The Lattice of Non-crossing Partitions and the Birkhoff-Lewis Equations

A matrix associated with the chromatic join of non-crossing partitions has been introduced by Tutte to generalise the Birkhoff-Lewis equations. A conjecture is given for its determinant in terms of polynomials having the Beraha numbers among their roots. Corresponding results for join and meet on the lattices of partitions and non-crossing partitions are obtained in terms of the combinatorial invariants of the lattices. The conjecture is restated in terms the lattice of non-crossing partitions alone.