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Numbers k such that the k-th composition in standard order has its own run-lengths as a subsequence (not necessarily consecutive).
10

%I #9 May 16 2022 17:24:15

%S 0,1,10,21,26,43,53,58,107,117,174,186,292,314,346,348,349,373,430,

%T 442,570,585,586,629,676,693,696,697,698,699,804,826,858,860,861,885,

%U 954,1082,1141,1173,1210,1338,1353,1387,1392,1393,1394,1396,1397,1398,1466

%N Numbers k such that the k-th composition in standard order has its own run-lengths as a subsequence (not necessarily consecutive).

%C First differs from A353432 (the consecutive case) in having 0 and 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 53: 110101 (1,2,2,1)

%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 rosQ[y_]:=Length[y]==0||MemberQ[Subsets[y],Length/@Split[y]];

%t Select[Range[0,100],rosQ[stc[#]]&]

%Y The version for partitions is A325755, counted by A325702.

%Y These compositions are counted by A353390.

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

%Y The consecutive case is A353432, counted by A353392.

%Y A005811 counts runs in binary expansion.

%Y A011782 counts compositions.

%Y A066099 lists compositions in standard order, reverse A228351.

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

%Y Words with all distinct run-lengths: A032020, A044813, A098859, A130091, A329739, A351017.

%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, consecutive 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.

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

%Y - Knapsack compositions are A333223, counted by A325676.

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

%Y Cf. A114640, A165413, A181819, A318928, A325705, A329738, A333224/A333257, A333755, A353393, A353403, A353430.

%K nonn

%O 0,3

%A _Gus Wiseman_, May 15 2022