Self-Avoiding Walk

A self-avoiding walk is a path from one point to another which never intersects itself. Such paths are usually considered to occur on lattices, so that steps are only allowed in a discrete number of directions and of certain lengths.

Consider a self-avoiding walk on a two-dimensional square
grid (i.e., a lattice path which never visits the same lattice point twice) which
starts at the origin, takes first step in the positive horizontal direction, and
is restricted to nonnegative grid points only. The number of such paths of
, 2, ... steps
are 1, 2, 5, 12, 30, 73, 183, 456, 1151, ... (OEIS A046170).

Similarly, consider a self-avoiding walk which starts at the origin, takes first step in the positive horizontal direction, is not restricted to nonnegative
grid points only, but which is restricted to take an up step before taking
the first down step. The number of such paths of , 2, ... steps
are 1, 2, 5, 13, 36, 98, 272, 740, 2034, ... (OEIS A046171).

Self-avoiding rook walks are walks on an grid which
start from
, end at
, and are composed
of only horizontal and vertical steps. The following table gives the first few numbers
of such walks for small
and
. The values for
, 2, ... are 2, 12, 184, 8512, 1262816,
... (OEIS A007764).
![]() | 2 | 3 | 4 | 5 | 6 |
2 | 2 | ||||
3 | 4 | 12 | |||
4 | 8 | 38 | 184 | ||
5 | 16 | 125 | 976 | 8512 | |
6 | 32 | 414 | 5382 | 79384 | 1262816 |
There are a number of known formulas for computing for small
. For example,
![]() |
(1)
|
There is a recurrence relation for , given by
,
,
,
, and
![]() |
(2)
|
for , as well as the generating
function
![]() |
(3)
|
(Abbott and Hanson 1978, Finch 2003).
A related sequence is the number of shapes which can be formed by bending a piece of wire of length in the plane, where bends are of 0 or
and the wire may cross itself
at right angles but not pass over itself. The number of shapes for wires of length
1, 2, ... are 1, 2, 4, 10, 24, 66, 176, 493, ... (OEIS A001997).

Consider a self-avoiding walk on a two-dimensional square
grid from one corner to another such that no two consecutive steps are in the same
direction. The number of such paths for
, 2, ... are
1, 2, 2, 4, 10, 36, 188, ... (OEIS A034165;
counting the number of paths on the
point "lattice"
as 1), and the maximum lengths of these paths are 0, 2, 4, 10, 12, 26, 36, ... (OEIS
A034166).