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We prove an Amitsur--Levitzki-type theorem for Grassmann algebras, statingthat the minimal degree of a standard identity that is a polynomial identity ofthe ring of $n \times n$ matrices over the $m$-generated Grassmann algebra isat least $2\left\lfloor\frac{m}{2}\right\rfloor+4n-4$ for all $n,m\geq 2$ andthis bound is sharp for $m=2,3$ and any $n\geq 2$. The arguments are purelycombinatorial, based on computing sums of signs corresponding to Euleriantrails in directed graphs.

The Minimal Degree Standard Identity on $M_nE^2$ and $M_nE^3$