A bijective answer to a question of Zvonkin (Q1575087)

This paper provides an alternative proof (based on combinatorial arguments) of the following well-known result: If \(x_i\) \((i= 1,2,\dots,n)\) are independent random variables, each having a normal distribution with mean \(0\) and standard deviation \(1\), then the following is true: \[ E\Biggl[\prod_{1\leq i< j\leq n}(x_i- x_j)^2\Biggr]= (0!)(1!)(2!)\cdots(n!). \] The author provides a bijective proof of the above identity, the bijection being obtained by combining a bijection of Gessel and a bijection of Ehrenborg with the interpretation that the \(n\)th moment of standardized normal random variable counts the number of matchings on an \(n\) element set which is zero if \(n\) is odd, and \((n-1)\cdot(n-3)\cdot(n-5)\cdot\dots\cdot 1\) if \(n\) is even.