Coefficients of catalan states without nesting and maximal number of returns

Abstract

We find and study properties of the coefficients [Formula: see text] and [Formula: see text] of Catalan states with the maximal number of returns and no nesting in the Kauffman bracket sum of an [Formula: see text] lattice crossing [Formula: see text]. We show that, up to a power of [Formula: see text], [Formula: see text] and [Formula: see text] are polynomials in variable [Formula: see text] with unimodal and palindromic coefficients. Our results can be viewed as a step toward obtaining closed form formulas for coefficients of an arbitrary Catalan state obtained from [Formula: see text].