cp's OEIS Frontend

A124767 Number of level runs for compositions in standard order.

%I A124767 #8 Apr 18 2020 00:04:10
%S A124767 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 A124767 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 A124767 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 A124767 Number of level runs for compositions in standard order.
%C A124767 The standard order of compositions is given by A066099.
%C A124767 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 A124767 a(0) = 0, a(n) = 1 + Sum_{1<=i=1<k, b(i)!=b(i+1)} 1 for n > 0.
%F A124767 For n > 0, a(n) = A333382(n) + 1. - _Gus Wiseman_, Apr 08 2020
%e A124767 Composition number 11 is 2,1,1; the level runs are 2; 1,1; so a(11) = 2.
%e A124767 The table starts:
%e A124767   0
%e A124767   1
%e A124767   1 1
%e A124767   1 2 2 1
%e A124767   1 2 1 2 2 3 2 1
%e A124767   1 2 2 2 2 2 3 2 2 3 2 3 2 3 2 1
%e A124767   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 A124767 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 A124767 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A124767 Table[Length[Split[stc[n]]],{n,0,100}] (* _Gus Wiseman_, Apr 17 2020 *)
%Y A124767 Row-lengths are A011782.
%Y A124767 Compositions counted by number of runs are A238279 or A333755.
%Y A124767 All of the following pertain to compositions in standard order (A066099):
%Y A124767 - Length is A000120.
%Y A124767 - Sum is A070939.
%Y A124767 - Weakly decreasing compositions are A114994.
%Y A124767 - Adjacent equal pairs are counted by A124762.
%Y A124767 - Weakly decreasing runs are counted by A124765.
%Y A124767 - Weakly increasing runs are counted by A124766.
%Y A124767 - Equal runs are counted by A124767 (this sequence).
%Y A124767 - Weakly increasing compositions are A225620.
%Y A124767 - Strict compositions A233564.
%Y A124767 - Constant compositions are A272919.
%Y A124767 - Anti-runs are counted by A333381.
%Y A124767 - Adjacent unequal pairs are counted by A333382.
%Y A124767 - Anti-run compositions are A333489.
%Y A124767 - Runs-resistance is A333628.
%Y A124767 - Run-lengths are A333769 (triangle).
%Y A124767 Cf. A029931, A048793, A066099, A106356, A228351, A318928, A329744, A333217, A333219, A333627.
%K A124767 easy,nonn,tabf
%O A124767 0,6
%A A124767 _Franklin T. Adams-Watters_, Nov 06 2006