Geodesic
Enumerating parking completions using join and split
The electronic journal of combinatorics, Tome 27 (2020) no. 2
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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.
DOI : 10.37236/9194
Classification : 05E10, 05E18
Mots-clés : rational parking functions, \(q,t\)-Catalan numbers, rational Catalan numbers, diagonal harmonics, shuffle conjecture

Ayomikun Adeniran    ; Steve Butler    ; Galen Dorpalen-Barry    ; Pamela E. Harris  1   ; Cyrus Hettle    ; Qingzhong Liang    ; Jeremy L. Martin    ; Hayan Nam 

1 Williams College 
@article{10_37236_9194,
     author = {Ayomikun Adeniran and Steve Butler and Galen Dorpalen-Barry and Pamela E. Harris and Cyrus Hettle and Qingzhong Liang and Jeremy L. Martin and Hayan Nam},
     title = {Enumerating parking completions using join and split},
     journal = {The electronic journal of combinatorics},
     year = {2020},
     volume = {27},
     number = {2},
     doi = {10.37236/9194},
     zbl = {1441.05229},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/9194/}
}
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%A Hayan Nam
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Ayomikun Adeniran; Steve Butler; Galen Dorpalen-Barry; Pamela E. Harris; Cyrus Hettle; Qingzhong Liang; Jeremy L. Martin; Hayan Nam. Enumerating parking completions using join and split. The electronic journal of combinatorics, Tome 27 (2020) no. 2. doi: 10.37236/9194

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