OFFSET
0,3
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).
For the present sequence, the a(n)-th composition in standard order is obtained by compressing the n-th composition in standard order.
LINKS
EXAMPLE
The standard compositions and their compressions begin:
0: () --> 0: ()
1: (1) --> 1: (1)
2: (2) --> 2: (2)
3: (1,1) --> 1: (1)
4: (3) --> 4: (3)
5: (2,1) --> 5: (2,1)
6: (1,2) --> 6: (1,2)
7: (1,1,1) --> 1: (1)
8: (4) --> 8: (4)
9: (3,1) --> 9: (3,1)
10: (2,2) --> 2: (2)
11: (2,1,1) --> 5: (2,1)
12: (1,3) --> 12: (1,3)
13: (1,2,1) --> 13: (1,2,1)
14: (1,1,2) --> 6: (1,2)
15: (1,1,1,1) --> 1: (1)
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]]], {n, 0, 30}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 24 2024
STATUS
approved