Further Investigations Involving Rook Polynomials with Only Real Roots
نویسندگان
چکیده
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. Special cases are shown to follow from a theorem of Szegg o. We also present a weighted version of the Poset Conjecture of enumerative combinatorics.
منابع مشابه
Theorems and Conjectures Involving Rook Polynomials with Only Real Zeros
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.
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