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Chain Decompositions of q, t-Catalan Numbers via Local Chains

December 2020
Annals of Combinatorics 24(4):1-27

Authors:

Virginia Tech

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.

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Finding the minimum triangle sizes for the sequence (νi(γ):i≥0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\nu ^i(\gamma ): i\ge 0)$$\end{document}. The first-return point for each object is starred

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Structure of an ordinary local chain

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Ann. Comb. 24 (2020) 739–765

2020 Springer Nature Switzerland AG

Published online October 4, 2020

https://doi.org/10.1007/s00026-020-00512-5 Annals of Combinatorics

Chain Decompositions of q, t-Catalan

Numbers via Local Chains

Seongjune Han, Kyungyong Lee, Li Li and Nicholas A. Loehr

Abstract. The q, t-Catalan number Catn(q, t) enumerates integer parti-

tions contained in an n×ntriangle 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 prop-

erty Catn(q, t)=Cat

n(t, q) based on decomposing the set of all integer

partitions into inﬁnite chains. Each such global chain Cµhas an opposite

chain Cµ∗; these combine to give a new small slice of Catn(q, t)thatis

symmetric in qand 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 Cµfrom 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,thetermsinCat

n(q, t)of

degree at least n

2−11 are symmetric in qand t.

Mathematics Subject Classiﬁcation. 05A19, 05A17, 05E05.

Keywords. q, t-Catalan numbers, Dyck paths, Dinv statistic, Joint sym-

metry, Integer partitions, Chain decompositions.

1. Introduction

The q, t-Catalan numbers Catn(q, t) are polynomials in qand tthat reduce

to the ordinary Catalan numbers when q=t= 1. These polynomials play a

prominent role in modern algebraic combinatorics, with connections to rep-

resentation theory, algebraic geometry, symmetric functions, knot theory, and

Kyungyong Lee was supported by NSF Grant DMS 1800207, the Korea Institute for Ad-

vanced Study (KIAS), and the University of Alabama. This work was supported by a grant

from the Simons Foundation/SFARI (Grant #633564 to Nicholas A. Loehr).

Content courtesy of Springer Nature, terms of use apply. Rights reserved.

A Conjectured Formula for the Rational q,t

\boldsymbol{q},\boldsymbol{t}

-Catalan Polynomial

Article

Full-text available

Sep 2023
ANN COMB

Graham Hawkes

We conjecture a formula for the rational q, t-Catalan polynomial Cr/s

{\mathcal {C}}_{r/s}

that is symmetric in q and t by definition. The conjecture posits that Cr/s

{\mathcal {C}}_{r/s}

can be written in terms of symmetric monomial strings indexed by maximal Dyck paths. We show that for any finite d∗

d^*

, giving a combinatorial proof of our conjecture on the infinite set of functions {Cr/sd:r≡1mods,d≤d∗}

\{ {\mathcal {C}}_{r/s}^d: r\equiv 1 \mod s, \,\,\, d \le d^*\}

is equivalent to a finite counting problem.

A conjectured formula for the rational q,t-Catalan polynomial

Preprint

Full-text available

Jul 2022

Graham Hawkes

We conjecture a formula for the rational q,t-Catalan polynomial

\mathcal{C}_{r/s}

that is symmetric in q and t by definition. Denoting by

\mathcal{C}_{r/s}^d

the homogeneous part of degree d less than the maximum degree appearing in

\mathcal{C}_{r/s}

, we prove that our conjecture is correct on the set

\{\mathcal{C}_{r/s}^d: r \cong 1\mod s, \, d \leq 20\}

. In the process we show that for any finite

d^*

providing a combinatorial proof of the symmetry of the infinite set of functions

\{\mathcal{C}_{r/s}^d: r \cong 1\mod s, \, d \leq d^*\}

is equivalent to carrying out a finite number of base case computations depending only on

d^*

. We provide python code needed to carry out these computations for

d^*=20

(or any finite

d^*

) as well as python code that can be used to check the conjecture for any relatively prime (s,r) for all d.

Chain Decompositions of q, t-Catalan Numbers: Tail Extensions and Flagpole Partitions

Article

Full-text available

Sep 2022
ANN COMB

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.

Chain Decompositions of q,t-Catalan Numbers via Local Chains

Preprint

Mar 2020

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.

