Some Combinatorial Formulas Related to Diagonal Ramsey Numbers

We derive some combinatorial formulas related to the diagonal Ramsey numbers $R(k)$. Each formula is a statement of the form "$F(n,k) = 0$ if and only if $n \ge R(k)$," where $F(n,k)$ is a combinatorial expression which depends on $n$ and $k$. Our work is closely related to a recent work by De Loera and Wesley.