[seqfan] Re: Self avoiding polygons

[hv at crypt.org]

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.

Hugo