  The graph of zigzag diagrams is a close relative of Young's lattice. The
boundary problem for this graph amounts to describing coherent random
permutations with descent-set statistic, and is also related to certain
positive characters on the algebra of quasi-symmetric functions. We establish
connections to some further relatives of Young's lattice and solve the boundary
problem by reducing it to the classification of spreadable total orders on
integers, as recently obtained by Jacka and Warren.
