Further investigation involving rook polynomials with only real zeros (Q1590216)

\textit{J. Haglund, K. Ono} and \textit{D. G. Wagner} [Math. Appl., Dordr. 467, 207-221 (1999; Zbl 0928.05003)] formulated the monotone column permanent (MCP) conjecture. Let \(A\) be a real \(n\times n\) matrix whose entries increase weakly down the columns and let \(J_n\) denote the \(n\times n\) matrix whose entries are all 1. The MCP conjecture asserts that the zeroes of the permanent of \(zA+J_n\), considered as a polynomial in \(z\), are all real. This paper extends the earlier work, by giving several new conjectures related to the MCP conjecture and with the same flavour. Each predicts that a certain polynomial will have only real roots. They are formulated in the setting of matchings in weighted graphs and digraphs. One polynomial is a generalisation of the cover polynomial for digraphs, while another generalises the hit polynomial. Some special cases of the conjectures are proved; for example, the Ferrers board case, when all weights are either 0 or 1.