A373954 - OEIS

A373954

Excess run-compression of standard compositions. Sum of all parts minus sum of compressed parts of the n-th integer composition in standard order.

0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 1, 0, 0, 1, 3, 0, 0, 0, 1, 0, 2, 0, 2, 0, 0, 2, 1, 1, 1, 2, 4, 0, 0, 0, 1, 3, 0, 0, 2, 0, 0, 4, 3, 0, 0, 1, 3, 0, 0, 0, 1, 0, 2, 0, 2, 1, 1, 3, 2, 2, 2, 3, 5, 0, 0, 0, 1, 0, 0, 0, 2, 0, 3, 2, 1, 0, 0, 1, 3, 0, 0, 0, 1, 2, 4, 2

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OFFSET

0,8

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.

We define the (run-) compression of a sequence to be the anti-run obtained by reducing each run of repeated parts to a single part. Alternatively, compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has compression (1,2,1).

LINKS

Table of n, a(n) for n=0..86.

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

FORMULA

a(n) = A029837(n) - A373953(n).

EXAMPLE

The excess compression of (2,1,1,3) is 1, so a(92) = 1.

MATHEMATICA

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

Table[Total[stc[n]]-Total[First/@Split[stc[n]]], {n, 0, 100}]

CROSSREFS

For length instead of sum we have A124762, counted by A106356.