions used here are A066099, run-sums A353847/A353932.

These are the positions of zeros in A354578, firsts A354905.

These compositions are counted by A354909.

The complement is A354912, counted by A354910.

A003242 counts anti-run compositions, ranked by A333489.

A124767 counts runs in standard compositions.

A238279 and A333755 count compositions by number of runs.

A351014 counts distinct runs of standard compositions, firsts A351015.

A353852 ranks compositions with distinct run-sums, counted by A353850.

A353853-A353859 pertain to composition run-sum trajectory.

Cf. A000120, A000918, A005811, A029837, A333381, A334299, A353832, A353849, A353860, A354907.

Sequence in context: A243367 A145670 A004061 * A277009 A277019 A336407

Adjacent sequences: A354901 A354902 A354903 * A354905 A354906 A354907

Keyword

nonn

Author

Gus Wiseman, Jun 21 2022

Status

approved