In this question Anthony Quas asks about the expected absolute value of the determinant of an $n\times n$ row stochastic matrix $A$, where the rows are independently selected from the uniform distribution on the unit $(n-1)$-dimensional simplex $x_1+\cdots+x_n=1$, $x_i\geq 0$. I can show by a messy computation that the integral over all such matrices of $(\det A)^2$ is $1/(n+1)!^{n-1}$. Is there a noncomputational reason for such a simple value?