OFFSET
0,8
COMMENTS
The standard order of compositions is given by A066099.
A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of 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. This sequence gives the number of adjacent equal terms in the n-th composition in standard order. Alternatively, a(n) is one fewer than the number of maximal anti-runs in the same composition, where anti-runs are sequences without any adjacent equal terms. For example, the 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1) with anti-runs ((3,2,1,2),(2,1,2,5,1),(1),(1)), so a(1234567) = 4 - 1 = 3. - Gus Wiseman, Apr 08 2020
FORMULA
For a composition b(1),...,b(k), a(n) = Sum_{1<=i=1<k, b(i)=b(i+1)} 1.
For n > 0, a(n) = A333381(n) - 1. - Gus Wiseman, Apr 08 2020
EXAMPLE
Composition number 11 is 2,1,1; 2>1=1, so a(11) = 1.
The table starts:
0
0
0 1
0 0 0 2
0 0 1 1 0 0 1 3
0 0 0 1 0 1 0 2 0 0 1 1 1 1 2 4
0 0 0 1 1 0 0 2 0 0 2 2 0 0 1 3 0 0 0 1 0 1 0 2 1 1 2 2 2 2 3 5
MATHEMATICA
stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Length[Select[Partition[stc[n], 2, 1], SameQ@@#&]], {n, 0, 100}] (* Gus Wiseman, Apr 08 2020 *)
CROSSREFS
Anti-runs summing to n are counted by A003242(n).
A triangle counting maximal anti-runs of compositions is A106356.
A triangle counting maximal runs of compositions is A238279.
Partitions whose first differences are an anti-run are A238424.
All of the following pertain to compositions in standard order (A066099):
- Weakly decreasing runs are counted by A124765.
- Weakly increasing runs are counted by A124766.
- Equal runs are counted by A124767.
- Strictly increasing runs are counted by A124768.
- Strictly decreasing runs are counted by A124769.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Normal compositions are A333217.
- Adjacent unequal pairs are counted by A333382.
- Anti-runs are A333489.
KEYWORD
easy,nonn,tabf
AUTHOR
Franklin T. Adams-Watters, Nov 06 2006
STATUS
approved