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A353432 Numbers k such that the k-th composition in standard order has its own run-lengths as a consecutive subsequence.

%I A353432 #10 May 17 2022 17:47:30
%S A353432 0,1,10,21,26,43,58,107,117,174,186,292,314,346,348,349,373,430,442,
%T A353432 570,585,586,629,676,696,697,804,826,860,861,885,1082,1141,1173,1210,
%U A353432 1338,1387,1392,1393,1394,1396,1594,1653,1700,1720,1721,1882,2106,2165,2186
%N A353432 Numbers k such that the k-th composition in standard order has its own run-lengths as a consecutive subsequence.
%C A353432 First differs from A353402 (the non-consecutive version) in lacking 53.
%C A353432 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 A353432 The initial terms, their binary expansions, and the corresponding standard compositions:
%e A353432      0:          0  ()
%e A353432      1:          1  (1)
%e A353432     10:       1010  (2,2)
%e A353432     21:      10101  (2,2,1)
%e A353432     26:      11010  (1,2,2)
%e A353432     43:     101011  (2,2,1,1)
%e A353432     58:     111010  (1,1,2,2)
%e A353432    107:    1101011  (1,2,2,1,1)
%e A353432    117:    1110101  (1,1,2,2,1)
%e A353432    174:   10101110  (2,2,1,1,2)
%e A353432    186:   10111010  (2,1,1,2,2)
%e A353432    292:  100100100  (3,3,3)
%e A353432    314:  100111010  (3,1,1,2,2)
%e A353432    346:  101011010  (2,2,1,2,2)
%e A353432    348:  101011100  (2,2,1,1,3)
%e A353432    349:  101011101  (2,2,1,1,2,1)
%e A353432    373:  101110101  (2,1,1,2,2,1)
%e A353432    430:  110101110  (1,2,2,1,1,2)
%e A353432    442:  110111010  (1,2,1,1,2,2)
%t A353432 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A353432 rorQ[y_]:=Length[y]==0||MemberQ[Join@@Table[Take[y,{i,j}],{i,Length[y]},{j,i,Length[y]}],Length/@Split[y]];
%t A353432 Select[Range[0,10000],rorQ[stc[#]]&]
%Y A353432 These compositions are counted by A353392.
%Y A353432 This is the consecutive case of A353402, counted by A353390.
%Y A353432 The non-consecutive recursive version is A353431, counted by A353391.
%Y A353432 The recursive version is A353696, counted by A353430.
%Y A353432 A005811 counts runs in binary expansion.
%Y A353432 A011782 counts compositions.
%Y A353432 A066099 lists compositions in standard order, rev A228351, run-lens A333769.
%Y A353432 A329738 counts uniform compositions, partitions A047966.
%Y A353432 Statistics of standard compositions:
%Y A353432 - Length is A000120, sum A070939.
%Y A353432 - Runs are counted by A124767, distinct A351014.
%Y A353432 - Subsequences are counted by A334299, contiguous A124770/A124771.
%Y A353432 - Runs-resistance is A333628.
%Y A353432 Classes of standard compositions:
%Y A353432 - Partitions are A114994, strict A333255, rev A225620, strict rev A333256.
%Y A353432 - Runs are A272919, counted by A000005.
%Y A353432 - Golomb rulers are A333222, counted by A169942.
%Y A353432 - Anti-runs are A333489, counted by A003242.
%Y A353432 Cf. A044813, A165413, A181819, A318928, A325702, A325705, A325755, A333224, A333755, A353389, A353393, A353403.
%K A353432 nonn
%O A353432 1,3
%A A353432 _Gus Wiseman_, May 16 2022