Exercise 5, § 2

Let $A$ be a commutative ring and $n\geqslant 1$ an integer. For a matrix $M = (m_{i,j})$ of type $(n, n)$ with coefficients in A, the permanent of M is called the element

per(M) $=\sum_{\sigma\in S_n}m_{1,\sigma(1)}\cdots m_{n,\sigma(n)}$

of A.

Let $k\geqslant 1$ be an integer. We denote by $X_{n,k}$ the set of matrices M of type $(n, n)$ with real coefficients such that each column of M contains $k$ coefficients equal to 1, all the others being zero.

Now prove that the limit

lim$_{n\rightarrow+\infty}($inf$_{M\in X_{n,k}}$ per(M)$)^{1/n}$

exists.