A374757 Irregular triangle read by rows where row n lists the leaders of strictly decreasing runs in the n-th composition in standard order.

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

Offset

0,2

Comments

The leaders of strictly decreasing runs in a sequence are obtained by splitting it into maximal strictly decreasing subsequences and taking the first term of each.

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.

Links

Table of n, a(n) for n=0..83.

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Example

the 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1), with strictly decreasing runs ((3,2,1),(2),(2,1),(2),(5,1),(1),(1)), so row 1234567 is (3,2,2,2,5,1,1).
The nonnegative integers, corresponding compositions, and leaders of strictly decreasing runs begin:
    0:      () -> ()        15: (1,1,1,1) -> (1,1,1,1)
    1:     (1) -> (1)       16:       (5) -> (5)
    2:     (2) -> (2)       17:     (4,1) -> (4)
    3:   (1,1) -> (1,1)     18:     (3,2) -> (3)
    4:     (3) -> (3)       19:   (3,1,1) -> (3,1)
    5:   (2,1) -> (2)       20:     (2,3) -> (2,3)
    6:   (1,2) -> (1,2)     21:   (2,2,1) -> (2,2)
    7: (1,1,1) -> (1,1,1)   22:   (2,1,2) -> (2,2)
    8:     (4) -> (4)       23: (2,1,1,1) -> (2,1,1)
    9:   (3,1) -> (3)       24:     (1,4) -> (1,4)
   10:   (2,2) -> (2,2)     25:   (1,3,1) -> (1,3)
   11: (2,1,1) -> (2,1)     26:   (1,2,2) -> (1,2,2)
   12:   (1,3) -> (1,3)     27: (1,2,1,1) -> (1,2,1)
   13: (1,2,1) -> (1,2)     28:   (1,1,3) -> (1,1,3)
   14: (1,1,2) -> (1,1,2)   29: (1,1,2,1) -> (1,1,2)

Mathematica

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

Crossrefs

Row-leaders of nonempty rows are A065120.

Row-lengths are A124769.

The opposite version is A374683, sum A374684, length A124768.

The weak version is A374740, sum A374741, length A124765.

Row-sums are A374758.

Positions of identical rows are A374759 (counted by A374760).

Positions of distinct (strict) rows are A374767 (counted by A374761).

All of the following pertain to compositions in standard order:

- Length is A000120.

- Sum is A029837(n+1).

- Parts are listed by A066099.

- Number of adjacent equal pairs is A124762, unequal A333382.

- Run-length transform is A333627, sum A070939.

- Run-compression transform is A373948, sum A373953, excess A373954.

- Ranks of contiguous compositions are A374249, counted by A274174.

- Ranks of non-contiguous compositions are A374253, counted by A335548.

Six types of runs:

- Count: A124766, A124765, A124768, A124769, A333381, A124767.

- Leaders: A374629, A374740, A374683, A374757, A374515, A374251.

- Rank: A375123, A375124, A375125, A375126, A375127, A373948.

Cf. A051903, A106356, A188920, A189076, A233564, A238343, A333213, A373949, A374685, A374698, A374700, A374706.

Sequence in context: A308477 A079216 A181654 * A323756 A192710 A316831

Adjacent sequences: A374754 A374755 A374756 * A374758 A374759 A374760

Keyword

nonn

Author

Gus Wiseman, Jul 29 2024

Status

approved