A self-avoiding walk is a path from one point to another which never intersects itself. Such paths are usually considered to occur on point lattices, so that steps are only allowed in a discrete number of directions and of certain lengths.
An interacting growth walk is a random walk generated by choosing each next step from the unvisited neighbors of the current
lattice point (Narasimhan et al. 2002). The
initial point is marked visited. Let 
be the available neighbors, and let 
count the previously visited neighbors of 
, excluding the current point.
With a contact-bias parameter 
, the local transition probability
is 
.
Positive 
favors additional contacts; 
chooses uniformly from 
. If 
is empty, the walk is trapped and that attempt ends. The probability of a prescribed path is the product of its
local transition probabilities, which generally depend on the growth history. Even
at 
,
this gives a different law from choosing uniformly among all self-avoiding walks
of a specified length. In the illustrated local configuration, 
and the two available steps have contact counts 1
and 0. Their transition probabilities are therefore
and 
.
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 grid points with nonnegative Cartesian coordinates. The numbers 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
grid points with nonnegative Cartesian coordinates, but which is
restricted to take an up step before taking the first down step. The numbers of such
paths of 
, 2, ... steps are 1, 2, 5, 13, 36, 98, 272, 740, 2034, ...
(OEIS A046171).
Self-avoiding rook walks are lattice paths on the 
array of lattice points 
, where 
and 
are positive integers.
They start at 
,
end at 
,
use unit horizontal or vertical steps, and never visit the same lattice
point twice. The following table gives the first few numbers 
of such paths for small
and 
.
The values for 
,
3, ... 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 
and 
. For example,
 |
(1)
|
for 
.
There is a recurrence relation for 
, given by 
, 
, 
, 
, and
 |
(2)
|
for 
,
as well as the generating function
 |
(3)
|
for 
(Abbott and Hanson 1978, Finch 2003).
A related sequence counts shapes formed by bending a piece of wire of length 
in the plane, where 
is a positive integer.
The wire is divided into 
unit line segments, and bends
of 0 or 
are allowed only at the 
division marks. The wire may cross itself at right
angles, but no two line segments may overlap
over a positive length. Shapes are counted up to translations,
rotations, reflections,
and reversal of the wire (Foster 1937, pp. 50-51). The numbers 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 an 
array of lattice points
from one corner to the opposite corner such that no two consecutive steps are in
the same direction. The numbers 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).
