We consider the homogeneous components Ur of the map on R = k[x, y, z]/(x A , y B , z C ) that multiplies by x + y + z. We prove a relationship between the Smith normal forms of submatrices of an arbitrary Toeplitz matrix using Schur polynomials, and use this to give a relationship between Smith normal form entries of Ur. We also give a bijective proof of an identity proven by J. Li and F. Zanello equating the determinant of the middle homogeneous component Ur when (A, B, C) = (a + b, a + c, b + c) to the number of plane partitions in an a × b × c box. Finally, we prove that, for certain vector subspaces of R, similar identities hold relating determinants to symmetry classes of plane partitions, in particular classes 3, 6, and 8.2010 Mathematics Subject Classification. Primary: 05E40; Secondary: 05E05, 05E18, 13E10, 15A15. 2 , b := A−B+C 2 , c := −A+B+C 2 so that A = a + b, B = a + c, C = b + c. Their proof proceeded by evaluating det(U m ) = det(M m (A, B, C)) directly, and comparing the answer to known formulae for such plane partitions. We respond to their call for a more direct, combinatorial explanation (see [10]) with the following:Theorem 1.2. Expressed in the monomial Z-basis for R = Z[x, y, z]/(x a+b , y a+c , z b+c ) the map U m : R m → R m+1 has its determinant det(U m ) equal, up to sign, to its permanent perm(U m ), and each nonzero term in its permanent corresponds naturally to a plane partition in an a × b × c box.