On the monotonicity of the permanent

Abstract:

Let ${\Omega _n}$ denote the set of all $n \times n$ doubly stochastic matrices and let ${J_n} = {[1/n]_{n \times n}}$. For $A \in {\Omega _n}$, if ${f_A}(t) = {\text {per((1 - t)}}{J_n} + tA)$ is a nondecreasing function of $t$ on $[0,1]$, we say that the monotonicity of permanent (abb. MP) holds for $A$ . Friedland and Mine [3] proved MP for $(n{J_n} - {I_n})/(n - 1)$. In [6], Lih and Wang proposed a problem of determining whether MP holds for ${J_{{n_1}}} \otimes \cdots \otimes {J_{{n_k}}},{n_i} > 0$. In this note, we prove MP for $((m{J_m} - {I_m}) \otimes s{J_s})/(m - 1)s$, extending the result of Friedland and Mine, and give an affirmative answer to the Lih and Wang’s question.