Consider a partition of a natural number n. The partition is called a t-core if each of the hook numbers (one more than the number of squares to the right and below a certain node of n) from its Ferrers board is not divisible by t. [HOS98] conjectured in 1998 that if t ≥ 5, then there exists a constant Nt such that for every positive integer n ≥ Nt , there exist two distinct rook equivalent t-c...

We define the notion of the Fourier transform for the rook monoid (also called the symmetric inverse semigroup) and provide two efficient divideand-conquer algorithms (fast Fourier transforms, or FFTs) for computing it. This paper marks the first extension of group FFTs to non-group semigroups.

The q-rook monoid Rn(q) is a semisimple C(q)-algebra that specializes when q → 1 to C[Rn], where Rn is the monoid of n × n matrices with entries from {0, 1} and at most one nonzero entry in each row and column. We use a Schur-Weyl duality between Rn(q) and the quantum general linear group Uqgl(r ) to compute a Frobenius formula, in the ring of symmetric functions, for the irreducible characters...

in this note we exhibit an exam;rle of a new class c)f combinatorial array, A ROOK square i9j it: an r x t square each of whose I ells is either empty or contains a distinct uncrrdered pair with cniik I,,;_ += 4er of integers (! 3, . . ., I + 1) such that each integer appears exactly Once m each row and exactly once in eacii bL:!:~WC oresent an analagaus square whose entries arc unordered tripl...

Let $$\mathcal {P}$$ be a simple thin polyomino, namely polyomino that has no holes and does not contain square tetromino as subpolyomino. In this paper, we determine the reduced Hilbert–Poincaré series $$h(t)/(1-t)^d$$ of $$K[\mathcal {P}]$$ by proving h(t) is rook polynomial . As an application, characterize Gorenstein polyominoes.

Abstract A rook placement is a subset of root system consisting positive roots with pairwise non-positive inner products. To each in one can assign the coadjoint orbit Borel subgroup reductive algebraic group this system. Degenerations such orbits induce natural partial order on set placements. We study combinatorial structure placements n− 1 respect to slightly different and prove that poset g...