cp's OEIS Frontend

A353427 Numbers k such that the k-th composition in standard order has all run-lengths > 1.

%I A353427 #7 May 17 2022 07:24:41
%S A353427 0,3,7,10,15,31,36,42,43,58,63,87,122,127,136,147,170,171,175,228,234,
%T A353427 235,250,255,292,295,343,351,471,484,490,491,506,511,528,547,586,591,
%U A353427 676,682,683,687,698,703,904,915,938,939,943,983,996,1002,1003,1018
%N A353427 Numbers k such that the k-th composition in standard order has all run-lengths > 1.
%C A353427 The k-th composition in standard order (graded reverse-lexicographic, 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 A353427 The terms and corresponding compositions begin:
%e A353427      0: ()
%e A353427      3: (1,1)
%e A353427      7: (1,1,1)
%e A353427     10: (2,2)
%e A353427     15: (1,1,1,1)
%e A353427     31: (1,1,1,1,1)
%e A353427     36: (3,3)
%e A353427     42: (2,2,2)
%e A353427     43: (2,2,1,1)
%e A353427     58: (1,1,2,2)
%e A353427     63: (1,1,1,1,1,1)
%e A353427     87: (2,2,1,1,1)
%e A353427    122: (1,1,1,2,2)
%e A353427    127: (1,1,1,1,1,1,1)
%t A353427 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A353427 Select[Range[0,100],!MemberQ[Length/@Split[stc[#]],1]&]
%Y A353427 The version for partitions is A001694, counted by A007690.
%Y A353427 The version for parts instead of lengths is A022340, counted by A212804.
%Y A353427 These compositions are counted by A114901.
%Y A353427 A subset of A348612 (counted by A261983).
%Y A353427 The case of all run-lengths = 2 is A351011.
%Y A353427 The case of all run-lengths > 2 is counted by A353400.
%Y A353427 A005811 counts runs in binary expansion.
%Y A353427 A011782 counts compositions.
%Y A353427 A066099 lists compositions in standard order, reverse A228351.
%Y A353427 Statistics of standard compositions:
%Y A353427 - Length is A000120, sum A070939.
%Y A353427 - Runs are counted by A124767.
%Y A353427 - Runs-resistance is A333628.
%Y A353427 - Run-lengths are A333769.
%Y A353427 Cf. A044813, A128695, A165413, A240085, A244164, A274174, A318928, A333489, A333755, A353402, A353432.
%K A353427 nonn
%O A353427 1,2
%A A353427 _Gus Wiseman_, May 16 2022