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A353848 Numbers k such that the k-th composition in standard order (row k of A066099) has all equal run-sums.

%I A353848 #6 May 31 2022 11:38:54
%S A353848 0,1,2,3,4,7,8,10,11,14,15,16,31,32,36,39,42,46,59,60,63,64,127,128,
%T A353848 136,138,143,168,170,175,187,238,248,250,255,256,292,316,487,511,512,
%U A353848 528,543,682,750,955,1008,1023,1024,2047,2048,2080,2084,2090,2111,2184
%N A353848 Numbers k such that the k-th composition in standard order (row k of A066099) has all equal run-sums.
%C A353848 Every sequence can be uniquely split into non-overlapping runs, read left-to-right. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4).
%C A353848 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.
%H A353848 Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%F A353848 A353849(a(n)) = 1.
%e A353848 The terms together with their binary expansions and corresponding compositions begin:
%e A353848      0:       0  ()
%e A353848      1:       1  (1)
%e A353848      2:      10  (2)
%e A353848      3:      11  (1,1)
%e A353848      4:     100  (3)
%e A353848      7:     111  (1,1,1)
%e A353848      8:    1000  (4)
%e A353848     10:    1010  (2,2)
%e A353848     11:    1011  (2,1,1)
%e A353848     14:    1110  (1,1,2)
%e A353848     15:    1111  (1,1,1,1)
%e A353848     16:   10000  (5)
%e A353848     31:   11111  (1,1,1,1,1)
%e A353848     32:  100000  (6)
%e A353848     36:  100100  (3,3)
%e A353848     39:  100111  (3,1,1,1)
%e A353848     42:  101010  (2,2,2)
%e A353848     46:  101110  (2,1,1,2)
%e A353848     59:  111011  (1,1,2,1,1)
%e A353848     60:  111100  (1,1,1,3)
%e A353848 For example:
%e A353848 - The 59th composition in standard order is (1,1,2,1,1), with run-sums (2,2,2), so 59 is in the sequence.
%e A353848 - The 2298th composition in standard order is (4,1,1,1,1,2,2), with run-sums (4,4,4), so 2298 is in the sequence.
%e A353848 - The 2346th composition in standard order is (3,3,2,2,2), with run-sums (6,6), so 2346 is in the sequence.
%t A353848 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A353848 Select[Range[0,100],SameQ@@Total/@Split[stc[#]]&]
%Y A353848 Standard compositions are listed by A066099.
%Y A353848 For equal lengths instead of sums we have A353744, counted by A329738.
%Y A353848 The version for partitions is A353833, counted by A304442.
%Y A353848 These compositions are counted by A353851.
%Y A353848 The distinct instead of equal version is A353852, counted by A353850.
%Y A353848 The run-sums themselves are listed by A353932, with A353849 distinct terms.
%Y A353848 A005811 counts runs in binary expansion.
%Y A353848 A300273 ranks collapsible partitions, counted by A275870.
%Y A353848 A351014 counts distinct runs in standard compositions, firsts A351015.
%Y A353848 A353840-A353846 pertain to partition run-sum trajectory.
%Y A353848 A353847 represents the run-sum transformation for compositions.
%Y A353848 A353853-A353859 pertain to composition run-sum trajectory.
%Y A353848 A353860 counts collapsible compositions.
%Y A353848 A353863 counts run-sum-complete partitions.
%Y A353848 Cf. A003242, A047966, A106356, A140690, A238279, A274174, A333381, A333489, A333755, A353832, A353864.
%K A353848 nonn
%O A353848 0,3
%A A353848 _Gus Wiseman_, May 30 2022