  We present a multinomial theorem for elliptic commuting variables. This
result extends the author's previously obtained elliptic binomial theorem to
higher rank. Two essential ingredients are a simple elliptic star-triangle
relation, ensuring the uniqueness of the normal form coefficients, and, for the
recursion of the closed form elliptic multinomial coefficients, the
Weierstra{\ss} type $\mathsf A$ elliptic partial fraction decomposition. From
our elliptic multinomial theorem we obtain, by convolution, an identity that is
equivalent to Rosengren's type $\mathsf A$ extension of the Frenkel--Turaev
${}_{10}V_9$ summation, which in the trigonometric or basic limiting case
reduces to Milne's type $\mathsf A$ extension of the Jackson ${}_8\phi_7$
summation. Interpreted in terms of a weighted counting of lattice paths in the
integer lattice $\mathbb Z^r$, our derivation of the $\mathsf A_r$
Frenkel--Turaev summation constitutes the first combinatorial proof of that
fundamental identity, and, at the same time, of important special cases
including the $\mathsf A_r$ Jackson summation.
