I am reading a book on Stochastic Models, and I don't understand this practice questions:
A doubly stochastic n × n matrix S has all entries equal to 1/n.
The permament of a n × n matrx A is defined as the sum of all possible permutations (α . . . ω) of the integers (1 . . . n).
Where α is the first row, ω is last row, 1 is the first column, n is the last column.
How do I find the permament of S as a function of n, and evaluate the permanent of the 7 × 7 matrix with all elements equal to 1/7?
Well, if all elements are equal, then each term in the permanent is the same: $(1/7)^n$, and as there are $n!$ permutations, the answer is $\frac{n!}{7^n}$.
For a general matrix the question is infinitely more involved. In fact, a conjecture due to Van der Waerdon that $\mbox{perm}(M)\geq n!/n^n$ for any doubly-stochastic matrix took 50 years to prove. Notice that when all elements are equal to $1/n$, you achieve equality.