Generalized Staircase Tableaux: Symmetry and Applications

We define a number of related combinatorial objects, each of which possesses a surprising symmetry. We include several applications such as a combinatorial explanation for certain fixed points of the involution $\omega$ on the ring of symmetric functions, as well as a relationship between certain skew Schur functions and skew $Q$-Schur functions. We give a $t$-deformation of these $Q$-Schur functions, and show that it is Schur positive, including a combinatorial description of the Schur coefficients. A corollary of our results is the equality of skew $Q$-Schur functions: $Q_{\lambda+\delta/\mu + \delta}=Q_{\lambda'+\delta/\mu' + \delta}$ for $\mu \subseteq \lambda$ and $\delta=(n,\ldots,1)$ for some $n > l(\lambda)$.