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%I #8 Nov 04 2021 11:09:11
%S 3,7,10,11,14,15,19,21,23,26,27,28,29,30,31,35,36,39,42,43,46,47,51,
%T 53,55,56,57,58,59,60,61,62,63,67,71,73,74,75,78,79,83,84,85,86,87,90,
%U 91,92,93,94,95,99,100,103,106,107,110,111,112,113,114,115,116
%N Numbers k such that the k-th composition in standard order is not an anti-run, i.e., has adjacent equal parts.
%C First differs from A345168 in lacking 37, corresponding to the composition (3,2,1).
%C 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.
%e The terms and corresponding standard compositions begin:
%e 3: (1,1) 35: (4,1,1) 61: (1,1,1,2,1)
%e 7: (1,1,1) 36: (3,3) 62: (1,1,1,1,2)
%e 10: (2,2) 39: (3,1,1,1) 63: (1,1,1,1,1,1)
%e 11: (2,1,1) 42: (2,2,2) 67: (5,1,1)
%e 14: (1,1,2) 43: (2,2,1,1) 71: (4,1,1,1)
%e 15: (1,1,1,1) 46: (2,1,1,2) 73: (3,3,1)
%e 19: (3,1,1) 47: (2,1,1,1,1) 74: (3,2,2)
%e 21: (2,2,1) 51: (1,3,1,1) 75: (3,2,1,1)
%e 23: (2,1,1,1) 53: (1,2,2,1) 78: (3,1,1,2)
%e 26: (1,2,2) 55: (1,2,1,1,1) 79: (3,1,1,1,1)
%e 27: (1,2,1,1) 56: (1,1,4) 83: (2,3,1,1)
%e 28: (1,1,3) 57: (1,1,3,1) 84: (2,2,3)
%e 29: (1,1,2,1) 58: (1,1,2,2) 85: (2,2,2,1)
%e 30: (1,1,1,2) 59: (1,1,2,1,1) 86: (2,2,1,2)
%e 31: (1,1,1,1,1) 60: (1,1,1,3) 87: (2,2,1,1,1)
%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t Select[Range[100],MatchQ[stc[#],{___,x_,x_,___}]&]
%Y Constant run compositions are counted by A000005, ranked by A272919.
%Y Counting these compositions by sum and length gives A131044.
%Y These compositions are counted by A261983.
%Y The complement is A333489, counted by A003242.
%Y The non-alternating case is A345168, complement A345167.
%Y A011782 counts compositions, strict A032020.
%Y A238279 counts compositions by sum and number of maximal runs.
%Y A274174 counts compositions with equal parts contiguous.
%Y A336107 counts non-anti-run permutations of prime factors.
%Y A345195 counts non-alternating anti-runs, ranked by A345169.
%Y For compositions in standard order (rows of A066099):
%Y - Length is A000120.
%Y - Sum is A070939
%Y - Maximal runs are counted by A124767.
%Y - Strict compositions are ranked by A233564.
%Y - Maximal anti-runs are counted by A333381.
%Y - Runs-resistance is A333628.
%Y Cf. A029931, A048793, A106356, A114901, A167606, A178470, A228351, A244164, A262046, A335452, A335464.
%K nonn
%O 1,1
%A _Gus Wiseman_, Nov 03 2021
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