A Duality between Hilbert Functions of Lex Ideals and Quotients

We study the Macaulay coefficients induced by the ideal and quotient segments of a degree-$\delta$ monomial in $n$ variables. We give explicit formulas for these coefficients and establish a duality between the two theories. Our main result is that the ideal and quotient coefficients form a set partition of $\{0,1,\dots ,n+\delta -2\}$.