Enumerating Parking Completions Using Join and Split

  • Ayomikun Adeniran
  • Steve Butler
  • Galen Dorpalen-Barry
  • Pamela E. Harris
  • Cyrus Hettle
  • Qingzhong Liang
  • Jeremy L. Martin
  • Hayan Nam


Given a strictly increasing sequence $\mathbf{t}$ with entries from $[n]:=\{1,\ldots,n\}$, a parking completion is a sequence $\mathbf{c}$ with $|\mathbf{t}|+|\mathbf{c}|=n$ and $|\{t\in \mathbf{t}\mid t\leqslant i\}|+|\{c\in \mathbf{c}\mid c\leqslant i\}|\geqslant i$ for all $i$ in $[n]$.  We can think of $\mathbf{t}$ as a list of spots already taken in a street with $n$ parking spots and $\mathbf{c}$ as a list of parking preferences where the $i$-th car attempts to park in the $c_i$-th spot and if not available then proceeds up the street to find the next available spot, if any.  A parking completion corresponds to a set of preferences $\mathbf{c}$ where all cars park.

We relate parking completions to enumerating restricted lattice paths and give formulas for both the ordered and unordered variations of the problem by use of a pair of operations termed Join and Split.  Our results give a new volume formula for most Pitman-Stanley polytopes, and enumerate the \emph{signature parking functions} of Ceballos and González D'León.

Author Biography

Pamela E. Harris, Williams College

Department of Mathematics and Statistics

Assistant Professor

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