cp's OEIS Frontend

A334968 Number of possible sums of subsequences (not necessarily contiguous) of the n-th composition in standard order (A066099).

%I A334968 #10 Jun 04 2020 06:40:11
%S A334968 1,2,2,3,2,4,4,4,2,4,3,5,4,5,5,5,2,4,4,6,4,6,6,6,4,6,6,6,6,6,6,6,2,4,
%T A334968 4,6,3,7,7,7,4,7,4,7,7,7,7,7,4,6,7,7,7,7,7,7,6,7,7,7,7,7,7,7,2,4,4,6,
%U A334968 4,8,8,8,4,6,6,8,6,8,8,8,4,8,6,8,6,8,8
%N A334968 Number of possible sums of subsequences (not necessarily contiguous) of the n-th composition in standard order (A066099).
%C A334968 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.
%F A334968 a(n) = A299701(A333219(n)).
%e A334968 The 139th composition is (4,2,1,1), with possible sums of subsequences {0,1,2,3,4,5,6,7,8}, so a(139) = 9.
%e A334968 Triangle begins:
%e A334968   1
%e A334968   2
%e A334968   2 3
%e A334968   2 4 4 4
%e A334968   2 4 3 5 4 5 5 5
%e A334968   2 4 4 6 4 6 6 6 4 6 6 6 6 6 6 6
%e A334968   2 4 4 6 3 7 7 7 4 7 4 7 7 7 7 7 4 6 7 7 7 7 7 7 6 7 7 7 7 7 7 7
%t A334968 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A334968 Table[Length[Union[Total/@Subsets[stc[n]]]],{n,0,100}]
%Y A334968 Row lengths are A011782.
%Y A334968 Dominated by A124771 (number of contiguous subsequences).
%Y A334968 Dominates A333257 (the contiguous case).
%Y A334968 Dominated by A334299 (number of subsequences).
%Y A334968 Golomb rulers are counted by A169942 and ranked by A333222.
%Y A334968 Positive subset-sums of partitions are counted by A276024 and A299701.
%Y A334968 Knapsack partitions are counted by A108917 and ranked by A299702
%Y A334968 Knapsack compositions are counted by A325676 and ranked by A333223.
%Y A334968 Contiguous subsequence-sums are counted by A333224 and ranked by A333257.
%Y A334968 Knapsack compositions are counted by A334268 and ranked by A334967.
%Y A334968 Cf. A000120, A029931, A048793, A066099, A070939, A108917, A325769, A325770, A325778, A334300, A335279.
%K A334968 nonn,tabf
%O A334968 0,2
%A A334968 _Gus Wiseman_, Jun 02 2020