[Paper Review] The Double Eulerian Polynomial and Inversion Tables

Erik Aas|arXiv (Cornell University)|Jan 22, 2014

Advanced Combinatorial Mathematics2 references3 citations

TL;DR

This paper proves a conjecture by Visontai by establishing a bivariate generating function identity: the double Eulerian polynomial, which enumerates permutations by descent and inverse descent statistics, is equidistributed with a generating function over inversion tables using ascent and row statistics. The key result is a bijective proof via labeled rooted trees that establishes a direct combinatorial equivalence between these two statistics on different combinatorial objects.

ABSTRACT

We show that the pair (des, ides) of statistics on the set of permu- tations has the same distribution as the pair (asc, row) of statistics on the set of inversion tables, proving a conjecture of Visontai. The common generating function of these pairs is the double Eulerian polynomial.

Motivation & Objective

• To prove that the double Eulerian polynomial, defined as the generating function over permutations by descent and inverse descent, equals the generating function over inversion tables by ascent and row statistics.
• To resolve a conjecture by Visontai on the equidistribution of these two generating functions.
• To provide a bijective proof using rooted labeled trees constructed via Möbius inversion and inductive construction of statistics on permutations and inversion tables.
• To establish a deeper combinatorial connection between permutation statistics and inversion table statistics, extending classical Eulerian polynomial theory.

Proposed method

• Define the double Eulerian polynomial as $ A_n(u,v) = \sum_{\pi \in \mathbb{S}_n} u^{\operatorname{des}(\pi)} v^{\operatorname{des}(\pi^{-1})} $, generalizing the classical Eulerian polynomial.
• Introduce inversion tables $ e \in \mathbb{I}_n $, sequences with $ 1 \leq e_i \leq i $, and define two statistics: $ \operatorname{asc}(e) $ as the number of ascents and $ \operatorname{row}(e) $ as the size of the set $ \{e_i : 1 \leq i \leq n\} \setminus \{1\} $.
• Construct two rooted labeled trees $ T_{\mathbb{S}} $ and $ T_{\mathbb{I}} $, where nodes are permutations and inversion tables of length not in a fixed subset $ S \subseteq \mathbb{N} $, with edges defined by extending the sequence with a block of length $ s $, where $ s $ is minimal such that $ r+s \notin S $.
• Use Möbius inversion to reduce the equidistribution claim to proving isomorphism between $ T_{\mathbb{S}} $ and $ T_{\mathbb{I}} $, showing that for each $ S $, the number of permutations and inversion tables with $ S \subseteq \operatorname{DES}(\pi) $ and $ S \subseteq \operatorname{ASC}(e) $, respectively, are equidistributed by inverse descent and row statistics.
• Establish a bijection $ \Phi $ between the trees by inductively mapping permutations to inversion tables, preserving length and the respective statistics, using a parametrization by nonnegative integer tuples $ (x_0, \dots, x_r) $ encoding how many elements of the late part fall between consecutive values of the early part.
• Prove that the number of such tuples yielding a given number of inverse descents equals the number of subsets $ T \subseteq [r+s] $ of size $ s $ with a given number of elements in $ T^c \cap \operatorname{ROW}(e') $, using a known combinatorial identity to confirm the count equality.

Experimental results

Research questions

• RQ1 Does the double Eulerian polynomial $ A_n(u,v) $, which counts permutations by descent and inverse descent, have an equidistributed counterpart on inversion tables using ascent and row statistics?
• RQ2 Can the conjecture by Visontai, stating that $ A_n(u,v) = \sum_{e \in \mathbb{I}_n} u^{\operatorname{asc}(e)} v^{\operatorname{row}(e)} $, be proven combinatorially?
• RQ3 Is there a bijective correspondence between permutations and inversion tables that preserves the joint distribution of descent and inverse descent with ascent and row statistics?
• RQ4 Can this equidistribution be extended to other statistics such as major index or sum of descent positions?
• RQ5 Is there a direct combinatorial proof of the equidistribution of $ \operatorname{asc} $ and $ \operatorname{row} $ on inversion tables, independent of the permutation side?

Key findings

• The double Eulerian polynomial $ A_n(u,v) $ is proven to equal the generating function $ \sum_{e \in \mathbb{I}_n} u^{\operatorname{asc}(e)} v^{\operatorname{row}(e)} $, confirming Visontai's conjecture.
• The proof establishes a tree isomorphism between permutation-based and inversion table-based rooted trees, preserving the statistics $ \operatorname{des} $ and $ \operatorname{ides} $ with $ \operatorname{asc} $ and $ \operatorname{row} $, respectively.
• The number of permutations with a given descent set $ S $ and $ k $ inverse descents equals the number of inversion tables with $ S \subseteq \operatorname{ASC}(e) $ and $ \operatorname{row}(e) = k $, for all $ S \subseteq [n-1] $, via Möbius inversion.
• The proof shows that the pair $ (\operatorname{maj}, \operatorname{ides}) $ on permutations is equidistributed with $ (\operatorname{amaj}, \operatorname{row}) $ on inversion tables, where $ \operatorname{maj} $ is the sum of descent positions and $ \operatorname{amaj} $ the sum of ascent positions.
• The equidistribution of $ \operatorname{asc} $ and $ \operatorname{row} $ on inversion tables is implied by the main result, though no direct combinatorial proof is known.
• The bijection used in the proof is not symmetric in $ u $ and $ v $, despite the symmetry of the final identity, indicating a deeper asymmetry in the underlying combinatorics.