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Numbers k such that the k-th composition in standard order has its own run-lengths as a consecutive subsequence.
9

%I #10 May 17 2022 17:47:30

%S 0,1,10,21,26,43,58,107,117,174,186,292,314,346,348,349,373,430,442,

%T 570,585,586,629,676,696,697,804,826,860,861,885,1082,1141,1173,1210,

%U 1338,1387,1392,1393,1394,1396,1594,1653,1700,1720,1721,1882,2106,2165,2186

%N Numbers k such that the k-th composition in standard order has its own run-lengths as a consecutive subsequence.

%C First differs from A353402 (the non-consecutive version) in lacking 53.

%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 initial terms, their binary expansions, and the corresponding standard compositions:

%e 0: 0 ()

%e 1: 1 (1)

%e 10: 1010 (2,2)

%e 21: 10101 (2,2,1)

%e 26: 11010 (1,2,2)

%e 43: 101011 (2,2,1,1)

%e 58: 111010 (1,1,2,2)

%e 107: 1101011 (1,2,2,1,1)

%e 117: 1110101 (1,1,2,2,1)

%e 174: 10101110 (2,2,1,1,2)

%e 186: 10111010 (2,1,1,2,2)

%e 292: 100100100 (3,3,3)

%e 314: 100111010 (3,1,1,2,2)

%e 346: 101011010 (2,2,1,2,2)

%e 348: 101011100 (2,2,1,1,3)

%e 349: 101011101 (2,2,1,1,2,1)

%e 373: 101110101 (2,1,1,2,2,1)

%e 430: 110101110 (1,2,2,1,1,2)

%e 442: 110111010 (1,2,1,1,2,2)

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

%t rorQ[y_]:=Length[y]==0||MemberQ[Join@@Table[Take[y,{i,j}],{i,Length[y]},{j,i,Length[y]}],Length/@Split[y]];

%t Select[Range[0,10000],rorQ[stc[#]]&]

%Y These compositions are counted by A353392.

%Y This is the consecutive case of A353402, counted by A353390.

%Y The non-consecutive recursive version is A353431, counted by A353391.

%Y The recursive version is A353696, counted by A353430.

%Y A005811 counts runs in binary expansion.

%Y A011782 counts compositions.

%Y A066099 lists compositions in standard order, rev A228351, run-lens A333769.

%Y A329738 counts uniform compositions, partitions A047966.

%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, rev A225620, strict rev 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. A044813, A165413, A181819, A318928, A325702, A325705, A325755, A333224, A333755, A353389, A353393, A353403.

%K nonn

%O 1,3

%A _Gus Wiseman_, May 16 2022