[Paper Review] Crossings and Nestings of Two Edges in Set Partitions

Svetlana Poznanovik, Catherine H. Yan|ArXiv.org|Oct 9, 2007

Advanced Combinatorial Mathematics6 references4 citations

TL;DR

This paper establishes a Klazar-type equivalence for crossings and nestings in set partitions, proving that if two partitions have identical distributions of crossings and nestings over the first two levels of their partition trees, then this equivalence extends to all levels and block counts. The authors use a novel tree structure on set partitions, generating functions, and continued fractions to derive a closed-form continued fraction expansion for the joint generating function of crossings and nestings, generalizing results from matchings to general set partitions with arbitrary block structures.

ABSTRACT

Let $π$ and $λ$ be two set partitions with the same number of blocks. Assume $π$ is a partition of $[n]$. For any integer $l, m \geq 0$, let $\mathcal{T}(π, l)$ be the set of partitions of $[n+l]$ whose restrictions to the last $n$ elements are isomorphic to $π$, and $\mathcal{T}(π, l, m)$ the subset of $\mathcal{T}(π,l)$ consisting of those partitions with exactly $m$ blocks. Similarly define $\mathcal{T}(λ, l)$ and $\mathcal{T}(λ, l,m)$. We prove that if the statistic $cr$ ($ne$), the number of crossings (nestings) of two edges, coincides on the sets $\mathcal{T}(π, l)$ and $\mathcal{T}(λ, l)$ for $l =0, 1$, then it coincides on $\mathcal{T}(π, l,m)$ and $\mathcal{T}(λ, l,m)$ for all $l, m \geq 0$. These results extend the ones obtained by Klazar on the distribution of crossings and nestings for matchings.

Motivation & Objective

• To extend Klazar's results on crossings and nestings in matchings to general set partitions.
• To define a new infinite tree structure on set partitions that generalizes the matching tree but respects the richer block structure of partitions.
• To prove that if two partitions have identical crossing and nesting statistics at levels l=0 and l=1 in their respective trees, then this equality holds for all levels and all block counts.
• To derive a continued fraction expansion for the joint generating function of crossings and nestings rooted at any given partition.
• To provide explicit formulas for the generating functions using Motzkin paths, Charlier diagrams, and binary sequences as combinatorial tools.

Proposed method

• Define a rooted tree $\mathcal{T}(\Pi)$ on all set partitions, where a partition $\pi$ of $[n+1]$ is a child of $\lambda$ of $[n]$ if the restriction of $\pi$ to $\{2,\dots,n+1\}$ is order-isomorphic to $\lambda$.
• Introduce $\mathcal{T}(\pi,l)$ as the set of partitions at level $l$ in the tree rooted at $\pi$, and $\mathcal{T}(\pi,l,m)$ as those with exactly $m$ blocks.
• Use generating functions $b_{l,r}$ to encode the multiset of $cr(\lambda)\alpha + ne(\lambda)\beta$ values over partitions in $\mathcal{T}(\pi,l)$, with recurrence relations involving $[r+1]_{q,p} = \frac{q^{r+1}-p^{r+1}}{q-p}$.
• Establish a recurrence for $b_{l,r}$: $b_{l,r} = b_{l-1,r-1} + (1 + [r+1]_{q,p})b_{l-1,r} + [r+1]_{q,p}b_{l-1,r+1}$, linking levels via weighted path enumeration.
• Connect the generating function $S_\pi(q,p,z)$ to continued fractions by modeling paths from $(l,0)$ to $(0,s)$ with steps $(-1,0)$, $(-1,1)$, $(-1,-1)$, assigning weights based on $[r+1]_{q,p}$ and $1+[r+1]_{q,p}$.
• Derive the continued fraction expansion $S_\pi(q,p,z) = \sum_{s=0}^{k-1} b_{0,s} K_s(z)$, where $K_s(z)$ is a continued fraction involving $[r]_{q,p}$ and $[r]_{q,p}+1$ coefficients.

Experimental results

Research questions

• RQ1 If two set partitions $\pi$ and $\lambda$ have the same number of crossings and nestings at levels $l=0$ and $l=1$ in their respective trees, does this equality extend to all levels $l \geq 0$ and all block counts $m \geq 0$?
• RQ2 Can the joint generating function for crossings and nestings over partitions rooted at a given $\pi$ be expressed as a continued fraction?
• RQ3 What is the structure of the generating function $S_\pi(q,p,z)$ for a partition $\pi$ with $k$ blocks, and how does it depend on the sequence of block contributions $x_i = u_i\alpha + v_i\beta$?
• RQ4 How do combinatorial objects like Motzkin paths and binary sequences encode the recurrence and generating function structure for $b_{l,r}$?
• RQ5 Is there a transformation between different continued fraction forms for the full generating function over all set partitions, and if so, what identities underlie it?

Key findings

• If the statistics $cr$ and $ne$ coincide on $\mathcal{T}(\pi,l)$ and $\mathcal{T}(\lambda,l)$ for $l=0,1$, then they coincide on $\mathcal{T}(\pi,l,m)$ and $\mathcal{T}(\lambda,l,m)$ for all $l,m \geq 0$, establishing a strong invariance property.
• The generating function $S_\pi(q,p,z) = \sum_{l \geq 0} b_{l,0} z^l$ admits a continued fraction expansion $S_\pi(q,p,z) = \sum_{s=0}^{k-1} b_{0,s} K_s(z)$, where $K_s(z)$ is a continued fraction with coefficients $[r+1]_{q,p}$ and $1 + [r+1]_{q,p}$.
• For a singleton block partition $\pi = \{1\}$, the generating function simplifies to $S_{\{1\}}(q,p,z) = \cfrac{1}{1 - ([1]_{q,p}+1)z - \cfrac{[1]_{q,p}z^2}{1 - ([2]_{q,p}+1)z - \cfrac{[2]_{q,p}z^2}{\ddots}}}$.
• The full generating function over all set partitions is $\sum_{n \geq 0} \sum_{\lambda \in \Pi_n} q^{cr(\lambda)} p^{ne(\lambda)} z^n = 1 + \cfrac{z}{1 - ([1]_{q,p}+1)z - \cfrac{[1]_{q,p}z^2}{1 - ([2]_{q,p}+1)z - \cfrac{[2]_{q,p}z^2}{\ddots}}}$.
• The generating function $b_{l,r}$ satisfies a recurrence involving $[r+1]_{q,p}$, which allows the construction of the continued fraction via path enumeration on weighted lattices.
• The continued fraction forms (5.5) and (5.6) are equivalent under a known contraction formula, showing consistency across different parametrizations of the generating function.