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A353696 Numbers k such that the k-th composition in standard order (A066099) is empty, a singleton, or has run-lengths that are a consecutive subsequence that is already counted.

%I A353696 #7 May 23 2022 17:46:52
%S A353696 0,1,2,4,8,10,16,32,43,58,64,128,256,292,349,442,512,586,676,697,826,
%T A353696 1024,1210,1338,1393,1394,1396,1594,2048,2186,2234,2618,2696,2785,
%U A353696 2786,2792,3130,4096,4282,4410,4666,5178,5569,5570,5572,5576,5584,6202,8192
%N A353696 Numbers k such that the k-th composition in standard order (A066099) is empty, a singleton, or has run-lengths that are a consecutive subsequence that is already counted.
%C A353696 First differs from the non-consecutive version A353431 in lacking 22318, corresponding to the binary word 101011100101110 and standard composition (2,2,1,1,3,2,1,1,2), whose run-lengths (2,2,1,1,2,1) are a subsequence but not a consecutive subsequence.
%C A353696 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 A353696 The terms together with their corresponding compositions begin:
%e A353696     0: ()
%e A353696     1: (1)
%e A353696     2: (2)
%e A353696     4: (3)
%e A353696     8: (4)
%e A353696    10: (2,2)
%e A353696    16: (5)
%e A353696    32: (6)
%e A353696    43: (2,2,1,1)
%e A353696    58: (1,1,2,2)
%e A353696    64: (7)
%e A353696   128: (8)
%e A353696   256: (9)
%e A353696   292: (3,3,3)
%e A353696   349: (2,2,1,1,2,1)
%e A353696   442: (1,2,1,1,2,2)
%e A353696   512: (10)
%e A353696   586: (3,3,2,2)
%e A353696   676: (2,2,3,3)
%e A353696   697: (2,2,1,1,3,1)
%e A353696   826: (1,3,1,1,2,2)
%t A353696 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A353696 yoyQ[y_]:=Length[y]<=1||MemberQ[Join@@Table[Take[y,{i,j}],{i,Length[y]},{j,i,Length[y]}],Length/@Split[y]]&&yoyQ[Length/@Split[y]];
%t A353696 Select[Range[0,1000],yoyQ[stc[#]]&]
%Y A353696 Non-recursive non-consecutive for partitions: A325755, counted by A325702.
%Y A353696 Non-consecutive: A353431, counted by A353391.
%Y A353696 Non-consecutive for partitions: A353393, counted by A353426.
%Y A353696 Non-recursive non-consecutive: A353402, counted by A353390.
%Y A353696 Counted by: A353430.
%Y A353696 Non-recursive: A353432, counted by A353392.
%Y A353696 A005811 counts runs in binary expansion.
%Y A353696 A011782 counts compositions.
%Y A353696 A066099 lists compositions in standard order, run-lengths A333769.
%Y A353696 Statistics of standard compositions:
%Y A353696 - Length is A000120, sum A070939.
%Y A353696 - Runs are counted by A124767, distinct A351014.
%Y A353696 - Subsequences are counted by A334299, contiguous A124770/A124771.
%Y A353696 - Runs-resistance is A333628.
%Y A353696 Classes of standard compositions:
%Y A353696 - Partitions are A114994, strict A333255, multisets A225620, sets A333256.
%Y A353696 - Runs are A272919, counted by A000005.
%Y A353696 - Golomb rulers are A333222, counted by A169942.
%Y A353696 - Anti-runs are A333489, counted by A003242.
%Y A353696 Cf. A032020, A114640, A181819, A228351, A329739, A318928, A325705, A329738, A333224, A353427, A353403.
%K A353696 nonn
%O A353696 1,3
%A A353696 _Gus Wiseman_, May 22 2022