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.
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, run-compression removes all parts equal to the part immediately to their left. For example, (1,1,2,2,1) has run-compression (1,2,1).
LINKS
EXAMPLE
The standard compositions and their run-compressions begin:
0: () --> ()
1: (1) --> (1)
2: (2) --> (2)
3: (1,1) --> (1)
4: (3) --> (3)
5: (2,1) --> (2,1)
6: (1,2) --> (1,2)
7: (1,1,1) --> (1)
8: (4) --> (4)
9: (3,1) --> (3,1)
10: (2,2) --> (2)
11: (2,1,1) --> (2,1)
12: (1,3) --> (1,3)
13: (1,2,1) --> (1,2,1)
14: (1,1,2) --> (1,2)
15: (1,1,1,1) --> (1)
MATHEMATICA
stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[First/@Split[stc[n]], {n, 100}]
CROSSREFS
Last column is A001511.
First column is A065120.
Row-lengths are A124767.
Row n has A334028(n) distinct elements.
Rows are ranked by A373948 (standard order).
Row-sums are A373953.
A007947 (squarefree kernel) represents run-compression of multisets.
A066099 lists the parts of compositions in standard order.
A116861 counts partitions by sum of run-compression.
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Jul 09 2024
STATUS
approved