A390675 - OEIS

A390675

Number of integer compositions whose first sums are the (2n)-th composition in standard order.

1, 2, 0, 3, 1, 0, 0, 4, 1, 1, 0, 0, 0, 0, 0, 5, 1, 2, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, 2, 0, 2, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 1, 2, 0, 3, 1, 0, 0, 2, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0

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OFFSET

1,2

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.

The first sums of a nonempty sequence (a, b, c, d, ...) are (a+b, b+c, c+d, ...).

LINKS

Table of n, a(n) for n=1..87.

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

FORMULA

a(2^k) = k+1.

EXAMPLE

The 68-th composition in standard order is (4,3), which is the first sums of both (2,2,1) and (3,1,2), so a(34) = 2.

MATHEMATICA

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

uncha[tar_, ini_]:=(cur={ini}; Do[AppendTo[cur, tar[[k]]-If[k==1, ini, cur[[k]]]], {k, Length[tar]}]; cur);

Table[Length[Select[Table[uncha[stc[n], i], {i, Max[stc[n]]}], Min@@#>=1&]], {n, 2, 100, 2}]

CROSSREFS

Positions of first appearances are A000079.

Positive terms (before bisection) are indicated by A390676, complement A390677.

Positions of 1 (before bisection) are A390745.

A011782 counts compositions.

A066099 lists all compositions in standard order.

A357213 counts compositions by sum of first sums.

A390432 lists first sums of standard compositions, rows ranked by A390568.

A390673 ranks compositions with distinct first sums, counted by A390567.

A390674 ranks compositions with equal first sums, counted by A342527.

Cf. A000120, A001511, A008965, A029837, A029931, A070939, A357135, A358133, A390448, A390678.

Sequence in context: A240606 A335889 A324379 * A035165 A290256 A303110

Adjacent sequences: A390672 A390673 A390674 * A390676 A390677 A390678

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 22 2025

STATUS

approved