%I #7 Jun 24 2022 17:19:49

%S 0,1,11,119,5615,251871

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

%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.

%e The terms together with their corresponding compositions begin:

%e 0: ()

%e 1: (1)

%e 11: (2,1,1)

%e 119: (1,1,2,1,1,1)

%e 5615: (2,2,1,1,1,2,1,1,1,1)

%e 251871: (1,1,1,2,2,1,1,1,1,2,1,1,1,1,1)

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t pd=Table[Length[Union[Length/@Split[stc[n]]]],{n,0,10000}];

%t Table[Position[pd,n][[1,1]]-1,{n,0,Max@@pd}]

%Y The standard compositions used here are A066099, run-sums A353847/A353932.

%Y The version for partitions is A006939, for run-sums A002110.

%Y For run-sums instead of run-lengths we have A246534 (firsts in A353849).

%Y For runs instead of run-lengths we have A351015 (firsts in A351014).

%Y These are the positions of first appearances in A354579.

%Y A005811 counts runs in binary expansion.

%Y A333627 ranks the run-lengths of standard compositions.

%Y A351596 ranks compositions with distinct run-lengths, counted by A329739.

%Y A353744 ranks compositions with equal run-lengths, counted by A329738.

%Y A353852 ranks compositions with distinct run-sums, counted by A353850.

%Y A353853-A353859 are sequences pertaining to composition run-sum trajectory.

%Y A353860 counts collapsible compositions.

%Y Cf. A003242, A029837, A071625, A124767, A182857, A238279/A333755, A325278, A333381, A333489, A333629.

%K nonn,more

%O 0,3

%A _Gus Wiseman_, Jun 23 2022