Jacobi–Trudi Identities for Boolean Tableaux and Ideal-tableaux of Zigzag Posets

Aboolean tableauis an arrayT = (Tij) of the elements of a finite boolean algebra with several rows and infinitely many columns, where the entries increase from left to right and downwards. We study the generating functions for various classes of boolean tableaux. Applying the Gessel–Viennot method to certain nonplanar digraphs, we have determinantal formulas for the generating functions, which are regarded as generalized Jacobi–Trudi identities. By this theorem, we can also deal withideal-tableaux of zigzags, and give some new totally positive matrices.