cp's OEIS Frontend

A354579 Number of distinct lengths of runs in the n-th composition in standard order.

%I A354579 #9 Jun 12 2022 22:52:33
%S A354579 0,1,1,1,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,2,1,2,1,2,1,1,2,2,2,2,2,1,1,1,
%T A354579 1,2,1,1,1,2,1,1,1,1,1,1,2,2,1,1,1,2,1,2,1,2,2,2,1,2,2,2,2,1,1,1,1,2,
%U A354579 1,1,1,2,1,2,2,2,1,1,2,2,1,1,1,2,2,2,2
%N A354579 Number of distinct lengths of runs in the n-th composition in standard order.
%C A354579 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 lengths (2,3,1,2).
%C A354579 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 A354579 The positions of first appearances together with the corresponding compositions begin:
%e A354579        1: (1)
%e A354579       11: (2,1,1)
%e A354579      119: (1,1,2,1,1,1)
%e A354579     5615: (2,2,1,1,1,2,1,1,1,1)
%e A354579   251871: (1,1,1,2,2,1,1,1,1,2,1,1,1,1,1)
%t A354579 stc[n_]:=Differences[Prepend[Join@@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A354579 Table[Length[Union[Length/@Split[stc[n]]]],{n,0,100}]
%Y A354579 Standard compositions are listed by A066099.
%Y A354579 The version for partitions is A071625.
%Y A354579 For runs instead of run-lengths we have A351014, firsts A351015.
%Y A354579 Positions of 0's and 1's are A353744, counted by A329738.
%Y A354579 For sums instead of lengths we have A353849, ones at A353848.
%Y A354579 Positions of first appearances are A354906.
%Y A354579 A003242 counts anti-run compositions, ranked by A333489.
%Y A354579 A005811 counts runs in binary expansion.
%Y A354579 A333627 ranks the run-lengths of standard compositions.
%Y A354579 A351596 ranks compositions with distinct run-lengths, counted by A329739.
%Y A354579 A353835 counts distinct run-sums of prime indices, weak A353861.
%Y A354579 A353847 ranks the run-sums of standard compositions.
%Y A354579 A353852 ranks compositions with distinct run-sums, counted by A353850.
%Y A354579 A353860 counts collapsible compositions.
%Y A354579 Cf. A029837, A124767, A181819, A238279, A333381, A333755, A353839, A353851.
%K A354579 nonn
%O A354579 0,12
%A A354579 _Gus Wiseman_, Jun 11 2022