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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.
2

%I #7 May 23 2022 17:46:52

%S 0,1,2,4,8,10,16,32,43,58,64,128,256,292,349,442,512,586,676,697,826,

%T 1024,1210,1338,1393,1394,1396,1594,2048,2186,2234,2618,2696,2785,

%U 2786,2792,3130,4096,4282,4410,4666,5178,5569,5570,5572,5576,5584,6202,8192

%N 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 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 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 The terms together with their corresponding compositions begin:

%e 0: ()

%e 1: (1)

%e 2: (2)

%e 4: (3)

%e 8: (4)

%e 10: (2,2)

%e 16: (5)

%e 32: (6)

%e 43: (2,2,1,1)

%e 58: (1,1,2,2)

%e 64: (7)

%e 128: (8)

%e 256: (9)

%e 292: (3,3,3)

%e 349: (2,2,1,1,2,1)

%e 442: (1,2,1,1,2,2)

%e 512: (10)

%e 586: (3,3,2,2)

%e 676: (2,2,3,3)

%e 697: (2,2,1,1,3,1)

%e 826: (1,3,1,1,2,2)

%t stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t 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 Select[Range[0,1000],yoyQ[stc[#]]&]

%Y Non-recursive non-consecutive for partitions: A325755, counted by A325702.

%Y Non-consecutive: A353431, counted by A353391.

%Y Non-consecutive for partitions: A353393, counted by A353426.

%Y Non-recursive non-consecutive: A353402, counted by A353390.

%Y Counted by: A353430.

%Y Non-recursive: A353432, counted by A353392.

%Y A005811 counts runs in binary expansion.

%Y A011782 counts compositions.

%Y A066099 lists compositions in standard order, run-lengths A333769.

%Y Statistics of standard compositions:

%Y - Length is A000120, sum A070939.

%Y - Runs are counted by A124767, distinct A351014.

%Y - Subsequences are counted by A334299, contiguous A124770/A124771.

%Y - Runs-resistance is A333628.

%Y Classes of standard compositions:

%Y - Partitions are A114994, strict A333255, multisets A225620, sets A333256.

%Y - Runs are A272919, counted by A000005.

%Y - Golomb rulers are A333222, counted by A169942.

%Y - Anti-runs are A333489, counted by A003242.

%Y Cf. A032020, A114640, A181819, A228351, A329739, A318928, A325705, A329738, A333224, A353427, A353403.

%K nonn

%O 1,3

%A _Gus Wiseman_, May 22 2022