Generalized staircase partitions and Macdonald principal specializations
Generalized staircase partitions are obtained by replacing each box of an ordinary staircase with a fixed rectangle. Between consecutive generalized staircases, we determine the unique shortest sequence with horizontal strips as successive differences. The conjugate sequence is the unique shortest one with vertical strips as successive differences. We derive explicit ratios along both sequences from the finite principal specialization formula for monic Macdonald polynomials with parameters $q$ and $t$. More generally, for two partitions related by inclusion, we prove that coefficientwise nonnegativity of their ratio after removal of its monomial factor forces independence of $q$. Except for sequences of single columns, coefficientwise nonnegativity along the horizontal sequence is equivalent to complementation in the rectangle determined by the number of variables. In the conjugate vertical sequence, coefficientwise nonnegativity occurs exactly at the smallest possible number of variables, except for the ratio from the empty partition to a column. Iterating the ratios between consecutive generalized staircases yields a triangular product formula. Each such ratio is expressed through a centered product satisfying exchange, reciprocity, and inversion identities. In the Hall-Littlewood specialization, the ratio from the initial generalized staircase to any partition in the horizontal sequence is a monomial times a Gaussian polynomial enumerating partitions in a rectangle. In the Jack limit, the ratios admit product formulas, and an additive analogue of the centered product satisfies parity and exchange identities. At the Schur specialization with all variables equal to one, we obtain product formulas for ratios of semistandard tableau counts. We derive ratios of hook products and of standard Young tableau counts from a shift between consecutive generalized staircase diagrams.