Greenberger-Horne-Zeilinger paradoxes with symmetric multiport beam splitters

In a gedanken experiment N particles in a generalized Greenberger-Horne-Zeilinger-(GHZ)-type beam-entangled state (each particle can be in one of M beams) are fed into N symmetric $2M$-port beam splitters (spatially separated). Correlation functions for such a process (using the Bell numbers value assignment approach) reveal a remarkable symmetry. For $N=M+1\mathrm{}>~4$ a series of GHZ paradoxes are shown.