Order by static disorder in the Ising chain magnet Ca$_3$Co$_{2-x}$Mn$_x$O$_6$

Ising chain compound Ca$_3$Co$_{2-x}$Mn$_x$O$_6$ exhibits up-up- down-down long-range magnetic order (LRO) in a broad range of 0.75 $<$$x$$<$1. The LRO is abruptly lost in the narrow vicinity of $x$=1, and the magnetic state becomes incommensurate. The commensurate state (but not the LRO) is recovered for larger $x$. This is surprising because the stoichiometric $x$=1 state exhibits the best Co/Mn ionic order, and the magnetic LRO appears only in the samples with reduced ionic order. We argue that this ``order-by-static-disorder'' phenomenon may be related to the disruption of the long-range magnetic interactions by the magnetic-site disorder, reducing magnetic frustration.