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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...
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In these notes we give a brief introduction to decomposition theory and we summarize some classical and well-known results. The main question is that if a partitioning of a topological space (in other words a decomposition) is given, then what is the topology of the quotient space. The main result is that an upper semi-continuous decomposition yiel...
Citations
... Let us now outline a combinatorial procedure that would (we will carry out this procedure in some but not all cases) prove our conjecture: As noted in [7,8] it is convenient to consider the homogeneous parts of C r/s separately. Therefore, let C d r/s be the part of C r/s of total degree M − d in q and t where M is the number of boxes fully contained in the triangle (0, 0), (s, 0), (s, r). ...
... The first step of Procedure 1 is inspired by previous attempts to create a symmetric string decomposition of Dyck paths such as were made [7,8]. The second step is inspired by previous attempts to solve the problem by giving an explicit bijection that interchanges area and dinv. ...
... We choose d * = 20 because the time needed to actually complete the base case check is still reasonably small. We note that the approach of bounding d was used to combinatorially prove the symmetry of the classical (when r = s + 1) q, t-Catalan polynomial in [7,8]. The former of these restricted to d ∈ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} and the latter extended this to include d = 10 and d = 11. ...
We conjecture a formula for the rational q, t-Catalan polynomial Cr/s that is symmetric in q and t by definition. The conjecture posits that Cr/s can be written in terms of symmetric monomial strings indexed by maximal Dyck paths. We show that for any finite d∗, giving a combinatorial proof of our conjecture on the infinite set of functions {Cr/sd:r≡1mods,d≤d∗} is equivalent to a finite counting problem.
... One important tool for building the chains C μ is the successor map (called nu 1 in this paper, and called ν in [3,6]). For each partition γ with deficit k and dinv i, nu 1 (γ) (if defined) is a partition with deficit k and dinv i + 1. ...
... We now review the definition and basic properties of the original next-up map nu 1 (called ν in [3,6]). For any integer partition γ, recall γ 1 is the first (longest) part of γ, and (γ) is the length (number of positive parts) of γ. ...
... We now define the next-down map nd 1 (called ν −1 in [3,6]). The domain of nd 1 is the set C 1 = {γ : γ 1 ≥ (γ)}. ...
This article is part of an ongoing investigation of the combinatorics of q, t-Catalan numbers . 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 indexed by partitions of size k. Among other structural properties, these chains can be paired to give refinements of the famous symmetry property . Previously, we introduced a map that builds the tail part of each chain . 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 for a (generalized) flagpole partition , assuming that the chains indexed by certain specific smaller partitions (depending on ) are already known. We also give some enumerative and asymptotic results for flagpole partitions and their generalized versions.
... In particular it gives a combinatorial proof of the symmetry of the classical q, t-Catalan polynomial for all d ≤ 20. We note that this had been done up to d ≤ 9 in [7] and up to d ≤ 11 in [8]. In fact, when we reduce our constructions to the m = 1 case there are certain overlaps with the methodologies of these papers. ...
... Remark 1. When m = 1, the maps right and left essentially reduce to the maps ν −1 and ν of [7] and [8]. ...
... Example 4. Let m = 5 and let x 0 = [0, 4,9,12,6,6,6,7,12,8]. For i < 0 define ...
We conjecture a formula for the rational q,t-Catalan polynomial that is symmetric in q and t by definition. Denoting by the homogeneous part of degree d less than the maximum degree appearing in , we prove that our conjecture is correct on the set . In the process we show that for any finite providing a combinatorial proof of the symmetry of the infinite set of functions is equivalent to carrying out a finite number of base case computations depending only on . We provide python code needed to carry out these computations for (or any finite ) as well as python code that can be used to check the conjecture for any relatively prime (s,r) for all d.
... This appendix lists the global chains and values of a, m, h for all deficit partitions µ with 7 ≤ |µ| ≤ 9. The online extended appendix [9] presents this information for k = 10 and k = 11. In the data below, initial objects that do not start new local chains are marked N. ...
... . a = (7,9,11,13,15,17,19,21,23), m = (0, 0, 0, 0, 0, 0, 0, 0, 0), h = (9,9,9,9,9,9,9,9,9). ...
... . a = (7,9,11,13,15,17,19,21,23), m = (0, 0, 0, 0, 0, 0, 0, 0, 0), h = (9,9,9,9,9,9,9,9,9). ...
The q,t-Catalan number enumerates integer partitions contained in an 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 based on decomposing the set of all integer partitions into infinite chains. Each such global chain has an opposite chain ; these combine to give a new small slice of 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 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 of degree at least are symmetric in q and t.
