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Tatsushi Shimazaki

Publications and source records attributed to Tatsushi Shimazaki.

6 recordsLinked to original sources
arXiv2026

Generalized staircase partitions and Macdonald principal specializations

Generalized staircase partitions are obtained by replacing each box of an ordinary staircase with a fixed rectangle. We determine the unique shortest horizontal-strip sequence between consecutive generalized staircases and its conjugate vertical-strip sequence. For monic Macdonald polynomials, we derive explicit finite principal-specialization ratios along both sequences. For arbitrary nested partitions, coefficientwise nonnegativity after removal of the monomial factor forces independence of $q$. Along the horizontal sequence, this nonnegativity is characterized by rectangular complementation, apart from the one-column case. Along the vertical sequence, it occurs at the smallest admissible number of variables, apart from the initial column case. Endpoint ratios yield a triangular product formula and a centered product with exchange, reciprocity, and inversion identities. Hall-Littlewood, Jack, and Schur specializations give Gaussian-polynomial, finite-product, and tableau formulas, respectively.

math.CO↗
arXiv2026

Crystals and sign-reversing involutions for set-valued skew tableaux and decorated states of the five-vertex model

We construct bijections among semistandard set-valued tableaux of skew shape, marked Gelfand-Tsetlin patterns, and decorated states of the five-vertex model with boundary data determined by the skew shape. The bijections preserve monomial weight, identify tableau excess with the number of marks and nontrivial bumps, and intertwine the ordinary type A crystal operators with local transformations of decorated states. We prove that each fiber determined by a fixed maximum tableau is a graded Boolean lattice whose rank function is the excess. Its weighted generating function has a product formula. The sign-reversing involution changes one Boolean coordinate on every tableau that it does not fix, and we determine its commutation with all raising operators and with lowering operators except at the two colors involving the changed entry. We construct the dual involution associated with minimum entries and prove corresponding Boolean structures for fibers determined by a fixed minimum tableau and for fibers with prescribed minimum and maximum tableaux.

math.CO↗
arXiv2026

Staircase hook-length ratios and special values of Jacobi polynomials

We relate hook-length products for adjacent staircase partitions to special values of Jacobi polynomials. This connection expresses the number of semistandard tableaux in terms of Jacobi polynomials defined via Gauss hypergeometric functions. From this identity, we derive the special values of stable Grothendieck polynomials and $K$-theoretic Schur $P$-functions indexed by adjacent staircase partitions. These values provide ratios of the numbers of set-valued and shifted set-valued semistandard tableaux. This connection is further clarified by the theory of excited Young diagrams, which characterizes the coefficients in these specializations.

math.CO↗
arXiv2025

Special values of $K$-theoretic Schur $P$- and $Q$-functions

We provide the special values of the skew version of the $K$-theoretic Schur $P$- and $Q$-functions. Using these special values, we show an oddness property of the number of shifted set-valued skew tableaux. Additionally, we generalize these special values to another skew case. Based on these special values, we give pairs among certain shifted set-valued skew tableaux.

math.CO↗
arXiv2024

Special values of Grothendieck polynomials in terms of hypergeometric functions

We give some special values of Grothendieck polynomials and an explicit formula for the number of set-valued tableaux. For Young diagrams consisting of a single row or a single column, both the value and number are written by the Gauss' hypergeometric function ${}_2F_1$. For general Young diagrams, the Holman hypergeometric function $F^{(n)}$ is used to represent both the value and count. As an application, we derive a summation formula for $F^{(n)}$.

math.CO↗