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
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
Compression of standard compositions is A373953.
Positions of ones are A373955.
A066099 lists the parts of all compositions in standard order.
A114901 counts compositions with no isolated parts.
A240085 counts compositions with no unique parts.
A333627 takes the rank of a composition to the rank of its run-lengths.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 27 2024
STATUS
approved