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LHS = number of pairs of squares on an n×n grid of squares

I have tried conditioning on the diagonals to enumerate the RHS but nothing I have tried results in a disjoint exhaustive enumeration.

Another method I have attempted is to consider selecting columns and rows and picking two of their intersections in some prescribed method but nothing I have come up with works.

Jack Tiger Lam
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1 Answers1

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As you mentioned LHS is the number of ways to pick two coordinate elements in an $n\times n$ matrix. Now, you can do this in three ways:

  1. Either the two coordinates belong to the same triangle (upper or lower, so $2$ options here) this can be done in $2\binom{\binom{n+1}{2}}{2}-\binom{n}{2}$ ways.
  2. Or they belong to opposite parts (without being the same coordinate reflected), if so select both coordinates in the same triangle in $\binom{\binom{n}{2}}{2}$ ways and reflect one of the two coordinates (in $2$ ways).
  3. The coordinates are literally in opposite sites of the main diagonal, this can be done in $\binom{n}{2}$ because you just have to pick one of them and reflect it.
Phicar
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    I think what really helps here is to recognize $\binom{n+1}{2}$ as triangle numbers! For a combinatorial proof of this: https://math.stackexchange.com/q/2478616/620957. – Milten Aug 18 '21 at 08:57
  • @Milten Thanks Milten, I was assuming Jack knows about that and the fact that $n^2=\binom{n}{2}+\binom{n+1}{2}$ by his same combinatorial setup. – Phicar Aug 18 '21 at 09:02
  • I like this answer, but I have one concern. For case 3, I am unsure because how does it avoid breaking down when $n$ is odd? (the central square could be one of the chosen squares and reflecting this on the diagonal leaves you with the same square.) – Jack Tiger Lam Aug 18 '21 at 11:02
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    @JackTigerLam the parity on $n$ is not a problem, $\binom{n}{2}$ just takes care of elements ${x,y}$ such that $x<y$. So if this set represents the coordinate $(x,y)$ then $(x,y)\neq (y,x)$. The case in which things lie in the main diagonal is taken care in 1. – Phicar Aug 18 '21 at 11:07