q-ROOK POLYNOMIALS AND MATRICES OVER FINITE FIELDS
نویسندگان
چکیده
Connections between q-rook polynomials and matrices over nite elds are exploited to derive a new statistic for Garsia and Remmel's q-hit polynomial. Both this new statistic mat and another statistic for the q-hit polynomial recently introduced by Dworkin are shown to induce di erent multiset Mahonian permutation statistics for any Ferrers board. In addition, for the triangular boards they are shown to generate di erent families of Euler-Mahonian statistics. For these boards the family includes Denert's statistic den, and gives a new proof of Foata and Zeilberger's Theorem that (exc; den) is equi-distributed with (des;maj). The mat family appears to be new. A proof is also given that the q-hit polynomials are symmetric and unimodal.
منابع مشابه
Splitting fields for characteristic polynomials of matrices with entries in a finite field
Let Mn(q) be the set of all n× n matrices with entries in the finite field Fq. With asymptotic probability one, the characteristic polynomial of a random A ∈ Mn(q) does not have all its roots in Fq. Let Xn(A) be the degree of the splitting field of the characteristic polynomial of A, and let μn be the average degree: μn = 1 |Mn(q)| ∑
متن کاملEigenvalues of Random Matrices over Finite Fields
We determine the limiting distribution of the number of eigenvalues of a random n×n matrix over Fq as n → ∞. We show that the q → ∞ limit of this distribution is Poisson with mean 1. The main tool is a theorem proved here on asymptotic independence for events defined by conjugacy class data arising from distinct irreducible polynomials. The proof of this theorem uses the cycle index for matrice...
متن کاملLinearized polynomial maps over finite fields
We consider polynomial maps described by so-called (multivariate) linearized polynomials. These polynomials are defined using a fixed prime power, say q. Linearized polynomials have no mixed terms. Considering invertible polynomial maps without mixed terms over a characteristic zero field, we will only obtain (up to a linear transformation of the variables) triangular maps, which are the most b...
متن کاملRelatively prime polynomials and nonsingular Hankel matrices over finite fields
The probability for two monic polynomials of a positive degree n with coefficients in the finite field Fq to be relatively prime turns out to be identical with the probability for an n × n Hankel matrix over Fq to be nonsingular. Motivated by this, we give an explicit map from pairs of coprime polynomials to nonsingular Hankel matrices that explains this connection. A basic tool used here is th...
متن کاملGenerating matrices of highest order over a finite field
Shift registers/Primitive polynomials find applications in various branches of Mathematics, Coding Theory and Cryptography. Matrix analogues of primitive polynomials do exist. In this paper, an algorithmic approach to generating all such matrices over GF(2) has been presented. A technique for counting all such n× n matrices over GF(2) is also presented. The technique may be easily extended to o...
متن کامل