The lattice of non-crossing partitions and the Birkhoff-Lewis equations (Q1329073)

A matrix associated with the chromatic join of non-crossing partitions, introduced by Tutte, can be used to generalize the Birkhoff-Lewis equations. Studying the determinant of this matrix gives an interesting conjecture (still open) in which the determinant is expressed in terms of some specific polynomials. The matrix can be described in terms of two lattices. The paper restates some results obtained already earlier by Baeza-Yates, Dahab and Younger. These results are concerned with the determinants of four closely related matrices (however, these matrices can be described in terms of only one lattice), and are proved directly in the incidence algebra of the related lattices. The results can be restated in terms of the combinatorial invariants for the lattices. The main result of the paper is the reformulation of the conjecture in terms of the determination of another matrix of which the description can be given in terms of only one lattice (the lattice of non-crossing partitions).