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A353853 Trajectory of the composition run-sum transformation (or condensation) of n, using standard composition numbers.

%I A353853 #7 Jun 01 2022 08:22:02
%S A353853 0,1,2,3,2,4,5,6,7,4,8,9,10,8,11,10,8,12,13,14,10,8,15,8,16,17,18,19,
%T A353853 18,20,21,17,22,23,20,24,25,26,24,27,26,24,28,20,29,21,17,30,18,31,16,
%U A353853 32,33,34,35,34,36,32,37,38,39,36,32,40,41,42,32
%N A353853 Trajectory of the composition run-sum transformation (or condensation) of n, using standard composition numbers.
%C A353853 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).
%C A353853 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.
%C A353853 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 given in row 11 corresponds to the trajectory (2,1,1) -> (2,2) -> (4).
%e A353853 Triangle begins:
%e A353853    0
%e A353853    1
%e A353853    2
%e A353853    3  2
%e A353853    4
%e A353853    5
%e A353853    6
%e A353853    7  4
%e A353853    8
%e A353853    9
%e A353853   10  8
%e A353853   11 10  8
%e A353853   12
%e A353853   13
%e A353853   14 10  8
%e A353853 For example, the trajectory of 29 is 29 -> 21 -> 17, corresponding to the compositions (1,1,2,1) -> (2,2,1) -> (4,1).
%t A353853 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A353853 stcinv[q_]:=Total[2^(Accumulate[Reverse[q]])]/2;
%t A353853 Table[NestWhileList[stcinv[Total/@Split[stc[#]]]&,n,MatchQ[stc[#],{___,x_,x_,___}]&],{n,0,50}]
%Y A353853 These sequences for partitions are A353840-A353846.
%Y A353853 This is the iteration of A353847, with partition version A353832.
%Y A353853 Row-lengths are A353854, counted by A353859.
%Y A353853 Final terms are A353855.
%Y A353853 Counting rows by weight of final term gives A353856.
%Y A353853 Rows ending in a power of 2 are A353857, counted by A353858.
%Y A353853 A003242 counts anti-run compositions, ranked by A333489, complement A261983.
%Y A353853 A005811 counts runs in binary expansion.
%Y A353853 A011782 counts compositions.
%Y A353853 A066099 lists compositions in standard order.
%Y A353853 A318928 gives runs-resistance of binary expansion.
%Y A353853 A329739 counts compositions with all distinct run-lengths.
%Y A353853 A333627 ranks the run-lengths of standard compositions.
%Y A353853 A351014 counts distinct runs in standard compositions, firsts A351015.
%Y A353853 A353853-A353859 pertain to composition run-sum trajectory.
%Y A353853 A353929 counts distinct runs in binary expansion, firsts A353930.
%Y A353853 A353932 lists run-sums of standard compositions.
%Y A353853 Cf. A237685, A238279, A304442, A325277, A333381, A333755, A353833, A353848, A353849, A353850, A353852, A353860.
%K A353853 nonn
%O A353853 0,3
%A A353853 _Gus Wiseman_, Jun 01 2022