Seongjune Han’s research while affiliated with University of Alabama and other places

This article is part of an ongoing investigation of the combinatorics of q, t-Catalan numbers ${{\,\mathrm{Cat}\,}}_n(q,t)$. We develop a structure theory for integer partitions based on the partition statistics dinv, deficit, and minimum triangle height. Our goal is to decompose the infinite set of partitions of deficit k into a disjoint union of chains ${\mathcal {C}}_{\mu }$ indexed by partitions of size k. Among other structural properties, these chains can be paired to give refinements of the famous symmetry property ${{\,\mathrm{Cat}\,}}_n(q,t)={{\,\mathrm{Cat}\,}}_n(t,q)$. Previously, we introduced a map that builds the tail part of each chain ${\mathcal {C}}_{\mu }$. Our first main contribution here is to extend this map to construct larger second-order tails for each chain. Second, we introduce new classes of partitions called flagpole partitions and generalized flagpole partitions. Third, we describe a recursive construction for building the chain ${\mathcal {C}}_{\mu }$ for a (generalized) flagpole partition $\mu$, assuming that the chains indexed by certain specific smaller partitions (depending on $\mu$) are already known. We also give some enumerative and asymptotic results for flagpole partitions and their generalized versions.

This article is part of an ongoing investigation of the combinatorics of q,t-Catalan numbers $\textrm{Cat}_n(q,t)$. We develop a structure theory for integer partitions based on the partition statistics dinv, deficit, and minimum triangle height. Our goal is to decompose the infinite set of partitions of deficit k into a disjoint union of chains $\mathcal{C}_{\mu}$ indexed by partitions of size k. Among other structural properties, these chains can be paired to give refinements of the famous symmetry property $\textrm{Cat}_n(q,t)=\textrm{Cat}_n(t,q)$. Previously, we introduced a map NU that builds the tail part of each chain $\mathcal{C}_{\mu}$. Our first main contribution here is to extend \NU and construct larger second-order tails for each chain. Second, we introduce new classes of partitions (flagpole partitions and generalized flagpole partitions) and give a recursive construction of the full chain $\mathcal{C}_{\mu}$ for generalized flagpole partitions $\mu$.

The q, t-Catalan number Catn(q,t) enumerates integer partitions contained in an n×n triangle by their dinv and external area statistics. The paper by Lee et al. (SIAM J Discr Math 32:191–232, 2018) proposed a new approach to understanding the symmetry property Catn(q,t)=Catn(t,q) based on decomposing the set of all integer partitions into infinite chains. Each such global chain Cμ has an opposite chain Cμ∗; these combine to give a new small slice of Catn(q,t) that is symmetric in q and t. Here, we advance the agenda of Lee et al. (SIAM J Discr Math 32:191–232, 2018) by developing a new general method for building the global chains Cμ from smaller elements called local chains. We define a local opposite property for local chains that implies the needed opposite property of the global chains. This local property is much easier to verify in specific cases compared to the corresponding global property. We apply this machinery to construct all global chains for partitions with deficit at most 11. This proves that for all n, the terms in Catn(q,t) of degree at least n2-11 are symmetric in q and t.

The q,t-Catalan number $\mathrm{Cat}_n(q,t)$ enumerates integer partitions contained in an $n\times n$ triangle by their dinv and external area statistics. The paper [LLL18 (Lee, Li, Loehr, SIAM J. Discrete Math. 32(2018))] proposed a new approach to understanding the symmetry property $\mathrm{Cat}_n(q,t)=\mathrm{Cat}_n(t,q)$ based on decomposing the set of all integer partitions into infinite chains. Each such global chain $\mathcal{C}_{\mu}$ has an opposite chain $\mathcal{C}_{\mu^*}$; these combine to give a new small slice of $\mathrm{Cat}_n(q,t)$ that is symmetric in q and t. Here we advance the agenda of [LLL18] by developing a new general method for building the global chains $\mathcal{C}_{\mu}$ from smaller elements called local chains. We define a local opposite property for local chains that implies the needed opposite property of the global chains. This local property is much easier to verify in specific cases compared to the corresponding global property. We apply this machinery to construct all global chains for partitions with deficit at most 11. This proves that for all n, the terms in $\mathrm{Cat}_n(q,t)$ of degree at least $\binom{n}{2}-11$ are symmetric in q and t.