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Number of level runs for compositions in standard order.
174

%I #8 Apr 18 2020 00:04:10

%S 0,1,1,1,1,2,2,1,1,2,1,2,2,3,2,1,1,2,2,2,2,2,3,2,2,3,2,3,2,3,2,1,1,2,

%T 2,2,1,3,3,2,2,3,1,2,3,4,3,2,2,3,3,3,3,3,4,3,2,3,2,3,2,3,2,1,1,2,2,2,

%U 2,3,3,2,2,2,2,3,3,4,3,2,2,3,3,3,2,2,3,2,3,4,3,4,3,4,3,2,2,3,3,3,2,4,4,3,3

%N Number of level runs for compositions in standard order.

%C The standard order of compositions is given by A066099.

%C For n > 0, a(n) is one more than the number of adjacent unequal terms in the n-th composition in standard order. Also the number of runs in the same composition. - _Gus Wiseman_, Apr 08 2020

%F a(0) = 0, a(n) = 1 + Sum_{1<=i=1<k, b(i)!=b(i+1)} 1 for n > 0.

%F For n > 0, a(n) = A333382(n) + 1. - _Gus Wiseman_, Apr 08 2020

%e Composition number 11 is 2,1,1; the level runs are 2; 1,1; so a(11) = 2.

%e The table starts:

%e 0

%e 1

%e 1 1

%e 1 2 2 1

%e 1 2 1 2 2 3 2 1

%e 1 2 2 2 2 2 3 2 2 3 2 3 2 3 2 1

%e 1 2 2 2 1 3 3 2 2 3 1 2 3 4 3 2 2 3 3 3 3 3 4 3 2 3 2 3 2 3 2 1

%e The 1234567th composition in standard order is (3,2,1,2,2,1,2,5,1,1,1) with runs ((3),(2),(1),(2,2),(1),(2),(5),(1,1,1)), so a(1234567) = 8. - _Gus Wiseman_, Apr 08 2020

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Table[Length[Split[stc[n]]],{n,0,100}] (* _Gus Wiseman_, Apr 17 2020 *)

%Y Row-lengths are A011782.

%Y Compositions counted by number of runs are A238279 or A333755.

%Y All of the following pertain to compositions in standard order (A066099):

%Y - Length is A000120.

%Y - Sum is A070939.

%Y - Weakly decreasing compositions are A114994.

%Y - Adjacent equal pairs are counted by A124762.

%Y - Weakly decreasing runs are counted by A124765.

%Y - Weakly increasing runs are counted by A124766.

%Y - Equal runs are counted by A124767 (this sequence).

%Y - Weakly increasing compositions are A225620.

%Y - Strict compositions A233564.

%Y - Constant compositions are A272919.

%Y - Anti-runs are counted by A333381.

%Y - Adjacent unequal pairs are counted by A333382.

%Y - Anti-run compositions are A333489.

%Y - Runs-resistance is A333628.

%Y - Run-lengths are A333769 (triangle).

%Y Cf. A029931, A048793, A066099, A106356, A228351, A318928, A329744, A333217, A333219, A333627.

%K easy,nonn,tabf

%O 0,6

%A _Franklin T. Adams-Watters_, Nov 06 2006