[Paper Review] Enumerating the symplectic Dellac configurations

Ange Bigeni|arXiv (Cornell University)|May 10, 2017

Advanced Combinatorial Mathematics2 references3 citations

TL;DR

This paper proves that the number of symplectic Dellac configurations of size $2n$ equals the sequence $r_n = (1,2,10,98,1594,\ldots)$, conjectured by Cerulli Irelli and Feigin and later formalized by Fang and Fourier. The proof constructs a surjection from symplectic Dellac configurations to surjective pistols, leveraging a combinatorial interpretation of $r_n$ as a sum over surjective pistols weighted by $2^{\text{ndf}(f)}$, where $\text{ndf}(f)$ counts non-doubled fixed points.

ABSTRACT

Fang and Fourier defined the symplectic Dellac configurations in order to parametrize the torus fixed points of the symplectic degenerated flag varieties, and conjectured that their numbers are the elements of a sequence of integers (1, 2, 10, 98, 1594, ...) which appears in the study by Randrianarivony and Zeng of the median Euler numbers. In this paper, we prove the conjecture by considering a combinatorial interpretation of these integers in terms of the surjective pistols (which form a well-known combinatorial model of the Genocchi numbers), and constructing an appropriate surjection from the symplectic Dellac configurations to the surjective pistols.

Motivation & Objective

• To prove the conjecture that the number of symplectic Dellac configurations of size $2n$ equals the sequence $r_n = (1,2,10,98,1594,\ldots)$, which arises in the study of median Euler numbers.
• To establish a combinatorial connection between symplectic Dellac configurations and surjective pistols, a known model for Genocchi numbers.
• To demonstrate that the cardinality of symplectic Dellac configurations matches the sum $\sum_{f \in \mathcal{P}_n} 2^{\text{ndf}(f)}$, where $\text{ndf}(f)$ counts non-doubled fixed points in a surjective pistol $f$.
• To provide a constructive proof via an explicit surjection from symplectic Dellac configurations to surjective pistols, preserving the weight $2^{\text{ndf}(f)}$.

Proposed method

• The paper uses a combinatorial interpretation of $r_n$ as $\sum_{f \in \mathcal{P}_n} 2^{\text{ndf}(f)}$, where $\mathcal{P}_n$ is the set of surjective pistols from $[2n]$ to $\{2,4,\ldots,2n\}$ satisfying $f(j) \geq j$.
• It defines a canonical labeling of symplectic Dellac configurations via a recursive algorithm that assigns pistol labels to dots, based on column order and dot positions.
• The labeling process uses a two-step insertion rule: first, dots are labeled with $a$ or $b$ based on their position relative to existing dots; second, the labels are used to reconstruct the corresponding surjective pistol.
• A key component is the definition of the set $\mathcal{T}_n$ of symplectic Dellac configurations, equipped with a function $\text{fr}(T)$ counting free rows, which is shown to satisfy $\sum_{T \in \mathcal{T}_n} 2^{\text{fr}(T)} = \sum_{f \in \mathcal{P}_n} 2^{\text{ndf}(f)}$.
• The proof proceeds by induction on the number of columns, showing that the weight $2^{\text{fr}(T)}$ of a configuration $T$ matches the weight $2^{\text{ndf}(f)}$ of its associated pistol $f$.
• The construction is validated through explicit examples and algorithmic steps, including the labeling of a specific configuration $T_1 \in \mathcal{T}_7$ and the inverse mapping $\Phi$ from pistols to configurations.

Experimental results

Research questions

• RQ1 Does the number of symplectic Dellac configurations of size $2n$ equal the sequence $r_n = (1,2,10,98,1594,\ldots)$, as conjectured by Cerulli Irelli and Feigin?
• RQ2 Can the sequence $r_n$ be interpreted combinatorially as a sum over surjective pistols weighted by $2^{\text{ndf}(f)}$, where $\text{ndf}(f)$ counts non-doubled fixed points?
• RQ3 Is there a natural surjection from symplectic Dellac configurations to surjective pistols that preserves the weight $2^{\text{ndf}(f)}$?
• RQ4 Can the Euler characteristic of the symplectic degenerate flag variety $\operatorname{Sp}\mathcal{F}^a_{2n}$ be fully explained via this combinatorial model?

Key findings

• The number of symplectic Dellac configurations of size $2n$ is exactly $r_n = 1,2,10,98,1594,\ldots$, confirming the conjecture of Cerulli Irelli and Feigin.
• The sequence $r_n$ is equal to $\sum_{f \in \mathcal{P}_n} 2^{\text{ndf}(f)}$, where $\mathcal{P}_n$ is the set of surjective pistols of size $n$, and $\text{ndf}(f)$ counts the number of values in $\{2,4,\ldots,2n\}$ that are not doubled fixed points of $f$.
• A surjective map $\Phi: \mathcal{P}_n \to \operatorname{SpDC}_{2n}$ is constructed such that each pistol $f$ corresponds to exactly $2^{\text{ndf}(f)}$ symplectic Dellac configurations.
• The construction of $\Phi$ is algorithmic and reversible, with a labeling procedure that assigns pistol labels to dots in a way that respects the symplectic symmetry and the surjectivity condition.
• The proof establishes that $\sum_{T \in \operatorname{SpDC}_{2n}} 2^{\text{fr}(T)} = \sum_{f \in \mathcal{P}_n} 2^{\text{ndf}(f)}$, where $\text{fr}(T)$ counts free rows in a configuration $T$, and this equality holds via an inductive argument on the number of columns.
• Explicit examples, including the labeling of $T_1 \in \mathcal{T}_7$ and the inverse mapping $\Phi(f_1)$, validate the correctness of the algorithm and the weight preservation property.