%I #7 Sep 01 2021 03:08:55
%S 1,2,1,1,2,3,2,1,2,3,1,2,1,3,3,1,2,1,3,2,1,2,3,4,3,2,1,3,4,2,1,3,2,4,
%T 1,3,2,1,4,4,2,3,1,2,4,3,1,2,3,4,1,2,3,1,4,4,2,1,3,2,4,1,3,2,1,4,3,2,
%U 1,3,4,4,3,1,2,3,4,1,2,3,1,4,2,3,1,2,4,4,1,3,2,1,4,3,2,1,3,4,2,1,3,2,4,4,1,2,3,1,4,2,3,1,2,4,3,1,2,3,4
%N Front-to-back insertion-permutation sequence.
%C Contains every finite sequence of distinct numbers infinitely many times.
%F Write 1. Then place 2 before 1 and then 2 after 1, yielding 21 and 12, as well as the first 5 terms of the sequence. Next, generate the 6 permutations of 1, 2, 3 by inserting 3 into 21 and then 12, from front-to-back, like this: 321, 231, 213 then 213, 132, 123. Next, generate the 24 permutations of 1, 2, 3, 4 by inserting 4 into the permutations of 1, 2, 3. Continue forever.
%e The permutations can be written as
%e 1,
%e 21, 12,
%e 321, 231, 213, 312, 132, 123, etc.
%e Write them in order and insert commas.
%o (PARI) tabf(nn) = my(v=[[1]], w); print(v); for(n=2, nn, w=List([]); for(k=1, #v, for(i=1, n, listput(w, concat([v[k][1..i-1], n, v[k][i..n-1]])))); print(Vec(v=w))); \\ _Jinyuan Wang_, Sep 01 2021
%Y Cf. A030298, A098281.
%K nonn
%O 1,2
%A _Clark Kimberling_, Sep 01 2004