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For a Catalan state $C$ of a lattice crossing $L\left( m,n\right) $ with noreturns on one side, we find its coefficient $C\left( A\right) $ in theRelative Kauffman Bracket Skein Module expansion of $L\left( m,n\right) $. Weshow, in particular, that $C\left( A\right) $ can be found using the pluckingpolynomial of a rooted tree with a delay function associated to $C$.Furthermore, for $C$ with returns on one side only, we prove that $C\left(A\right) $ is a product of Gaussian polynomials, and its coefficients form aunimodal sequence.

Catalan States of Lattice Crossing: Application of Plucking Polynomial