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A353431 Numbers k such that the k-th composition in standard order is empty, a singleton, or has its own run-lengths as a subsequence (not necessarily consecutive) that is already counted.

%I A353431 #8 May 17 2022 17:48:00
%S A353431 0,1,2,4,8,10,16,32,43,58,64,128,256,292,349,442,512,586,676,697,826,
%T A353431 1024,1210,1338,1393,1394,1396,1594,2048,2186,2234,2618,2696,2785,
%U A353431 2786,2792,3130,4096,4282,4410,4666,5178,5569,5570,5572,5576,5584,6202,8192
%N A353431 Numbers k such that the k-th composition in standard order is empty, a singleton, or has its own run-lengths as a subsequence (not necessarily consecutive) that is already counted.
%C A353431 First differs from A353696 (the consecutive version) in having 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 subsequence but not a consecutive subsequence.
%C A353431 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 A353431 The initial terms, their binary expansions, and the corresponding standard compositions:
%e A353431      0:           0  ()
%e A353431      1:           1  (1)
%e A353431      2:          10  (2)
%e A353431      4:         100  (3)
%e A353431      8:        1000  (4)
%e A353431     10:        1010  (2,2)
%e A353431     16:       10000  (5)
%e A353431     32:      100000  (6)
%e A353431     43:      101011  (2,2,1,1)
%e A353431     58:      111010  (1,1,2,2)
%e A353431     64:     1000000  (7)
%e A353431    128:    10000000  (8)
%e A353431    256:   100000000  (9)
%e A353431    292:   100100100  (3,3,3)
%e A353431    349:   101011101  (2,2,1,1,2,1)
%e A353431    442:   110111010  (1,2,1,1,2,2)
%e A353431    512:  1000000000  (10)
%e A353431    586:  1001001010  (3,3,2,2)
%e A353431    676:  1010100100  (2,2,3,3)
%e A353431    697:  1010111001  (2,2,1,1,3,1)
%t A353431 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A353431 rorQ[y_]:=Length[y]<=1||MemberQ[Subsets[y],Length/@Split[y]]&& rorQ[Length/@Split[y]];
%t A353431 Select[Range[0,100],rorQ[stc[#]]&]
%Y A353431 The non-recursive version for partitions is A325755, counted by A325702.
%Y A353431 These compositions are counted by A353391.
%Y A353431 The version for partitions A353393, counted by A353426, w/o primes A353389.
%Y A353431 The non-recursive version is A353402, counted by A353390.
%Y A353431 The non-recursive consecutive case is A353432, counted by A353392.
%Y A353431 The consecutive case is A353696, counted by A353430.
%Y A353431 A005811 counts runs in binary expansion.
%Y A353431 A011782 counts compositions.
%Y A353431 A066099 lists compositions in standard order, rev A228351, run-lens A333769.
%Y A353431 A329738 counts uniform compositions, partitions A047966.
%Y A353431 Statistics of standard compositions:
%Y A353431 - Length is A000120, sum A070939.
%Y A353431 - Runs are counted by A124767, distinct A351014.
%Y A353431 - Subsequences are counted by A334299, contiguous A124770/A124771.
%Y A353431 - Runs-resistance is A333628.
%Y A353431 Classes of standard compositions:
%Y A353431 - Partitions are A114994, multisets A225620, strict A333255, sets A333256.
%Y A353431 - Constant compositions are A272919, counted by A000005.
%Y A353431 - Golomb rulers are A333222, counted by A169942.
%Y A353431 - Knapsack compositions are A333223, counted by A325676.
%Y A353431 - Anti-runs are A333489, counted by A003242.
%Y A353431 Cf. A032020, A044813, A114640, A165413, A181819, A329739, A318928, A325705, A333224, A353427, A353403.
%K A353431 nonn
%O A353431 1,3
%A A353431 _Gus Wiseman_, May 16 2022