Core Label Orders of the Lattices W(m,n)

We determine the core label orders of the lattices W(m,n). Although these orders cease to be join-semilattices when m ≥ 2 and n > 0, every interval admits an explicit product decomposition into a Boolean lattice and one-letter shuffle lattices. The decomposition follows from a terminal-block description of the core labels and yields gradedness, formulas for interval invariants, and real-rooted rank polynomials. Under Mühle's bijection with two-toned tilings, the core order becomes inclusion of a signature recording internal blue cells and terminal blue strips. We also obtain a closed generating function for the M-triangles of the family, recovering the Boolean and Hochschild cases.