Related papers: Combinatorics of Tableau Inversions

Let $x_{1},x_{2},\ldots,x_{n}$ be $n$ numbers, and $y_{1},y_{2},\ldots,y_{n}$ be $n$ further numbers chosen such that all $n^{2}$ pairwise sums $x_{i}+y_{j}$ are nonzero. Consider the $n\times n$-matrix \[ C:=\left(…

The Sherman-Woodbury-Morrison (SWM) formula gives an explicit formula for the inverse perturbation of a matrix in terms of the inverse of the original matrix and the perturbation. This formula is useful for numerical applications. We have…

We introduce a family of univariate polynomials indexed by integer partitions. At prime powers, they count the number of subspaces in a finite vector space that transform under a regular diagonal matrix in a specified manner. This…

This paper deals with those positive integers N such that, for given integers g and k with 1< k<g, the base-g digits of N and kN appear in reverse order. Such N are called (g, k) reverse multiples. Anne Ludington Young, in 1992, developed a…

Let $D$ be an oriented graph. The inversion of a set $X$ of vertices in $D$ consists in reversing the direction of all arcs with both ends in $X$. The inversion number of $D$, denoted by ${\rm inv}(D)$, is the minimum number of inversions…

A fundamental identity in the representation theory of the partition algebra is $n^k = \sum_{\lambda} f^\lambda m_k^\lambda$ for $n \geq 2k$, where $\lambda$ ranges over integer partitions of $n$, $f^\lambda$ is the number of standard Young…

Counting inversions is a classic and important problem in databases. The number of inversions, $K^*$, in a list $L=(L(1),L(2),\ldots,L(n))$ is defined as the number of pairs $i < j$ with $L(i) > L(j)$. In this paper, new results for this…

A new matrix operation based on inserting columns and rows, similarly to the mediant operation between fractions, gives rise to the Farey determinants matrix or, equivalently, the matrix of the numerators of the differences of Farey…

Let $X$ be a smooth polarized algebraic surface over the compex number field. We discuss the invariants obtained from the moduli stacks of semistable sheaves of arbitrary ranks on $X$. For that purpose, we construct the virtual fundamental…

Egecioglu and Remmel gave an interpretation for the entries of the inverse Kostka matrix K^{-1} in terms of special rim-hook tableaux. They were able to use this interpretation to give a combinatorial proof that KK^{-1}=I but were unable to…

We propose a generalization of the topological vertex, which we call the "non-commutative topological vertex". This gives open BPS invariants for a toric Calabi-Yau manifold without compact 4-cycles, where we have D0/D2/D6-branes wrapping…

Motivated by the $Z$-polynomials of matroids, Ferroni, Matherne, Stevens, and Vecchi introduced the inverse $Z$-polynomial of a matroid. In this paper, we prove several fundamental properties of the inverse $Z$-polynomial, including…

The inversion problem for rational B\'ezier curves is addressed by using resultant matrices for polynomials expressed in the Bernstein basis. The aim of the work is not to construct an inversion formula but finding the corresponding value…

A new family of asymmetric matrices of Walsh-Hadamard type is introduced. We study their properties and, in particular, compute their determinants and discuss their eigenvalues. The invertibility of these matrices implies that certain…

We give new bounds and asymptotic estimates on the number of standard Young tableaux of skew shape in a variety of special cases. Our approach is based on Naruse's hook-length formula. We also compare our bounds with the existing bounds on…

Let $\delta=(\delta_1,\ldots,\delta_n)$ be a string of letters $h$ and $v$. We define a Young tableau to be $\delta$-semistandard if the entries are weakly increasing along rows and columns, and the entries $i$ form a horizontal strip if…

The study of permutation and partition statistics is a classical topic in enumerative combinatorics. The major index statistic on permutations was introduced a century ago by Percy MacMahon in his seminal works. In this extended abstract,…

$k$-diagonal circulant matrices and cyclic banded matrices are widely used in numerical simulations and signal processing of circular linear systems. Algorithms that directly involve or specify linear or quadratic complexity for the…

Transformation coefficients between standard bases for irreducible representations of the Brauer centralizer algebra $\mathfrak{B}_f(x)$ and split bases adapted to the $\mathfrak{B}_{f_1} (x) \times \mathfrak{B}_{f_2} (x) \subset…

In this paper we study ideas which have proved useful in topological network theory in the context of lattices of numbers. A number lattice $L_S$ is a collection of row vectors, over $\mathbb{Q}$ on a finite column set $S,$ generated by…