Parking functions, valet functions and priority queues

Abstract

Parking functions on [ n] = {1, …, n} are those functions p: [ n] → [ n] satisfying the condition |{ i: p( i) ⩽ r}| ⩾ r for each r, and are ( n + 1) n − 1 in number. These are equinumerate with allowable input-output pairs of permutations of [ n] in a priority queue. We present a new bijection between parking functions and allowable pairs which has many interesting invariance properties. We extend our bijection to allowable pairs of multisets and introduce valet functions as the corresponding extension of parking functions. Using our bijection, we interpret the inversion enumerator for trees in the case of allowable pairs. We end with a comparison of our bijection with other known bijections involving these combinatorial structures, including a new bijection between parking functions and labelled trees.