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Imagine I have an $N$ by $M$ rectangular lattice where I randomly assign one of $k$ colors to every vertex in the lattice. I then write down a list of the ${N*M}\choose{2}$ possible unordered pairs of vertices, noting the color of each vertex in a pair and the Euclidean distance between them. Finally, I count the number of pairs of vertices with the both the same vertex color combination and Euclidean distance as a previously recorded pair, $P$. For example, if I've previously recorded a "red" and a "blue" vertex with Euclidean distance $d_i$ between them, which we'll write as a tuple: {$c_1, c_2, d_i$}, another example of a "red" and "blue" vertex pairing with the same Euclidean distance, {$c_1,c_2,d_i$} or {$c_2,c_1,d_i$} (order of the colors does not matter), would increase $P$ by one.

With a hat tip to Gerhard Paseman, we can write that $P = $${N*M}\choose{2}$$-T$, where $T$ is the total number of distinct tuples: {$c_i,c_j,d_k$}, where , again, the order of the colors do not matter and vertices of the same color are allowed.

As a function of $N$, $M$, and $k$, what is the expected value of $P$?

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Imagine I have an $N$ by $M$ rectangular lattice where I randomly assign one of $k$ colors to every vertex in the lattice. I then write down a list of the ${N*M}\choose{2}$ possible unordered pairs of vertices, noting the color of each vertex in a pair and the Euclidean distance between them. Finally, I count the number of pairs of vertices with the both the same vertex color combination and Euclidean distance as a previously recorded pair, $P$. For example, if I've previously recorded a "red" and a "blue" vertex with Euclidean distance $d_i$ between them, which we'll write as a tuple: {$c_1, c_2, d_i$}, another example of a "red" and "blue" vertex pairing with the same Euclidean distance$d_i$ , {$c_1,c_2,d_i$} or {$c_2,c_1,d_i$} (order of the colors does not matter), would be included in increase $P$ by one.

With a hat tip to Gerhard Paseman, we can write that $P = $${N*M}\choose{2}$$-T$, where $T$ is the counttotal number of distinct tuples: {$c_i,c_j,d_k$}, where, again, the order of the colors do not matter and vertices of the same color are allowed.

As a function of $N$, $M$, and $k$, what is the expected value of $P$?

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Imagine I have an $N$ by $M$ rectangular lattice where I randomly assign one of $k$ colors to every vertex in the lattice. I then write down a list of the $\frac{(N*M)^2}{2}$ {N*M}\choose{2}$possible unordered pairs of vertices, noting the color of each vertex in a pair and the Euclidean distance between them. Finally, I count the number of pairs of vertices with the both the same vertex color combination and Euclidean distance as a previously recorded pair,$P$. For example, if I've previously recorded a "red" and a "blue" vertex with Euclidean distance$d_i$between them, another example of a "red" and "blue" vertex pairing with the same Euclidean distance$d_i$would be included in the count. As a function of$N$,$M$, and$k$, what is the expected value of$P\$?

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