A Combinatorial Approach to the Symmetry of q,t-Catalan Numbers

Article

Full-text available

Feb 2016

The \emph{q,t-Catalan numbers}

C_n(q,t)

are polynomials in q and t that reduce to the ordinary Catalan numbers when q=t=1. These polynomials have important connections to representation theory, algebraic geometry, and symmetric functions. Haglund and Haiman discovered combinatorial formulas for

C_n(q,t)

as weighted sums of Dyck paths (or equivalently, integer partitions contained in a staircase shape). This paper undertakes a combinatorial investigation of the joint symmetry property

C_n(q,t)=C_n(t,q)

. We conjecture some structural decompositions of Dyck objects into "mutually opposite" subcollections that lead to a bijective explanation of joint symmetry in certain cases. A key new idea is the construction of infinite chains of partitions that are independent of n but induce the joint symmetry for all n simultaneously. Using these methods, we prove combinatorially that for

0\leq k\leq 9

and all n, the terms in

C_n(q,t)

of total degree

\binom{n}{2}-k

have the required symmetry property.

Rational Parking Functions and Catalan Numbers

Article

Full-text available

Mar 2016
ANN COMB

The “classical” parking functions, counted by the Cayley number (n+1)n−1, carry a natural permutation representation of the symmetric group S n in which the number of orbits is the Catalan number

{\frac{1}{n+1} \left( \begin{array}{ll} 2n \\ n \end{array} \right)}

. In this paper, we will generalize this setup to “rational” parking functions indexed by a pair (a, b) of coprime positive integers. These parking functions, which are counted by b a−1, carry a permutation representation of S a in which the number of orbits is the “rational” Catalan number

{\frac{1}{a+b} \left( \begin{array}{ll} a+b \\ a \end{array} \right)}

. First, we compute the Frobenius characteristic of the S a -module of (a, b)-parking functions, giving explicit expansions of this symmetric function in the complete homogeneous basis, the power-sum basis, and the Schur basis. Second, we study q-analogues of the rational Catalan numbers, conjecturing new combinatorial formulas for the rational q-Catalan numbers

{\frac{1}{[a+b]_{q}} {{\left[ \begin{array}{ll} a+b \\ a \end{array} \right]}_{q}}}

and for the q-binomial coefficients

{{{\left[ \begin{array}{ll} n \\ k \end{array} \right]}_{q}}}

. We give a bijective explanation of the division by [a+b]q that proves the equivalence of these two conjectures. Third, we present combinatorial definitions for q, t-analogues of rational Catalan numbers and parking functions, generalizing the Shuffle Conjecture for the classical case. We present several conjectures regarding the joint symmetry and t = 1/q specializations of these polynomials. An appendix computes these polynomials explicitly for small values of a and b.

Combinatorics of certain higher q,t-Catalan polynomials: Chains, joint symmetry, and the Garsia-Haiman formula

Article

Full-text available

Nov 2012

The higher q,t-Catalan polynomial

C^{(m)}_n(q,t)

can be defined combinatorially as a weighted sum of lattice paths contained in certain triangles, or algebraically as a complicated sum of rational functions indexed by partitions of n. This paper proves the equivalence of the two definitions for all

m\geq 1

and all

n\leq 4

. We also give a bijective proof of the joint symmetry property

C^{(m)}_n(q,t)=C^{(m)}_n(t,q)

for all

m\geq 1

and all

n\leq 4

. The proof is based on a general approach for proving joint symmetry that dissects a collection of objects into chains, and then passes from a joint symmetry property of initial points and terminal points to joint symmetry of the full set of objects. Further consequences include unimodality results and specific formulas for the coefficients in

C^{(m)}_n(q,t)

for all

m\geq 1

and all

n\leq 4

. We give analogous results for certain rational-slope q,t-Catalan polynomials.

Toric braids and (m,n)-parking functions

Article

Apr 2016
DUKE MATH J

Anton Mellit

The Dyck path algebra construction of Carlsson and Mellit from arXiv:1508.06239 is interpreted as a representation of "the positive part" of the group of toric braids. Then certain sums over (m,n)-parking functions are related to evaluations of this representation on some special braids. The compositional (km,kn)-shuffle conjecture of Bergeron, Garsia, Leven and Xin from arXiv:1404.4616 is then shown to be a corollary of this relation.

