|
%I #6 Mar 31 2020 10:54:35
%S 0,1,3,5,7,14,11,13,15,30,43,29,23,46,27,45,31,62,122,61,87,117,59,
%T 118,47,94,107,93,55,110,91,109,63,126,250,125,343,245,123,246,175,
%U 350,235,349,119,238,347,237,95,190,378,189,215,373,187,374,111,222,363
%N Least STC-number of a composition whose sequence of run-lengths has STC-number n.
%C All terms belong to A003754.
%C A composition of n is a finite sequence of positive integers summing to n. The composition with STC-number k (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.
%e The sequence together with the corresponding compositions begins:
%e 0: ()
%e 1: (1)
%e 3: (1,1)
%e 5: (2,1)
%e 7: (1,1,1)
%e 14: (1,1,2)
%e 11: (2,1,1)
%e 13: (1,2,1)
%e 15: (1,1,1,1)
%e 30: (1,1,1,2)
%e 43: (2,2,1,1)
%e 29: (1,1,2,1)
%e 23: (2,1,1,1)
%e 46: (2,1,1,2)
%e 27: (1,2,1,1)
%e 45: (2,1,2,1)
%e 31: (1,1,1,1,1)
%e 62: (1,1,1,1,2)
%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t seq=Table[Total[2^(Accumulate[Reverse[Length/@Split[stc[n]]]])]/2,{n,0,1000}];
%t Table[Position[seq,i][[1,1]],{i,First[Split[Union[seq],#1+1==#2&]]}]-1
%Y Position of first appearance of n in A333627.
%Y All of the following pertain to compositions in standard order (A066099):
%Y - The length is A000120.
%Y - Compositions without terms > 2 are A003754.
%Y - Compositions without ones are ranked by A022340.
%Y - The partial sums from the right are A048793.
%Y - The sum is A070939.
%Y - Adjacent equal pairs are counted by A124762.
%Y - Equal runs are counted by A124767.
%Y - Strict compositions are ranked by A233564.
%Y - The partial sums from the left are A272020.
%Y - Constant compositions are ranked by A272919.
%Y - Normal compositions are ranked by A333217.
%Y - Heinz number is A333219.
%Y - Anti-runs are counted by A333381.
%Y - Adjacent unequal pairs are counted by A333382.
%Y - Runs-resistance is A333628.
%Y - First appearances of run-resistances are A333629.
%Y Cf. A029931, A098504, A114994, A225620, A228351, A238279, A242882, A318928, A329744, A329747, A333489.
%K nonn
%O 0,3
%A _Gus Wiseman_, Mar 31 2020
|
