[Paper Review] The peak statistics on simsun permutations

Shi-Mei Ma, Yeong‐Nan Yeh|arXiv (Cornell University)|Jan 25, 2016

Advanced Combinatorial Mathematics19 references3 citations

TL;DR

This paper investigates peak statistics—specifically left peaks, interior peaks, and up-down runs—on simsun permutations, establishing connections to generating functions, recurrence relations, and real-rootedness. It introduces simsun permutations of the second kind and derives a bivariate generating function that links them to André permutations and cycle-up-down structures, revealing deep combinatorial symmetries via exponential generating functions and differential equations.

ABSTRACT

In this paper, we study the relationship among left peaks, interior peaks and up-down runs of simsun permutations. Properties of the generating polynomials, including the recurrence relation, generating function and real-rootedness are studied. Moreover, we introduce and study simsun permutations of the second kind.

Motivation & Objective

• To analyze the distribution of left peaks, interior peaks, and up-down runs in simsun permutations.
• To establish recurrence relations and generating functions for peak statistics on simsun permutations.
• To introduce and study simsun permutations of the second kind, extending the combinatorial framework.
• To connect the generating functions of simsun permutations to those of André permutations and cycle-up-down structures.
• To investigate real-rootedness and structural properties of the associated polynomials.

Proposed method

• Uses a constructive approach to relate simsun permutations to permutations with a given number of interior peaks via insertion techniques.
• Derives a recurrence relation for the generating polynomial $ S_n(x) $: $ S_{n+1}(x) = (1 + nx)S_n(x) + x(1 - 2x)S_n'(x) $, with $ S_0(x) = 1 $.
• Establishes a closed-form generating function: $ S(x,z) = \left( \frac{\sqrt{2x-1}\sec(\frac{z}{2}\sqrt{2x-1})}{\sqrt{2x-1} - \tan(\frac{z}{2}\sqrt{2x-1})} \right)^2 $.
• Introduces simsun permutations of the second kind via cycle structure and defines a bivariate generating function $ S(x,q;z) = S(x,z)^q $.
• Applies differential equations to the generating function, deriving $ (1 - xz)S_z = qS + x(1 - 2x)S_x $, which characterizes the generating function.
• Uses combinatorial bijections and exponential generating functions to relate simsun permutations to cycle-up-down permutations and André permutations.

Experimental results

Research questions

• RQ1 How are left peaks in simsun permutations related to the descent and peak statistics of other permutation classes?
• RQ2 What is the generating function for the number of simsun permutations by left peaks, and what are its structural properties?
• RQ3 How do interior peak statistics on simsun permutations relate to those on general permutations?
• RQ4 What is the role of up-down runs in the enumeration of simsun permutations?
• RQ5 What are the combinatorial and algebraic properties of simsun permutations of the second kind?

Key findings

• The generating polynomial $ S_n(x) $ for left peaks on simsun permutations satisfies the recurrence $ S_{n+1}(x) = (1 + nx)S_n(x) + x(1 - 2x)S_n'(x) $, with $ S_0(x) = 1 $.
• The exponential generating function $ S(x,z) $ is given by $ \left( \frac{\sqrt{2x-1}\sec(\frac{z}{2}\sqrt{2x-1})}{\sqrt{2x-1} - \tan(\frac{z}{2}\sqrt{2x-1})} \right)^2 $, linking it to trigonometric functions.
• The generating function for simsun permutations of the second kind satisfies $ S(x,q;z) = S(x,z)^q $, establishing a connection to cycle-up-down permutations.
• For $ q > 0 $, the polynomials $ S_n(x,q) $ have nonpositive and simple zeros, indicating real-rootedness.
• The Euler number $ E_{n+1} $ satisfies $ E_{n+1} = \frac{1}{2^n} \sum_{k=0}^n \binom{n}{k} S_k S_{n-k} $, where $ S_n $ is the $ n $th Springer number.
• The generating function $ S(1,q;z) = \frac{1}{(1 - \sin z)^q} $ counts cycle-up-down permutations by the number of cycles, confirming a deep connection to André permutations.