This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A360302 #39 Feb 06 2023 10:18:31
%S A360302 0,0,1,0,1,2,3,0,1,2,4,3,5,6,7,0,1,2,5,3,6,8,11,4,7,9,12,10,13,14,15,
%T A360302 0,1,2,6,3,7,10,16,4,8,11,17,13,19,22,26,5,9,12,18,14,20,23,27,15,21,
%U A360302 24,28,25,29,30,31,0,1,2,7,3,8,12,22,4,9,13,23,16
%N A360302 T(n,k) is the position of the set encoded in the binary expansion of k within the shortlex order for the powerset of [n]; triangle T(n,k), n>=0, 0<=k<=2^n-1, read by rows.
%C A360302 In shortlex order for 2^[n] the subsets are primarily sorted by cardinality and then into lexicographical order.
%C A360302 The set encoded by k consists of the indices of 1-bits (rightmost index is 1).
%C A360302 Row n is a permutation of {0, 1, ..., 2^n-1} whose inverse is in row n of A359941.
%H A360302 Alois P. Heinz, <a href="/A360302/b360302.txt">Rows n = 0..13, flattened</a>
%H A360302 Wikipedia, <a href="https://en.wikipedia.org/wiki/Shortlex_order">Shortlex order</a>
%F A360302 T(n,A359941(n,k)) = k = A359941(n,T(n,k)).
%e A360302 The subsets of [4] listed in shortlex order (starting at position 0) are: {}, {1}, {2}, {3}, {4}, {1,2}, {1,3}, {1,4}, {2,3}, {2,4}, {3,4}, {1,2,3}, {1,2,4}, {1,3,4}, {2,3,4}, {1,2,3,4}.
%e A360302 T(4,0) = T(4,0000_2) = 0: {} is at position 0.
%e A360302 T(4,3) = T(4,0011_2) = 5: {1,2} is at position 5.
%e A360302 T(4,6) = T(4,0110_2) = 8: {2,3} is at position 8.
%e A360302 T(4,7) = T(4,0111_2) = 11: {1,2,3} is at position 11.
%e A360302 T(4,15) = T(4,1111_2) = 15: {1,2,3,4} is at position 15.
%e A360302 Triangle T(n,k) begins:
%e A360302 0;
%e A360302 0, 1;
%e A360302 0, 1, 2, 3;
%e A360302 0, 1, 2, 4, 3, 5, 6, 7;
%e A360302 0, 1, 2, 5, 3, 6, 8, 11, 4, 7, 9, 12, 10, 13, 14, 15;
%e A360302 ...
%p A360302 T:= proc(n) option remember; local h, i, l;
%p A360302 l:= map(x-> add(2^(i-1), i=x),
%p A360302 [seq(combinat[choose]([$1..n], i)[], i=0..n)]);
%p A360302 h(0):=0; for i to nops(l) do h(l[i]):= (i-1) od:
%p A360302 seq(h(i), i=0..2^n-1)
%p A360302 end:
%p A360302 seq(T(n), n=0..6);
%Y A360302 Columns k=0-1 give: A000004, A057427.
%Y A360302 Row sums give A006516(n) = A000217(A000225(n)).
%Y A360302 Row lengths are A000079.
%Y A360302 Cf. A082185, A193360, A359941.
%K A360302 nonn,look,tabf
%O A360302 0,6
%A A360302 _Alois P. Heinz_, Feb 03 2023