We know that if the transition probability matrix of a Markov chain is doubly stochastic (column sum also 1) then the stationary distribution is known ($\frac{1}{\text{no. of states}}$) although the exact matrix is not known.
I am interested in knowing similar conditions on the matrix s.t. the matrix is not known explicitly although the stationary distribution is known.
Let $P$ be a $n\times n$ (row-)stochastic matrix such that $\pi P = \pi$ where $\pi_i=\frac1n$, $1\leqslant i\leqslant n$. Then for each column $j$, we have $$\sum_{i=1}^n \pi_i P_{ij} = \pi_j, $$ hence $$\sum_{i=1}^n \frac1n P_{ij} = \frac1n, $$ and multiplying by $n$ we see that the columns sum to $1$. So the stationary distribution is uniform if and only if $P$ is doubly stochastic.