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\title{The double Eulerian polynomial in terms of inversion tables}
\author{Erik Aas}
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\abstract{We prove a conjecture on double Eulerian polynomials due to Visontai \cite{V}, relating the number of descents and the number of inverse descents of permutations to the number of occupied rows and number of ascents of inversion tables.}



The double Eulerian polynomial $A_n(t,s)$ enumerates the number of descents and the number of inverse descents of a permutation,
$$
A_n(t,s) = \sum_{\pi\in\mathbb{S}_n} t^{\mathtt{des}{\pi}} s^{\mathtt{des}{\pi^{-1}}}.
$$

It is a natural generalization of the classical Eulerian polynomial $A_n(t, 1)$.
One interesting property of the classical Eulerian polynomial is its {\it unimodality}, which is an easy consequence of being nonnegative in the basis $\{t^i(1+t)^{n-i}\}_{i=0} ^n$.
One way of proving this is to note that $A_n(t,1)$ is the $h$-polynomial of the Coxeter complex (see for instance \cite{BR}).
Foata and Strehl gave a bijective proof involving 'valley-hopping' - see \cite{P} for a nice exposition.
Gessel \cite{B} conjectured that $A_n(t,s)$ similarly has nonnegative coefficients when expressed in the basis $B_{ij} = (ts)^i(t+s)^j(1+ts)^{n-2i-j}$.
This conjecture has motivated some of the work on $A_n(t,s)$, including this.

For any $n \geq 0$, we denote the set of permutations of $[n] = \{1,\dots,n\}$ (the {\it $n$-permutations}) by $\mathbb{S}_n$.
We think of permutations as {\it words}.
For example, the permutation mapping $1\mapsto 3$, $2\mapsto 1$, $3\mapsto 4$, $4\mapsto 2$ is identified with the word {\tt 3142}.
An {\it ($n$-)inversion table} is a sequence $e_1, \dots, e_n$ of integers satisfying $0 \leq e_i \leq i-1$ for each $i$.
The set of $n$-inversion tables is denoted $\mathbb{I}_n$. We now define some statistics.

\begin{itemize}
\item  $\mathtt{DES}(\pi) = \{i : \pi(i) > \pi(i+1) \}$
\item $\mathtt{IDES}(\pi) = \{i : \pi^{-1}(i) > \pi^{-1}(i+1) \}$
\item  $\mathtt{ASC}(e) = \{i : e_i < e_{i+1} \}$ (note the strict inequality).
\item  $\mathtt{ROW}(e) = \{e_i : 1 \leq i \leq n\} \backslash \{0\}$.
\end{itemize}

Moreover, $\mathtt{des} = \#\mathtt{DES}, \mathtt{ides}=\#\mathtt{IDES}, \mathtt{asc} = \#\mathtt{ASC}, \mathtt{row} = \#\mathtt{ROW}$.

Visontai \cite{V} conjectured that
$$
\sum_{\pi \in \mathbb{S}_n} t^{\mathtt{des}{\pi}} s^{\mathtt{ides}{\pi}} = \sum_{e\in \mathbb{I}_n} t^{\mathtt{asc}{e}} s^{\mathtt{row}{e}}
$$ for all $n$.

\subsection*{Examples}

For example, the inversion tables of length $5$ (written as words $e_1e_2e_3e_4e_5$) with $1$ ascent and two occupied rows are $00221, 00211, 00210,00021,00032,00031$.
The permutations of length $5$ with $1$ descent and two inverse descents are $24135,13524,23514,25134,35124,24513$.

We have $\mathtt{DES}(24135) = \{2\}$, $\mathtt{IDES}(24135) = \{1, 3\}$, $\mathtt{ASC}(00210) = \{2\}$, $\mathtt{ROW}(00210) = \{1,2\}$.



\begin{thebibliography}{9}
\bibitem{BR} Matthias Beck and Sinai Robins: Computing the continuous discretely: Integer-point enumeration in polyhedra, Undergraduate Texts in Mathematics, Springer, New York, 2007.
\bibitem{B} Petter Br\"{a}nd\'{e}n: Actions on permutations and unimodality of descent polynomials. European J. Combin., 29(2):514-531, 2008.
\bibitem{P} Kyle Petersen: Two-sided Eulerian numbers via balls in boxes. Math. Mag. (to appear). Preprint available at arXiv:1209.6273v1, 2012.
\bibitem{V} Mirk\'{o} Visontai: Some remarks on the joint distribution of descents and inverse descents, Electron. J. Combin., Volume 20, Issue 1, 2013, Research article 52, 12pp.
\end{thebibliography}
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