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

Offset: 0

Author

Gus Wiseman, Jun 27 2024

Keywords

• nonn

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).

Examples

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

Links

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

Crossrefs

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

The opposite for length is A124767, counted by A238279 and A333755.

Positions of zeros are A333489, counted by A003242.

Positions of nonzeros are A348612, counted by A131044.

Compositions counted by this statistic are A373951, opposite A373949.

Compression of standard compositions is A373953.

Positions of ones are A373955.

A037201 gives compression of first differences of primes, halved A373947.

A066099 lists the parts of all compositions in standard order.

A114901 counts compositions with no isolated parts.

A116861 counts partitions by this statistic, by length A116608.

A240085 counts compositions with no unique parts.

A333627 takes the rank of a composition to the rank of its run-lengths.

Cf. A070939, A238130, A238343, A272919, A285981, A333213, A333381, A333382.

Programs

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}]

Formula

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