A353744 - OEIS

%I #7 Jun 12 2022 22:52:21

%S 0,1,2,3,4,5,6,7,8,9,10,12,13,15,16,17,18,20,22,24,25,31,32,33,34,36,

%T 37,38,40,41,42,43,44,45,48,49,50,52,54,58,63,64,65,66,68,69,70,72,76,

%U 77,80,81,82,88,89,96,97,98,101,102,104,105,108,109,127,128

%N Numbers k such that the k-th composition in standard order has all equal run-lengths.

%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 Composition 2362 in standard order is (3,3,1,1,2,2), with run-lengths (2,2,2), so 2362 is in the sequence.

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

%t Select[Range[0,100],SameQ@@Length/@Split[stc[#]]&]

%Y Standard compositions are listed by A066099.

%Y The version for partitions is A072774, counted by A047966.

%Y These compositions are counted by A329738.

%Y For distinct instead of equal run-lengths we have A351596.

%Y For run-sums instead of lengths we have A353848, counted by A353851.

%Y For distinct run-sums we have A353852, counted by A353850.

%Y A003242 counts anti-run compositions, ranked by A333489.

%Y A005811 counts runs in binary expansion.

%Y A300273 ranks collapsible partitions, counted by A275870.

%Y A353838 ranks partitions with all distinct run-sums, counted by A353837.

%Y A353847 represents the composition run-sum transformation.

%Y A353853-A353859 pertain to composition run-sum trajectory.

%Y A353860 counts collapsible compositions.

%Y A353833 ranks partitions with all equal run-sums, counted by A304442.

%Y Cf. A029837, A071625, A124767, A175413, A238279, A333381, A333755, A353834, A353849.

%K nonn

%O 1,3

%A _Gus Wiseman_, Jun 11 2022