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Chain Decompositions of q,t-Catalan Numbers via Local Chains
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Abstract
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.
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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 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 \emph{q,t-Catalan numbers} 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 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 . 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 and all n, the terms in of total degree have the required symmetry property.
The higher q,t-Catalan polynomial 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 and all . We also give a bijective proof of the
joint symmetry property for all and
all . 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 for all
and all . We give analogous results for certain
rational-slope q,t-Catalan polynomials.
The q,t-Catalan numbers can be defined using rational functions, geometry
related to Hilbert schemes, symmetric functions, representation theory, Dyck
paths, partition statistics, or Dyck words. After decades of intensive study,
it was eventually proved that all these definitions are equivalent. In this
paper, we study the similar situation for higher q,t-Catalan numbers, where
the equivalence of the algebraic and combinatorial definitions is still
conjectural. We compute the limits of several versions of the modified higher
q,t-Catalan numbers and show that these limits equal the generating function
for integer partitions. We also identify certain coefficients of the higher
q,t-Catalan numbers as enumerating suitable integer partitions, and we make
some conjectures on the homological significance of the Bergeron-Garsia nabla
operator.
Let R_n be the ring of coinvariants for the diagonal action of the symmetric group S_n. It is known that the character of R_n as a doubly-graded S_n module can be expressed using the Frobenius characteristic map as \nabla e_n, where e_n is the n-th elementary symmetric function, and \nabla is an operator from the theory of Macdonald polynomials. We conjecture a combinatorial formula for \nabla e_n and prove that it has many desirable properties which support our conjecture. In particular, we prove that our formula is a symmetric function (which is not obvious) and that it is Schur positive. These results make use of the theory of ribbon tableau generating functions of Lascoux, Leclerc and Thibon. We also show that a variety of earlier conjectures and theorems on \nabla e_n are special cases of our conjecture. Finally, we extend our conjectures on \nabla e_n and several of the results supporting them to higher powers \nabla^m e_n.
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
whose degree zero part is the ring of symmetric functions Sym[X] over
. We then extend these operators to two larger algebras
and acting on this space, and interpret the right
generalization of the operator as an intertwiner between the two
actions, which is antilinear with respect to the conjugation .
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.
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.
. 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...
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.
We outline 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 coefficients. This has been an open problem since 1994. Because 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 the collection (n) of Dyck paths Pi of length 2n yielding C(n)(q, t) = summation operator(pi) t(a(Pi))q(b(Pi)). Our proof is based on a recursion for C(n)(q, t) suggested by a pair of statistics recently proposed by J. Haglund. One of the byproducts of our results is a proof of the validity of Haglund's conjecture.
In an earlier paper [14], we showed that the Hilbert scheme of points in the plane H
n
=Hilbn
(ℂ2) can be identified with the Hilbert scheme of regular orbits ℂ2n
//S
n
. Using this result, together with a recent theorem of Bridgeland, King and Reid [4] on the generalized McKay correspondence, we prove vanishing theorems for tensor powers of tautological bundles on the Hilbert scheme. We apply the vanishing theorems to establish (among other things) the character formula for diagonal harmonics conjectured by Garsia and the author in [9]. In particular we prove that the dimension of the space of diagonal harmonics is equal to (n+1)n
-1.
- Anton Mellit
Anton Mellit, "Toric braids and (m, n)-parking functions," arXiv:1604.07456 (2016).