The Real-rootedness Of Chain Polynomials On Distributive Lattice J(mn)

The real-rootedness of a polynomial,a property closely related to important combinatorial properties(such as unimodality and log-concavity)of its coefficient,has become an active topic in combinatorics in recent decades.Chain polynomial is one of the most important polynomials defined on posets.This paper mainly studies the real-rootedness of chain polynomial on distributive lattice J(mn).Add all the elements with rank less than or equal to n in J(mn)and then add the greatest element to get a new poset Jm,n,and then prove that the chain polynomial and h-polynomial of this new poset are real-rooted.