cp's OEIS Frontend

A353932 Irregular triangle read by rows where row k lists the run-sums of the k-th composition in standard order.

%I A353932 #6 Jun 12 2022 22:52:25
%S A353932 1,2,2,3,2,1,1,2,3,4,3,1,4,2,2,1,3,1,2,1,2,2,4,5,4,1,3,2,3,2,2,3,4,1,
%T A353932 2,1,2,2,3,1,4,1,3,1,1,4,1,2,2,2,3,2,2,1,3,2,5,6,5,1,4,2,4,2,6,3,2,1,
%U A353932 3,1,2,3,3,2,4,2,3,1,6,4,2,2,1,3
%N A353932 Irregular triangle read by rows where row k lists the run-sums of the k-th composition in standard order.
%C A353932 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).
%C A353932 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 A353932 Triangle begins:
%e A353932   1
%e A353932   2
%e A353932   2
%e A353932   3
%e A353932   2 1
%e A353932   1 2
%e A353932   3
%e A353932   4
%e A353932   3 1
%e A353932   4
%e A353932   2 2
%e A353932   1 3
%e A353932   1 2 1
%e A353932 For example, composition 350 in standard order is (2,2,1,1,1,2), so row 350 is (4,3,2).
%t A353932 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A353932 Table[Total/@Split[stc[n]],{n,0,30}]
%Y A353932 Row-sums are A029837.
%Y A353932 Standard compositions are listed by A066099.
%Y A353932 Row-lengths are A124767.
%Y A353932 These compositions are ranked by A353847.
%Y A353932 Row k has A353849(k) distinct parts.
%Y A353932 The version for partitions is A354584, ranked by A353832.
%Y A353932 A005811 counts runs in binary expansion.
%Y A353932 A300273 ranks collapsible partitions, counted by A275870.
%Y A353932 A353838 ranks partitions with all distinct run-sums, counted by A353837.
%Y A353932 A353851 counts compositions with all equal run-sums, ranked by A353848.
%Y A353932 A353840-A353846 pertain to partition run-sum trajectory.
%Y A353932 A353852 ranks compositions with all distinct run-sums, counted by A353850.
%Y A353932 A353853-A353859 pertain to composition run-sum trajectory.
%Y A353932 A353860 counts collapsible compositions.
%Y A353932 Cf. A003242, A175413, A181819, A238279, A274174, A333381, A333489, A333755, A353835, A353839, A353863, A353864.
%K A353932 nonn,tabf
%O A353932 1,2
%A A353932 _Gus Wiseman_, Jun 10 2022