A353857 - OEIS

%I #6 Jun 03 2022 07:42:48

%S 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 138,139,142,143,168,170,174,175,184,186,187,232,238,239,248,250,251,

%U 255,256,292,316,487,511,512,528,543,682,750,955,1008,1023,1024,2047

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

%C 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 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 terms together with their binary expansions and corresponding compositions begin:

%e    1:        1  (1)

%e    2:       10  (2)

%e    3:       11  (1,1)

%e    4:      100  (3)

%e    7:      111  (1,1,1)

%e    8:     1000  (4)

%e   10:     1010  (2,2)

%e   11:     1011  (2,1,1)

%e   14:     1110  (1,1,2)

%e   15:     1111  (1,1,1,1)

%e   16:    10000  (5)

%e   31:    11111  (1,1,1,1,1)

%e   32:   100000  (6)

%e   36:   100100  (3,3)

%e   39:   100111  (3,1,1,1)

%e   42:   101010  (2,2,2)

%e   46:   101110  (2,1,1,2)

%e   59:   111011  (1,1,2,1,1)

%e   60:   111100  (1,1,1,3)

%e   63:   111111  (1,1,1,1,1,1)

%t stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Select[Range[100],Length[FixedPoint[Total/@Split[#]&,stc[#]]]==1&]

%Y The version for partitions is A353844.

%Y The trajectory length is A353854, firsts A072639, for partitions A353841.

%Y The last part of the trajectory is A353855, for partitions A353842.

%Y These compositions are counted by A353858.

%Y A005811 counts runs in binary expansion.

%Y A011782 counts compositions.

%Y A066099 lists compositions in standard order.

%Y A318928 gives runs-resistance of binary expansion.

%Y A325268 counts partitions by omicron, rank statistic A304465.

%Y A333627 ranks the run-lengths of standard compositions.

%Y A351014 counts distinct runs in standard compositions, firsts A351015.

%Y A353840-A353846 pertain to partition run-sum trajectory.

%Y A353847 represents composition run-sum transformation, partitions A353832.

%Y A353853-A353859 pertain to composition run-sum trajectory.

%Y A353932 lists run-sums of standard compositions.

%Y Cf. A237685, A238279, A304442, A325277, A333381, A333755, A353848, A353849, A353850, A353852.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jun 01 2022