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%I #9 Jun 03 2022 07:42:39
%S 0,1,2,2,4,5,6,4,8,9,8,8,12,13,8,8,16,17,18,18,20,17,22,20,24,25,24,
%T 24,20,17,18,16,32,33,34,34,32,37,38,32,40,41,32,34,44,45,32,40,48,49,
%U 50,50,52,49,54,52,40,41,40,32,32,37,34,32,64,65,66,66,68
%N Last term of the trajectory of the composition run-sum transformation (condensation) of the n-th composition in standard order.
%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, corresponding to (2,1,1) -> (2,2) -> (4), has last term a(11) = 8.
%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 trajectory 139 -> 138 -> 136 -> 128 ends with a(139) = 128.
%t stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t Table[Total[2^Accumulate[Reverse[FixedPoint[Total/@Split[#]&,stc[n]]]]/2],{n,0,100}]
%Y The version for partitions is A353842.
%Y This trajectory has length A353854, firsts A072639, partitions A353841.
%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 a partition's run-sum trajectory.
%Y A353847 represents a composition's run-sums, partitions A353832.
%Y A353853-A353859 pertain to a composition's run-sum trajectory.
%Y Cf. A237685, A304442, A333381, A333489, A333755, A353848, A353849, A353850, A353852, A353860.
%K nonn
%O 0,3
%A _Gus Wiseman_, Jun 01 2022
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