... grid¹

Due to limitations of space, we cannot show all of it here.

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... heuristic²

i.e. hand-waving.

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...machinery³

It could be a mathematical, virtual, quantum...it really doesn't matter.

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... way⁴

Other than giving up.

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... list⁵

Any method that is ``slower'' than this really is worthless.

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... evaluate''⁶

Since $c_n = \frac{1}{2\pi i} \oint \frac{f(q)}{q^{n+1}} {\mathrm{d}} q$, we can recover $c_n$ from the expression of $f(q)$ or $e(q)$.

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... bounds⁷

This topic is evolving rapidly; see Steve Finch's web page on mathematical constants for up-to-date information ( http://www.mathsoft.com/asolve/constant/constant.html )

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...space⁸

i.e. the amount of computer memory required to compute $c_n$ also grows exponentially with $n$.

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... Ideally⁹

In a perfect world where our brains are larger...well even this may not be enough; no-one has proved that a good solution exists.

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... frustrated¹⁰

This is an understatement of the extreme difficulty that has been encountered by those venturing into this area of enumerative combinatorics.

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... exactly¹¹

In which case $N$ is probably a small integer, probably well under $100$.

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... approximation¹²

In which case $N$ is probably much larger, but the error bounds on the estimates of the coefficients also become larger and larger with $N$.

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... size¹³

i.e. on average how wide are they? And how does this width change with $n$?

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...vice-versa¹⁴

Extracting asymptotic behaviour from a solution is not necessarily trivial -- indeed for the problems described in chapter 6, we expect the asymptotics to be rather simple, but the form of the solution makes proving this rather difficult.

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... lattice¹⁵

The square grid is dual to itself, while the dual of triangular grid is the hexagonal grid (and vice versa).

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... two¹⁶

Spiral walks on the square and triangular lattices [22,27,42,5,28] and $3$-choice polygons on the square lattice [8].

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... polyominoes¹⁷

One can also solve directed polyominoes whose cells lie on or below the line $y=x$; these arise in the solution of directed polyominoes.

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... lattices¹⁸

Using lattice duality square and hexagonally celled directed polyominoes are also solved.

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... below¹⁹

A cell is supported only from below if there is another cell directly below it, but no cell directly on its left.

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... polyominoes²⁰

Column-convex polyominoes are always polygons.

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... interesting²¹

On the square lattice there are $4$ directions that each bond may take, and then $3$ possible directions for the following bond (since immediate reversal of direction would violate self-avoidance), and so there are $12$ possible two-step configurations. The different TSRW models are obtained by allowing or disallowing each of these $12$ configurations. Hence there are $2^{12} = 4096$ different TSRW models on the square lattice! Most of these are essentially either zero- or one-dimensional.

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... model²²

Since there are both $60^\circ$ and $120^\circ$ turns on the triangular lattice, there are three possible models of spiral walks (depending on which of these turns is allowed). -- the solved model allows only $120^\circ$ turns.

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... polygons²³

Polygons that are both row-convex and column-convex are called convex polygons -- this is an ambiguous convention.

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... functions²⁴

In the case of polygon models and bond-animals, this generating function enumerates animals according to the number of vertical and horizontal bonds.

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... transparent²⁵

In the sense of being a very direct application of animals and polyominoes.

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... size²⁶

Not the number of monomers, rather a measure of the space occupied by the polymer.

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... gyration²⁷

Other measures of the size of a polymer are expected to behave similarly.

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... any²⁸

Any linear polymer that is flexible and has short-range interactions.

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... solved²⁹

It has not been entirely solved; some properties of the model have been found -- the free energy and spontaneous magnetisation -- while others -- most notably the susceptibility -- remain unknown.

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...³⁰

Since we consider only a finite portion of the lattice, the number of nearest-neighbour pairs is finite, and so the sum is convergent.

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