Zigzags, contingency tables, and quotient rings

Let be a matrix of variables and let be the polynomial ring in these variables. Given two weak compositions of lengths and , we study the ideal generated by row sums, column sums, monomials in row of degree , and monomials in column of degree . We prove results connecting algebraic properties of the quotient ring with the set of ‐contingency tables. The standard monomial basis of with respect to a diagonal term order is encoded by the matrix‐ball avatar of the Robinson–Schensted–Knuth correspondence. We describe the Hilbert series of in terms of a zigzag statistic on contingency tables. The ring carries a graded action of the product of symmetry groups of the sequences and ; we describe how to calculate the isomorphism type of this graded action. Our analysis regards the set as a locus in the affine space and applies orbit harmonics to this locus.