It is well-known that the increasing parking functions are counted by the Catalan numbers. The Catalan numbers also count the dominant alcoves in the Shi arrangement of type $A_{n}$. Athanasiadis-Linusson gave a bijection between these two (and in fact between the sets of all Shi alcoves and all parking functions.) In addition, Fishel-Vazirani gave a bijection from dominant Shi alcoves to partitions which are both $n$ and $n+1$-core. This (composed with Athanasiadis-Linusson) gives a bijection from such partitions to increasing parking functions. Has anyone made this direct map explicit?