Theorems and Conjectures Involving Rook Polynomials with Only Real Zeros
نویسندگان
چکیده
Let A = (aij) be a real n n matrix with non-negative entries which are weakly increasing down columns. If B = (bij) is the n n matrix where bij := aij+z; then we conjecture that all of the roots of the permanent of B, as a polynomial in z; are real. Here we establish several special cases of the conjecture.
منابع مشابه
Further Investigations Involving Rook Polynomials with Only Real Roots
We present a number of conjectures involving rook polynomials having only real zeros. Many of these generalize a previous conjecture of the author, K. Ono, and D. G. Wagner, namely that if A is a real n n matrix which is weakly increasing down columns, then the permanent of zA + Jn has only real zeros. In some cases these conjectures are motivated by factorization theorems for Ferrers boards. S...
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We study the zeros of two families of polynomials related to rook theory and matchings in graphs. One of these families is based on the cover polynomial of a digraph introduced by Chung and Graham [ChGr]. Another involves a version of the \hit polynomial" of rook theory, but which applies to weighted matchings in (non-bipartite) graphs. For both of these families we prove a result which is anal...
متن کاملTheorems and Conjectures Involving Rook Polynomials with Real Roots
Let A = (a ij) be a real n n matrix with non-negative entries which are weakly increasing down columns. If B = (b ij) is the nn matrix where b ij := a ij +z; then we conjecture that all of the roots of the permanent of B, as a polynomial in z; are real. Here we establish several special cases of the conjecture.
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