A proof of the shuffle conjecture

Article

Aug 2015

We present a proof of the compositional shuffle conjecture \cite{haglund2012compositional}, which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra \cite{haglund2005diagcoinv}. We first formulate the combinatorial side of the conjecture in terms of certain operators on a graded vector space

V_\cdot

whose degree zero part is the ring of symmetric functions Sym[X] over

\mathbb{Q}(q,t)

. We then extend these operators to two larger algebras

\mathbb{A}

and

\mathbb{A}^*

acting on this space, and interpret the right generalization of the

\nabla

operator as an intertwiner between the two actions, which is antilinear with respect to the conjugation

(q,t)\mapsto (q^{-1},t^{-1})

The q, t-Catalan Numbers and the Space of Diagonal Harmonics with an Appendix on the Combinatorics of

Article

James Haglund

A Remarkable q, t-Catalan Sequence and q-Lagrange Inversion

Article

Jul 1996

We introduce a rational function C n(q, t) and conjecture that it always evaluates to a polynomial in q, t with non-negative integer coefficients summing to the familiar Catalan number \frac{1}{{n + 1}}\left( {\begin{array}{*{20}c} {2n} \\ n \\ \end{array} } \right). We give supporting evidence by computing the specializations D_n \left( q \right) = C_n \left( {q{1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-\nulldelimiterspace} q}} \right)q^{\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)} and C n(q) = C n(q, 1) = C n(1,q). We show that, in fact, D n(q)q -counts Dyck words by the major index and C n(q) q -counts Dyck paths by area. We also show that C n(q, t) is the coefficient of the elementary symmetric function e n in a symmetric polynomial DHn(x; q, t) which is the conjectured Frobenius characteristic of the module of diagonal harmonic polynomials. On the validity of certain conjectures this yields that C n(q, t) is the Hilbert series of the diagonal harmonic alternants. It develops that the specialization DHn(x; q, 1) yields a novel and combinatorial way of expressing the solution of the q-Lagrange inversion problem studied by Andrews [2], Garsia [5] and Gessel [11]. Our proofs involve manipulations with the Macdonald basis {P μ(x; q, t)}μ which are best dealt with in λ-ring notation. In particular we derive here the λ-ring version of several symmetric function identities.

Conjectured statistics for the q,t-Catalan numbers

Article

Oct 2000

James Haglund

We introduce the distribution function Fn(q,t) of a pair of statistics on Catalan words and conjecture Fn(q,t) equals Garsia and Haiman's q,t-Catalan sequence Cn(q,t), which they defined as a sum of rational functions. We show that Fn,s(q,t), defined as the sum of these statistics restricted to Catalan words ending in s ones, satisfies a recurrence relation. As a corollary we are able to verify that Fn(q,t)=Cn(q,t) when t=1/q. We also show the partial symmetry relation Fn(q,1)=Fn(1,q). By modifying a proof of Haiman of a q-Lagrange inversion formula based on results of Garsia and Gessel, we obtain a q-analogue of the general Lagrange inversion formula which involves Catalan words grouped according to the number of ones at the end of the word.

A proof of the q,t-Catalan positivity conjecture

Article

Oct 2002
DISCRETE MATH

. We present here a proof that a certain rational function C n (q, t) which has come to be known as the "q, t-Catalan" is in fact a polynomial with positive integer coe#cients. This has been an open problem since 1994. The precise form of the conjecture is given in the J. Algebraic Combin. 5 (1996), no. 3, 191--244, where it is further conjectured that C n (q, t) is the Hilbert Series of the Diagonal Harmonic Alternants in the variables (x 1 ,x 2 ,...,x n ;y 1 ,y 2 ,...,y n) . Since C n (q, t) evaluates to the Catalan number at t = q =1, it has also been an open problem to find a pair of statistics a(#),b(#)on Dyck paths # in the nn square yielding C n (q, t)= P # t a(#) q b(#) . Our proof is based on a recursion for C n (q, t) suggested by a pair of statistics a(#),b(#) recently proposed by J. Haglund. Thus one of the byproducts of our developments is a proof of the validity of Haglund's conjecture. It should also be noted that our arguments rely and expand on the plethystic m...

A continuous family of partition statistics equidistributed with length

Article

Feb 2009

This article investigates a remarkable generalization of the generating function that enumerates partitions by area and number of parts. This generating function is given by the infinite product ∏i⩾11/(1−tqi). We give uncountably many new combinatorial interpretations of this infinite product involving partition statistics that arose originally in the context of Hilbert schemes. We construct explicit bijections proving that all of these statistics are equidistributed with the length statistic on partitions of n. Our bijections employ various combinatorial constructions involving cylindrical lattice paths, Eulerian tours on directed multigraphs, and oriented trees.

Chain Decompositions of q, t-Catalan Numbers via Local Chains

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