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A354906 Position of first appearance of n in A354579 = Number of distinct run-lengths of standard compositions.

%I A354906 #7 Jun 24 2022 17:19:49
%S A354906 0,1,11,119,5615,251871
%N A354906 Position of first appearance of n in A354579 = Number of distinct run-lengths of standard compositions.
%C A354906 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 A354906 The terms together with their corresponding compositions begin:
%e A354906        0: ()
%e A354906        1: (1)
%e A354906       11: (2,1,1)
%e A354906      119: (1,1,2,1,1,1)
%e A354906     5615: (2,2,1,1,1,2,1,1,1,1)
%e A354906   251871: (1,1,1,2,2,1,1,1,1,2,1,1,1,1,1)
%t A354906 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A354906 pd=Table[Length[Union[Length/@Split[stc[n]]]],{n,0,10000}];
%t A354906 Table[Position[pd,n][[1,1]]-1,{n,0,Max@@pd}]
%Y A354906 The standard compositions used here are A066099, run-sums A353847/A353932.
%Y A354906 The version for partitions is A006939, for run-sums A002110.
%Y A354906 For run-sums instead of run-lengths we have A246534 (firsts in A353849).
%Y A354906 For runs instead of run-lengths we have A351015 (firsts in A351014).
%Y A354906 These are the positions of first appearances in A354579.
%Y A354906 A005811 counts runs in binary expansion.
%Y A354906 A333627 ranks the run-lengths of standard compositions.
%Y A354906 A351596 ranks compositions with distinct run-lengths, counted by A329739.
%Y A354906 A353744 ranks compositions with equal run-lengths, counted by A329738.
%Y A354906 A353852 ranks compositions with distinct run-sums, counted by A353850.
%Y A354906 A353853-A353859 are sequences pertaining to composition run-sum trajectory.
%Y A354906 A353860 counts collapsible compositions.
%Y A354906 Cf. A003242, A029837, A071625, A124767, A182857, A238279/A333755, A325278, A333381, A333489, A333629.
%K A354906 nonn,more
%O A354906 0,3
%A A354906 _Gus Wiseman_, Jun 23 2022