cp's OEIS Frontend

A334299 Number of distinct subsequences (not necessarily contiguous) of compositions in standard order (A066099).

%I A334299 #10 Jun 04 2020 06:39:42
%S A334299 1,2,2,3,2,4,4,4,2,4,3,6,4,7,6,5,2,4,4,6,4,6,7,8,4,7,6,10,6,10,8,6,2,
%T A334299 4,4,6,3,8,8,8,4,8,4,9,8,12,11,10,4,7,8,10,8,11,12,13,6,10,9,14,8,13,
%U A334299 10,7,2,4,4,6,4,8,8,8,4,6,6,12,7,14,12,10,4
%N A334299 Number of distinct subsequences (not necessarily contiguous) of compositions in standard order (A066099).
%C A334299 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 A334299 a(n) = A334300(n) + 1.
%e A334299 Triangle begins:
%e A334299   1
%e A334299   2
%e A334299   2 3
%e A334299   2 4 4 4
%e A334299   2 4 3 6 4 7 6 5
%e A334299   2 4 4 6 4 6 7 8 4 7 6 10 6 10 8 6
%e A334299 If the k-th composition in standard order is c, then we say that the STC-number of c is k. The n-th column below lists the STC-numbers of the subsequences of the composition with STC-number n:
%e A334299   0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15
%e A334299      0  0  1  0  2  2  3  0  4   2   5   4   6   6   7
%e A334299            0     1  1  1     1   0   3   1   5   3   3
%e A334299                  0  0  0     0       2   0   3   2   1
%e A334299                                      1       2   1   0
%e A334299                                      0       1   0
%e A334299                                              0
%t A334299 stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
%t A334299 Table[Length[Union[Subsets[stc[n]]]],{n,0,100}]
%Y A334299 Row lengths are A011782.
%Y A334299 Looking only at contiguous subsequences gives A124771.
%Y A334299 Compositions where every subinterval has a different sum are A333222.
%Y A334299 Knapsack compositions are A333223.
%Y A334299 Contiguous positive subsequence-sums are counted by A333224.
%Y A334299 Contiguous subsequence-sums are counted by A333257.
%Y A334299 Disallowing empty subsequences gives A334300.
%Y A334299 Subsequence-sums are counted by A334968.
%Y A334299 Cf. A000120, A029931, A048793, A066099, A070939, A108917, A325676, A334967.
%K A334299 nonn
%O A334299 0,2
%A A334299 _Gus Wiseman_, Jun 01 2020