[seqfan] Re: Self avoiding polygons

[hv at crypt.org]

I wrote:
:John Mason <masonmilan33 at gmail.com> wrote:
::Hi seqfans.
::I would be grateful if anyone could tell me of an existing proof of the following assertion, relative to self-avoiding polygons on the square lattice (aka the boundaries of "profane" polyominoes):
::
::Premise: one way to build such a polygon of perimeter 2n+2 is to take a polygon of perimeter 2n, find within that polygon a segment of length s, and "shift" a component of that segment, of integral length, "outwards" by one unit, but only if such a manoeuvre does not touch, even at a corner, any existing part of the polygon.
::So for example:
::OOO
::OOO
::OOO
::Can generate (among others):
::OOO
::OOOO
::OOOO
::
::Assertion: any self-avoiding polygon on the square lattice, of size 2n+2, for n > 2, may be generated from some polygon of size 2n by using the above-described procedure.
:
:This is not my sort of area, but my immediate thought is that a square would
:not be generated by that procedure, since all its antecedents would have
:perimeter >= that of the square.

Oh, immediately after sending that I reread the original and saw that it
wants to shift "a segment of length s"; my comment was based on assuming
a segment of length 1, so apologies for the noise.

Hugo