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%I #9 May 31 2022 22:26:37
%S 0,1,1,1,1,2,2,1,1,2,1,1,2,2,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,1,2,
%T 2,2,1,3,3,1,2,3,1,2,3,2,1,2,2,2,3,3,3,2,2,3,2,3,2,1,1,3,2,1,1,2,2,2,
%U 2,3,3,2,2,2,2,2,2,3,2,2,2,3,2,2,2,2,3
%N Number of distinct positive run-sums 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).
%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.
%H Mathematics Stack Exchange, <a href="https://math.stackexchange.com/q/87559">What is a sequence run? (answered 2011-12-01)</a>
%e Composition 462903 in standard order is (1,1,4,7,1,2,1,1,1), with run-sums (2,4,7,1,2,3), of which a(462903) = 5 are distinct.
%t stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t Table[Length[Union[Total/@Split[stc[n]]]],{n,0,100}]
%Y Counting repeated runs also gives A124767.
%Y Positions of first appearances are A246534.
%Y For distinct runs instead of run-sums we have A351014 (firsts A351015).
%Y A version for partitions is A353835, weak A353861.
%Y Positions of 1's are A353848, counted by A353851.
%Y The version for binary expansion is A353929 (firsts A353930).
%Y The run-sums themselves are listed by A353932, with A353849 distinct terms.
%Y For distinct run-lengths instead of run-sums we have A354579.
%Y A005811 counts runs in binary expansion.
%Y A066099 lists compositions in standard order.
%Y A165413 counts distinct run-lengths in binary expansion.
%Y A297770 counts distinct runs in binary expansion, firsts A350952.
%Y A353847 represents the run-sum transformation for compositions.
%Y A353853-A353859 pertain to composition run-sum trajectory.
%Y Distinct runs: A032020, A175413, A351013, A351018, A329739, A351290.
%Y Selected statistics of standard compositions:
%Y - Length is A000120.
%Y - Sum is A070939.
%Y - Heinz number is A333219.
%Y - Number of distinct parts is A334028.
%Y Selected classes of standard compositions:
%Y - Partitions are A114994, strict A333256.
%Y - Multisets are A225620, strict A333255.
%Y - Strict compositions are A233564.
%Y - Constant compositions are A272919.
%Y Cf. A003242, A044813, A071625, A238279, A329738, A333381, A333489, A333755, A353744, A353832, A353850, A353852, A353866.
%K nonn
%O 0,6
%A _Gus Wiseman_, May 30 2022
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