Problem 532

51. (MON 3) Let $f_{1}=\left(a_{1}, a_{2}, \ldots, a_{n}\right), n>2$, be a sequence of integers. From $f_{1}$ one constructs a sequence $f_{k}$ of sequences as follows: if $f_{k}=$ $\left(c_{1}, c_{2}, \ldots, c_{n}\right)$, then $f_{k+1}=\left(c_{i_{1}}, c_{i_{2}}, c_{i_{2}}+1, c_{i_{4}}+1, \ldots, c_{i_{2}}+1\right)$, where $\left(c_{i_{1}}, c_{i_{2}}, \ldots, c_{i_{2}}\right)$ is a permutation of $\left(c_{1}, c_{2}, \ldots, c_{n}\right)$. Give a necessary and sufficient condition for $f_{1}$ under which it is possible for $f_{k}$ to be a constant sequence $\left(b_{1}, b_{2}, \ldots, b_{n}\right), b_{1}=b_{2}=\cdots=b_{m}$, for some $k$.