Results 11 to 20 of about 2,345 (191)

Rook placements in Young diagrams and permutation enumeration

Advances in Applied Mathematics, 2011
Given two operators $\hat D$ and $\hat E$ subject to the relation $\hat D\hat E -q \hat E \hat D =p$, and a word $w$ in $M$ and $N$, the rewriting of $w$ in normal form is combinatorially described by rook placements in a Young diagram. We give enumerative results about these rook placements, particularly in the case where $p=(1-q)/q^2$.
exaly   +4 more sources

Rook placements and cellular decomposition of partition varieties

Discrete Mathematics, 1997
The paper deals with partition varieties, i.e., projective varieties associated to some partition \(\lambda=(\lambda_1,\dots,\lambda_n)\). The main result is that the partition variety of \(\lambda\) is a CW-complex which has a cellular decomposition that can be described combinatorially in terms of rook placements of the Ferrers board of \(\lambda ...
exaly   +2 more sources

Combinatorial Insights into Numerical Semigroups: Rook Placements and Lah Numbers

Turkish Journal of Science and Technology
This paper investigates the interplay between numerical semigroups and enumerative combinatorics through the lens of Young diagrams, rook polynomials, and the Lah numbers. For a given numerical semigroup S, we associate a Young diagram constructed from the gap set of S.
exaly   +2 more sources

A Graph Theory of Rook Placements [PDF]

The Electronic Journal of Combinatorics, 2021
Two boards are rook equivalent if they have the same number of non-attacking rook placements for any number of rooks. Define a rook equivalence graph on an equivalence class of Ferrers boards by specifying that two boards are connected by an edge if you can obtain one of the boards by moving squares in the other board out of one column and into a ...
openaire   +2 more sources

Combinatorics of diagrams of permutations [PDF]

Discrete Mathematics & Theoretical Computer Science, 2014
There are numerous combinatorial objects associated to a Grassmannian permutation $w_λ$ that index cells of the totally nonnegative Grassmannian. We study some of these objects (rook placements, acyclic orientations, various restricted fillings) and ...
Joel Brewster Lewis, Alejandro Morales
doaj   +1 more source

On the Spectra of Simplicial Rook Graphs [PDF]

Discrete Mathematics & Theoretical Computer Science, 2013
The $\textit{simplicial rook graph}$ $SR(d,n)$ is the graph whose vertices are the lattice points in the $n$th dilate of the standard simplex in $\mathbb{R}^d$, with two vertices adjacent if they differ in exactly two coordinates.
Jeremy L. Martin, Jennifer D. Wagner
doaj   +1 more source

A Combinatorial Model for $q$-Generalized Stirling and Bell Numbers [PDF]

Discrete Mathematics & Theoretical Computer Science, 2008
We describe a combinatorial model for the $q$-analogs of the generalized Stirling numbers in terms of bugs and colonies. Using both algebraic and combinatorial methods, we derive explicit formulas, recursions and generating functions for these $q ...
Miguel Méndez, Adolfo Rodríguez
doaj   +1 more source

Modified Growth Diagrams, Permutation Pivots, and the BWX Map $\phi^*$ [PDF]

Discrete Mathematics & Theoretical Computer Science, 2012
In their paper on Wilf-equivalence for singleton classes, Backelin, West, and Xin introduced a transformation $\phi^*$, defined by an iterative process and operating on (all) full rook placements on Ferrers boards. Bousquet-Mélou and Steingrimsson proved
Jonathan Bloom, Dan Saracino
doaj   +1 more source

Gradedness of the set of rook placements in A _n−1 [PDF]

Communications in Mathematics, 2021
Abstract A rook placement is a subset of a root system consisting of positive roots with pairwise non-positive inner products. To each rook placement in a root system one can assign the coadjoint orbit of the Borel subgroup of a reductive algebraic group with this root system.
openaire   +2 more sources

Rook placements in $G_2$ and $F_4$ and associated coadjoint orbits

Communications in Mathematics, 2023
Let $\mathfrak{n}$ be a maximal nilpotent subalgebra of a simple complex Lie algebra with root system $\Phi$. A subset $D$ of the set $\Phi^+$ of positive roots is called a rook placement if it consists of roots with pairwise non-positive scalar products. To each rook placement $D$ and each map $\xi$ from $D$ to the set $\mathbb{C}^{\times}$ of nonzero
Ignatev, Mikhail V., Surkov, Matvey A.
openaire   +4 more sources