A098281 Back-to-front insertion-permutation sequence.

1, 1, 2, 2, 1, 1, 2, 3, 1, 3, 2, 3, 1, 2, 2, 1, 3, 2, 3, 1, 3, 2, 1, 1, 2, 3, 4, 1, 2, 4, 3, 1, 4, 2, 3, 4, 1, 2, 3, 1, 3, 2, 4, 1, 3, 4, 2, 1, 4, 3, 2, 4, 1, 3, 2, 3, 1, 2, 4, 3, 1, 4, 2, 3, 4, 1, 2, 4, 3, 1, 2, 2, 1, 3, 4, 2, 1, 4, 3, 2, 4, 1, 3, 4, 2, 1, 3, 2, 3, 1, 4, 2, 3, 4, 1, 2, 4, 3, 1, 4, 2, 3, 1, 3, 2, 1, 4, 3, 2, 4, 1, 3, 4, 2, 1, 4, 3, 2, 1

Offset

1,3

Comments

Contains every finite sequence of distinct numbers infinitely many times.

Links

Table of n, a(n) for n=1..119.

Formula

Write 1. Then place 2 after 1 and then 2 before 1, yielding 12 and 21, as well as the first 5 terms of the sequence. Next, generate the 6 permutations of 1, 2, 3 by inserting 3 into 12 and then 21, from back-to-front, like this: 123, 132, 312 then 213, 231, 321. Next, generate the 24 permutations of 1, 2, 3, 4 by inserting 4 into the permutations of 1, 2, 3. Continue forever.

Example

The permutations can be written as
1,
12, 21,
123, 132, 312, 213, 231, 321, etc.
Write them in order and insert commas.

Mathematica

perms[n_] := perms[n] = If[n == 1, {{1}}, Flatten[Table[Insert[#, n, pos], {pos, -1, -n, -1}]& /@ perms[n-1], 1]];
Table[perms[n], {n, 1, 4}] // Flatten (* Jean-François Alcover, Sep 02 2021 *)

Prog

(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..n-i], n, v[k][n-i+1..n-1]])))); print(Vec(v=w))); \\ Jinyuan Wang, Aug 31 2021

Crossrefs

Cf. A030298, A098280.

Sequence in context: A182592 A030298 A370221 * A207324 A352620 A103343

Adjacent sequences: A098278 A098279 A098280 * A098282 A098283 A098284

Keyword

nonn,tabf

Author

Clark Kimberling, Sep 01 2004

Status

approved