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A353857 Numbers k such that the k-th composition in standard order has run-sum trajectory ending in a singleton.

%I A353857 #6 Jun 03 2022 07:42:48
%S A353857 1,2,3,4,7,8,10,11,14,15,16,31,32,36,39,42,46,59,60,63,64,127,128,136,
%T A353857 138,139,142,143,168,170,174,175,184,186,187,232,238,239,248,250,251,
%U A353857 255,256,292,316,487,511,512,528,543,682,750,955,1008,1023,1024,2047
%N A353857 Numbers k such that the k-th composition in standard order has run-sum trajectory ending in a singleton.
%C A353857 Every sequence can be uniquely split into a sequence of non-overlapping runs. 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). The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353847) until the rank of an anti-run is reached. For example, the trajectory 11 -> 10 -> 8 corresponds to the trajectory (2,1,1) -> (2,2) -> (4).
%C A353857 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 A353857 The terms together with their binary expansions and corresponding compositions begin:
%e A353857    1:        1  (1)
%e A353857    2:       10  (2)
%e A353857    3:       11  (1,1)
%e A353857    4:      100  (3)
%e A353857    7:      111  (1,1,1)
%e A353857    8:     1000  (4)
%e A353857   10:     1010  (2,2)
%e A353857   11:     1011  (2,1,1)
%e A353857   14:     1110  (1,1,2)
%e A353857   15:     1111  (1,1,1,1)
%e A353857   16:    10000  (5)
%e A353857   31:    11111  (1,1,1,1,1)
%e A353857   32:   100000  (6)
%e A353857   36:   100100  (3,3)
%e A353857   39:   100111  (3,1,1,1)
%e A353857   42:   101010  (2,2,2)
%e A353857   46:   101110  (2,1,1,2)
%e A353857   59:   111011  (1,1,2,1,1)
%e A353857   60:   111100  (1,1,1,3)
%e A353857   63:   111111  (1,1,1,1,1,1)
%t A353857 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A353857 Select[Range[100],Length[FixedPoint[Total/@Split[#]&,stc[#]]]==1&]
%Y A353857 The version for partitions is A353844.
%Y A353857 The trajectory length is A353854, firsts A072639, for partitions A353841.
%Y A353857 The last part of the trajectory is A353855, for partitions A353842.
%Y A353857 These compositions are counted by A353858.
%Y A353857 A005811 counts runs in binary expansion.
%Y A353857 A011782 counts compositions.
%Y A353857 A066099 lists compositions in standard order.
%Y A353857 A318928 gives runs-resistance of binary expansion.
%Y A353857 A325268 counts partitions by omicron, rank statistic A304465.
%Y A353857 A333627 ranks the run-lengths of standard compositions.
%Y A353857 A351014 counts distinct runs in standard compositions, firsts A351015.
%Y A353857 A353840-A353846 pertain to partition run-sum trajectory.
%Y A353857 A353847 represents composition run-sum transformation, partitions A353832.
%Y A353857 A353853-A353859 pertain to composition run-sum trajectory.
%Y A353857 A353932 lists run-sums of standard compositions.
%Y A353857 Cf. A237685, A238279, A304442, A325277, A333381, A333755, A353848, A353849, A353850, A353852.
%K A353857 nonn
%O A353857 1,2
%A A353857 _Gus Wiseman_, Jun 01 2022