A375125 - OEIS

A375125

Strictly increasing run-leader transformation for standard compositions.

0, 1, 2, 3, 4, 5, 1, 7, 8, 9, 10, 11, 1, 3, 3, 15, 16, 17, 18, 19, 2, 21, 5, 23, 1, 3, 6, 7, 3, 7, 7, 31, 32, 33, 34, 35, 36, 37, 9, 39, 2, 5, 42, 43, 5, 11, 11, 47, 1, 3, 6, 7, 1, 13, 3, 15, 3, 7, 14, 15, 7, 15, 15, 63, 64, 65, 66, 67, 68, 69, 17, 71, 4, 73

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OFFSET

0,3

COMMENTS

The a(n)-th composition in standard order lists the leaders of strictly increasing runs in the n-th composition in standard order.

The leaders of strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing 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..73.

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

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

FORMULA

A000120(a(n)) = A124768(n).

A065120(a(n)) = A065120(n).

A070939(a(n)) = A374684(n).

EXAMPLE

The 813th composition in standard order is (1,3,2,1,2,1), with strictly increasing runs ((1,3),(2),(1,2),(1)), with leaders (1,2,1,1). This is the 27th composition in standard order, so a(813) = 27.

MATHEMATICA

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

stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;

Table[stcinv[First/@Split[stc[n], Less]], {n, 0, 100}]

CROSSREFS

Positions of elements of A233564 are A374698, counted by A374687.

Positions of elements of A272919 are A374685, counted by A374686.

Ranks of rows of A374683.

The weak version is A375123.

The weak opposite version is A375124.

The opposite version is A375126.

Other transformations: A375127, A373948.