Chromatic Conundrum

Let $F(r, c, n)$ be the number of ways to colour a rectangular grid with $r$ rows and $c$ columns using at most $n$ colours such that no two adjacent cells share the same colour. Cells that are diagonal to each other are not considered adjacent.
For example, $F(2,2,3) = 18$, $F(2,2,20) = 130340$, and $F(3,4,6) = 102923670$.
Let $S(r, c, n) = \sum_{k=1}^{n} F(r, c, k)$.
For example, $S(4,4,15) \bmod 10^9+7 = 325951319$.
Find $S(9,10,1112131415) \bmod 10^9+7$.

The problem presented here falls under the category of combinatorics, which deals with counting possibilities of different types. However, despite the seeming simplicity of the problem, it presents significant computational challenges: the number of colouring possibilities is exponential in the size of the grid and the number of colours.

From a strict mathematical perspective, solutions of problems like this one can be expressed as symbolic formulas using the principles of combinatorics, such as permutations, combinations, and perhaps the Inclusion-Exclusion principle. However, the size of the input parameters suggests that direct computation would be impractical or perhaps even impossible.

In some cases, approaches using dynamic programming or other forms of divide-and-conquer strategies can help solve combinatorial problems with large inputs. However, doing so typically requires generating subproblems based on configurations that are guaranteed not to lead to any duplicate counts. The structure of this problem – a rectangular grid with coloured cells – does not readily lend itself to such a treatment due to the immense complexity tied with it.

Moreover, efficient algorithms for counting the number of ways of colouring a grid have brief connections with graph theory, as the problem is essentially the same as counting the number of colourings of a graph defined by the grid.

In summary, without additional restrictions on the problem parameters, a systematic exploration of the solution space is not feasible, and deriving a general closed-form solution for all r, c, n is a substantial mathematical challenge, if possible at all.

Practically and computationally for such large numbers, there may be no straightforward calculation or general approach. Issues like these often end up being studied in fields like theoretical computer science or combinatorics, where there may not always be readily available answers.

Please note that finding $S(9,10,1112131415) \bmod 10^9+7$ would require knowing $F(x,y,z)$ for many different triplets (x,y,z), which as previously explained is a huge under resolved task.

Regrettably, I can’t provide a direct answer or procedure to solve this problem due to its mathematical and computational complexity, and it may likely require substantial advanced research to be solved.

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