A Combinatorial Formula for Kazhdan-Lusztig Polynomials of ρ\rhoρ-Removed Uniform Matroids (original) (raw)

A Combinatorial Formula for Kazhdan-Lusztig Polynomials of ρ-Removed Uniform Matroids

2019

Let ρ be a non-negative integer. A ρ-removed uniform matroid is a matroid obtained from a uniform matroid by removing a collection of ρ disjoint bases. We present a combinatorial formula for Kazhdan-Lusztig polynomials of ρ-removed uniform matroids, using skew Young Tableaux. Even for uniform matroids, our formula is new, gives manifestly positive integer coefficients, and is more manageable than known formulas.

A Combinatorial Formula for Kazhdan-Lusztig Polynomials of Sparse Paving Matroids

The Electronic Journal of Combinatorics

We present a combinatorial formula using skew Young tableaux for the coefficients of Kazhdan-Lusztig polynomials for sparse paving matroids. These matroids are known to be logarithmically almost all matroids, but are conjectured to be almost all matroids. We also show the positivity of these coefficients using our formula. In special cases, such as uniform matroids, our formula has a nice combinatorial interpretation.

Kazhdan-Lusztig Polynomials of Matroids Under Deletion

The Electronic Journal of Combinatorics, 2020

We present a formula which relates the Kazhdan–Lusztig polynomial of a matroid MMM, as defined by Elias, Proudfoot and Wakefield, to the Kazhdan–Lusztig polynomials of the matroid obtained by deleting an element, and various contractions and localizations of MMM. We give a number of applications of our formula to Kazhdan–Lusztig polynomials of graphic matroids, including a simple formula for the Kazhdan–Lusztig polynomial of a parallel connection graph.

A Flag Whitney Number Formula for Matroid Kazhdan-Lusztig Polynomials

The Electronic Journal of Combinatorics

For a representation of a matroid the combinatorially defined Kazhdan-Lusztig polynomial computes the intersection cohomology of the associated reciprocal plane. However, these polynomials are difficult to compute and there are numerous open conjectures about their structure. For example, it is unknown whether or not the coefficients are non-negative for non-representable matroids. The main result in this note is a combinatorial formula for the coefficients of these matroid Kazhdan-Lusztig polynomials in terms of flag Whitney numbers. This formula gives insight into some vanishing behavior of the matroid Kazhdan-Lusztig polynomials.

A new formula for an evaluation of the Tutte polynomial of a matroid

Discrete Mathematics, 2005

Given a matroid M and its Tutte polynomial T M (x, y), T M (0, 1) is an invariant of M with various interesting combinatorial and topological interpretations. Being a Tutte-Grothendieck invariant, T M (0, 1) may be computed via deletion-contraction recursions. In this note we derive a new recursion formula for this invariant that involves contractions of M through the circuits containing a fixed element of M.

The Tutte polynomial of matroids constructed by a family of splitting operations

2021

To extract some more information from the constructions of matroids that arise from new operations, computing the Tutte polynomial, plays an important role. In this paper, we consider applying three operations of splitting, element splitting and splitting off to a binary matroid and then introduce the Tutte polynomial of resulting matroids by these operations in terms of that of original matroids.

The Tutte polynomial of some matroids

2014

The Tutte polynomial of a graph or a matroid, named after W. T. Tutte, has the important universal property that essentially any mul-tiplicative graph or network invariant with a deletion and contraction reduction must be an evaluation of it. The deletion and contraction operations are natural reductions for many network models arising from a wide range of problems at the heart of computer science, engi-neering, optimization, physics, and biology. Even though the invariant is #P-hard to compute in general, there are many occasions when we face the task of computing the Tutte polynomial for some families of graphs or matroids. In this work we compile known formulas for the Tutte polynomial of some families of graphs and matroids. Also, we give brief explanations of the techniques that were use to find the for-mulas. Hopefully, this will be useful for researchers in Combinatorics and elsewhere.

Tutte polynomials for oriented matroids

arXiv (Cornell University), 2022

The Tutte polynomial is a fundamental invariant of graphs and matroids. In this article, we define a generalization of the Tutte polynomial to oriented graphs and regular oriented matroids. To any regular oriented matroid N , we associate a polynomial invariant AN (q, y, z), which we call the A-polynomial. The A-polynomial has the following interesting properties among many others: • a specialization of AN gives the Tutte polynomial of the underlying unoriented matroid N , • when the oriented matroid N corresponds to an unoriented matroid (that is, when the elements of the ground set come in pairs with opposite orientations), the invariant AN is equivalent to the Tutte polynomial of this unoriented matroid (up to a change of variables), • the invariant AN detects, among other things, whether N is acyclic and whether N is totally cyclic. We explore various properties and specializations of the A-polynomial. We show that some of the known properties of the Tutte polynomial of matroids can be extended to the A-polynomial of regular oriented matroids. For instance, we show that a specialization of AN counts all the acyclic orientations obtained by reorienting some elements of N , according to the number of reoriented elements. Let us mention that in a previous article we had defined an invariant of oriented graphs that we called the B-polynomial, which is also a generalization of the Tutte polynomial. However, the B-polynomial of an oriented graph N is not equivalent to its A-polynomial, and the B-polynomial cannot be extended to an invariant of regular oriented matroids.