In 1992, Elkies, Kuperberg, Larsen, and Propp introduced a bijection between domino
tilings of Aztec diamonds and certain pairs of alternating-sign matrices whose sizes
differ by one. In this paper we first study those smaller permutations which, when
viewed as matrices, are paired with the matrices for doubly alternating Baxter
permutations. We call these permutations snow leopard permutations, and we use a
recursive decomposition to show they are counted by the Catalan numbers. This
decomposition induces a natural map from Catalan paths to snow leopard
permutations; we give a simple combinatorial description of the inverse of this map.
Finally, we also give a set of transpositions which generates these permutations.