[Paper Review] Eulerian polynomials, perfect matchings and Stirling permutations of the second kind

Shi-Mei Ma, Yeong‐Nan Yeh|arXiv (Cornell University)|Jul 5, 2016

Advanced Combinatorial Mathematics22 references3 citations

TL;DR

This paper introduces Stirling permutations of the second kind and establishes combinatorial connections between Eulerian polynomials, perfect matchings, and cycle statistics. It provides generating functions and recurrence relations for polynomials enumerating these permutations by cycle ascent plateaus, fixed points, and cycles, revealing symmetry and real-rootedness properties, with key results linking them to derangements and $2n$-connected graphs via exponential generating functions and context-free grammars.

ABSTRACT

In this paper, we first present combinatorial proofs of a kind of expansions of the Eulerian polynomials of types A and B, and then we introduce Stirling permutations of the second kind. In particular, we count Stirling permutations of the second kind by their cycle ascent plateaus, fixed points and cycles.

Motivation & Objective

• To define and study Stirling permutations of the second kind as a generalization of classical Stirling permutations.
• To establish combinatorial interpretations of Eulerian polynomials of types A and B using perfect matchings.
• To enumerate Stirling permutations by cycle ascent plateaus, fixed points, and cycles using generating functions.
• To derive recurrence relations and symmetry properties for the resulting polynomials.
• To explore connections between Stirling derangements and $2n$-connected graphs through exponential generating functions.

Proposed method

• Introduces $s$-inversion sequences and uses them to express Eulerian polynomials $A_n(x)$ and $B_n(x)$ as generating functions over specific integer sequences.
• Defines Stirling permutations of the second kind via a generalization of the classical $k$-Stirling permutation condition.
• Uses context-free grammars and exponential generating functions to derive recurrence relations for the polynomials $R_n(x,q)$ and $N_n(x)$.
• Applies the exponential formula to derive the generating function $\sum_{n\geq 0} q_n \frac{z^n}{n!} = \frac{e^{-z}}{\sqrt{1-2z}}$ for the number of Stirling derangements.
• Establishes dual relations between generating functions of Stirling derangements and derangements polynomials, such as $S^2(x,z) = d(x,2z)$.
• Employs the operator $\frac{\partial}{\partial x}$ and combinatorial decomposition to derive recurrence $R_{n+1}(x,q) = 2nxR_n(x,q) + 2x(1-x)\frac{\partial}{\partial x}R_n(x,q) + 2nxqR_{n-1}(x,q)$.

Experimental results

Research questions

• RQ1 How can Stirling permutations of the second kind be defined and enumerated by cycle ascent plateaus, fixed points, and cycles?
• RQ2 What is the relationship between the generating functions of Stirling permutations and Eulerian polynomials of types A and B?
• RQ3 How do the polynomials $R_n(x,q)$ and $N_n(x)$ relate to perfect matchings and derangements?
• RQ4 What recurrence relations govern the enumeration of Stirling permutations by cycle structure?
• RQ5 What symmetry and root properties do the resulting polynomials exhibit?

Key findings

• The polynomial $R_n(x)$, enumerating Stirling derangements by cycle ascent plateaus, satisfies the recurrence $R_{n+1}(x,q) = 2nxR_n(x,q) + 2x(1-x)\frac{\partial}{\partial x}R_n(x,q) + 2nxqR_{n-1}(x,q)$ with initial conditions $R_1(x,q)=0$, $R_2(x,q)=2qx$, $R_3(x,q)=4qx(1+x)$.
• The generating function for $R_n(x)$ satisfies $S^2(x,z) = d(x,2z)$, establishing a duality between Stirling derangements and classical derangements.
• The polynomial $R_n(x)$ is symmetric and has only simple real zeros for $n \geq 2$, as shown via connection to known results on real-rootedness.
• Evaluating $S(-1,z)$ yields $\sqrt{\sec(2i z)}$, linking the signed sum of cycle ascent plateaus to the sequence $h_n$ with $h_0=1$, $h_1=2$, $h_2=28$, $h_3=1112$, $h_4=87568$, which counts specific permutations with cycle and excedance constraints.
• The number $q_n$ of Stirling derangements satisfies $q_{n+1} = 2n(q_n + q_{n-1})$ with $q_0=1$, $q_1=0$, $q_2=2$, and its exponential generating function is $\frac{e^{-z}}{\sqrt{1-2z}}$.
• The relation $M_n(x) = x^n N_n(1/x)$ links the polynomials $M_n(x)$ (counting odd-larger entries in perfect matchings) and $N_n(x)$ (counting even-larger entries), revealing a duality between even and odd cycle statistics in matchings.