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A354907 Number of distinct sums of contiguous constant subsequences (partial runs) of the n-th composition in standard order.

%I A354907 #6 Jun 24 2022 17:19:53
%S A354907 0,1,1,2,1,2,2,3,1,2,2,2,2,2,2,4,1,2,2,3,2,3,2,3,2,2,3,2,3,2,3,5,1,2,
%T A354907 2,3,2,3,3,3,2,3,3,3,3,2,2,4,2,2,3,3,3,3,2,3,3,3,3,2,3,3,4,6,1,2,2,3,
%U A354907 2,3,3,4,2,3,3,3,2,3,3,4,2,3,2,3,3,4,3
%N A354907 Number of distinct sums of contiguous constant subsequences (partial runs) of the n-th composition in standard order.
%C A354907 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 A354907 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).
%e A354907 Composition number 981 in standard order is (1,1,1,2,2,2,1), with partial runs (1), (2), (1,1), (2,2), (1,1,1), (2,2,2), with distinct sums {1,2,3,4,6}, so a(981) = 5.
%t A354907 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A354907 pre[y_]:=NestWhileList[Most,y,Length[#]>1&];
%t A354907 Table[Length[Union[Total/@Join@@pre/@Split[stc[n]]]],{n,0,100}]
%Y A354907 Positions of 1's are A000051.
%Y A354907 Positions of first appearances are A000079.
%Y A354907 The standard compositions used here are A066099, run-sums A353847/A353932.
%Y A354907 If we allow any subsequence we get A334968.
%Y A354907 The case of full runs is A353849, firsts A246534.
%Y A354907 A version for nonempty partitions is A353861, full A353835.
%Y A354907 Counting all distinct runs (instead of their distinct sums) gives A354582.
%Y A354907 A124767 counts runs in standard compositions.
%Y A354907 A238279 and A333755 count compositions by number of runs.
%Y A354907 A330036 counts distinct partial runs of prime indices, full A005811.
%Y A354907 A351014 counts distinct runs of standard compositions, firsts A351015.
%Y A354907 A353853-A353859 pertain to composition run-sum trajectory.
%Y A354907 A353860 counts collapsible compositions.
%Y A354907 A354584 lists run-sums of prime indices, rows ranked by A353832.
%Y A354907 Cf. A029837, A124771, A274174, A333381, A334299, A353864.
%K A354907 nonn
%O A354907 0,4
%A A354907 _Gus Wiseman_, Jun 23 2022