We consider properties of the box polynomials, a one variable polynomial defined over all integer partitions $λ$ whose Young diagrams fit in an $m$ by $n$ box. We show that these polynomials can be expressed by the finite difference operator applied to the power $x^{m+n}$. Evaluating box polynomials yields a variety of identities involving set partition enumeration. We extend the latter identities using restricted growth words and a new operator called the fast Fourier operator, and consider connections between set partition enumeration and the chromatic polynomial on graphs. We also give connections between the box polynomials and the excedance matrix, which encodes combinatorial data from a noncommutative quotient algebra motivated by the recurrence for the excedance set statistic on permutations.
Learning Read-Once Determinants and the Principal Minor Assignment Problem
Indian Institute of Science, Bangalore（印度科学学院）；Indian Institute of Technology Kanpur（坎普尔印度理工学院）；Indian Statistical Institute, Kolkata（加尔各答印度统计研究所）；Indian Institute of Technology Bombay（孟买印度理工学院）；Tata Institute of Fundamental Research, Mumbai（孟买塔塔基础研究所）
