[Paper Review] Square Partitions and Catalan Numbers

Matthew Bennett, Vyjayanthi Chari+2 more|arXiv (Cornell University)|Dec 25, 2009

Advanced Combinatorial Mathematics9 references3 citations

TL;DR

This paper introduces a novel algorithm that maps square partitions—those with maximal part $k$ and minimal part 1—through iterative application to generate sets of partitions whose cardinality is either a Catalan number $c_{\ell-k+1}$ or twice that value, depending on duality. The algorithm defines a non-isomorphic rooted tree structure and generalizes to ballot numbers via a two-parameter family, with implications for symmetric function modules in representation theory of $\mathfrak{sl}_2$ and affine Lie algebras.

ABSTRACT

For each integer $k\ge 1$, we define an algorithm which associates to a partition whose maximal value is at most $k$ a certain subset of all partitions. In the case when we begin with a partition $λ$ which is square, i.e $λ=λ_1\ge...\geλ_k>0$, and $λ_1=k,λ_k=1$, then applying the algorithm $\ell$ times gives rise to a set whose cardinality is either the Catalan number $c_{\ell-k+1}$ (the self dual case) or twice the Catalan number. The algorithm defines a tree and we study the propagation of the tree, which is not in the isomorphism class of the usual Catalan tree. The algorithm can also be modified to produce a two--parameter family of sets and the resulting cardinalities of the sets are the ballot numbers. Finally, we give a conjecture on the rank of a particular module for the ring of symmetric functions in $2\ell+m$ variables.

Motivation & Objective

• To define an algorithm that transforms square partitions into sets of partitions with cardinality related to Catalan numbers.
• To establish a non-isomorphic tree structure on partition sets, distinct from the standard Catalan tree, using an involution $\tau_k$.
• To generalize the algorithm to produce sets of size equal to ballot numbers $b_{\ell,m}$, extending the Catalan case.
• To conjecture that a certain module of symmetric functions in $2\ell + m$ variables is free of rank $b_{\ell,m-1}$, linking combinatorics to representation theory.

Proposed method

• Define a map $\tau_k$ on partitions with parts $\leq k$, which rotates the complement of a partition in an $n \times k$ rectangle by 180 degrees.
• Apply the algorithm iteratively to square partitions $\lambda$ with $\lambda_1 = k$, $\lambda_k = 1$, generating a tree of partitions.
• Use algebraic techniques (not combinatorial bijections) to prove that the size of the set after $\ell$ iterations is $c_{\ell-k+1}$ or $2c_{\ell-k+1}$.
• Generalize the algorithm to a two-parameter family by modifying the initial conditions and applying a recurrence involving ballot numbers.
• Prove an alternating identity for ballot numbers: $\sum_{j \geq 0} (-1)^j \binom{\ell - j}{j} b_{\ell - j, m} = \binom{m + \ell}{\ell}$.
• Construct a module $\mathbf{M}(\ell,m)$ over the ring of symmetric functions $\Lambda_r$, generated by $\mathbf{p}(\mu)$, and conjecture its freeness with basis indexed by $\cal P^\ell(\Omega_m)$.

Experimental results

Research questions

• RQ1 What is the structure of the set of partitions generated by iteratively applying the algorithm to a square partition?
• RQ2 How does the involution $\tau_k$ induce duality in the partition tree, and why does the resulting tree differ from the standard Catalan tree?
• RQ3 Can the algorithm be generalized to produce sets of size equal to ballot numbers $b_{\ell,m}$?
• RQ4 What is the connection between the combinatorics of these partition sets and the representation theory of $\mathfrak{sl}_2$ and affine Lie algebras?
• RQ5 Is the $\Lambda_r$-module $\mathbf{M}(\ell,m)$ free of rank $b_{\ell,m-1}$, as conjectured?

Key findings

• Applying the algorithm $\ell$ times to a square partition $\lambda$ with $\lambda_1 = k$ and $\lambda_k = 1$ yields a set of partitions of size $c_{\ell-k+1}$ in the self-dual case or $2c_{\ell-k+1}$ otherwise.
• The algorithm defines a rooted, ordered tree structure on partitions that is not isomorphic to the classical Catalan tree.
• The generalized algorithm produces $m$ rooted ordered trees, and the total number of partitions generated is the ballot number $b_{\ell,m}$.
• An alternating identity for ballot numbers is proven: $\sum_{j \geq 0} (-1)^j \binom{\ell - j}{j} b_{\ell - j, m} = \binom{m + \ell}{\ell}$.
• The conjecture is verified for $\ell = 1,2$ and $\ell = 3,4$ with $m = 0,1,2$, supporting the freeness of the module $\mathbf{M}(\ell,m)$.
• The module $\mathbf{M}(\ell,m)$ is conjectured to be free of rank $b_{\ell,m-1}$ over the ring of symmetric functions in $2\ell + m$ variables.