Let $X$ denote a finite nonempty set, and let $W$ denote a matrix whose rows and columns are indexed by $X$ and whose entries belong to some field $\fld{K}$. We study three planar algebras related to $W$. Briefly, a planar algebra is a graded vector space $\palg{V}=\cup_{n\in \fld{Z}^{+}\cup\{+,\,-\}} \palg{V}_{n}$ which is closed under ``planar'' operators.

The first planar algebra which we study, $\fp{}^{W}=\cup\fp{n}^{W}$, is defined by the group theoretic properties of $W$. For $n\in \fld{Z}^{+}$, $\fp{n}^{W}$ is the vector space of functions from $X^{n}$ to $\fld{K}$ which are constant on the $\aut(W)$-orbits of $X^{n}$, and $\fp{+}^{W}$, $\fp{-}^{W}$ are identified with $\fld{K}$. The second planar algebra, $\sg{}^{W}=\cup\sg{n}^{W}$, is the planar algebra generated $W$. We define it combinatorially: $\sg{n}^{W}$ is spanned by functions from $X^{n}$ to $\fld{K}$ defined via statistical mechanical sums on certain planar open graphs. The third planar algebra, $\og{}^{W}=\cup\og{n}^{W}$, differs from $\sg{}^{W}$ only in that the open graphs defining the functions need not be planar.

It turns out that $\sg{}^{W}\subseteq\og{}^{W}\subseteq\fp{}^{W}$. We show that $\sg{}^{W}=\og{}^{W}$ if and only if $\sg{4}^{W}$ contains a single special function known as the ``transposition''. We show that $\og{}^{W}=\fp{}^{W}$ whenever $|X|!$ is not divisible by the characteristic of $\fld{K}$.