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Guoce Xin

Publications and source records attributed to Guoce Xin.

At least 73 records · Page 4Linked to original sources
arXiv2018

On the Sweep Map for Fuss Rational Dyck Paths

Our main contribution here is the discovery of a new family of standard Young tableaux $ {\cal T}^k_n$ which are in bijection with the family ${\cal D}_{m,n}$ of Rational Dyck paths for $m=k\times n\pm 1$ (the so called "Fuss" case). Using this family we give a new proof of the invertibility of the sweep map in the Fuss case by means of a very simple explicit algorithm. This new algorithm has running time $O(m+n)$. It is independent of the Thomas-William algorithm.

math.CO↗
arXiv2018

A Proof the Functional Equation Conjecture

In the early 2000's the first and second named authors worked for a period of six years in an attempt of proving the Compositional Shuffle Conjecture [1]. Their approach was based on the discovery that all the Combinatorial properties predicted by the Compositional Shuffle Conjecture remain valid for each family of Parking Functions with prescribed diagonal cars. The validity of this property was reduced to the proof of a functional equation satisfied by a Catalan family of univariate polynomials. The main result in this paper is a proof of this functional equation. The Compositional Shuffle Conjecture was proved in 2015 by Eric Carlsson and Anton Mellit [3]. Our proof of the Functional Equation removes one of the main obstacles in the completion of the Garsia-Hicks approach to the proof of the Compositional Shuffle Conjecture. At the end of this writing we formulate a few further conjectures including what remains to be proved to complete this approach.

math.CO↗
arXiv2017

On the Schur positivity of $Δ_{e_2} e_n[X]$

Let $\mathbb{N}$ denote the set of non-negative integers. Haglund, Wilson, and the second author have conjectured that the coefficient of any Schur function $s_λ[X]$ in $Δ_{e_k} e_n[X]$ is a polynomial in $\mathbb{N}[q,t]$. We present four proofs of a stronger statement in the case $k=2$; We show that the coefficient of any Schur function $s_λ[X]$ in $Δ_{e_2} e_n[X]$ has a positive expansion in terms of $q,t$-analogs.

math.CO↗
arXiv2017

Inverting the Rational Sweep Map

We present a simple algorithm for inverting the sweep map on rational $(m,n)$-Dyck paths for a co-prime pair $(m,n)$ of positive integers. This work is inspired by Thomas-Williams work on the modular sweep map. A simple proof of the validity of our algorithm is included.

math.CO↗
arXiv2017

Dinv and Area

We give a new combinatorial proof of the well known result that the dinv of an $(m,n)$-Dyck path is equal to the area of its sweep map image. The first proof of this remarkable identity for co-prime $(m,n)$ is due to Loehr and Warrington. There is also a second proof (in the co-prime case) due to Gorsky and Mazin and a third proof due to Mazin.

math.CO↗
arXiv2017

Lecture notes on the Carlsson-Mellit proof of the shuffle conjecture

This note is based on the original proof of the shuffle conjecture by Carlsson and Mellit (arXiv:1508.06239, version 2), which seems to be too concise for the combinatorial community. James Haglund spent a semester to check through the proof line by line and took explicit note. Guoce Xin was asked by Adriano Garsia to elaborate the proof. The whole paper is thoroughly checked and filled in with details. Section 1 and the abstract of the original paper is included. Section 2 is rewritten for clarity. Other sections are filled with detailed proof and computation. We hope this note is accessible to graduate students.

math.CO↗
arXiv2015

An efficient search algorithm for inverting the sweep map on rational Dyck paths

Given a coprime pair $(m,n)$ of positive integers, rational $(m,n)$-Dyck paths are lattice paths in the $m\times n$ rectangle that never go below the diagonal. The sweep map of a rational $(m,n)$-Dyck paths $D$ is the rational Dyck path $Φ(D)$ obtained by sorting the steps of $D$ according to the ranks of their starting points, where the rank of $(a,b)$ is $bm-an$. It is conjectured to be a bijection, but to this date, $Φ$ is only known to be bijective for the Fuss case ($m=kn\pm 1$). In this paper we give an efficient search algorithm for inverting the $Φ$ map. Roughly speaking, given $σ\in \cal D_{m,n}$, by searching through a $d$-array tree of certain depth, we can output all $D$ such that $Φ(D)=σ$, where $d$ is the remainder of $m$ when divided by $n$. In particular, we show that $Φ$ is invertible for the Fuss case by giving a simple recursive construction for $Φ^{-1} (σ)$.

math.CO↗
arXiv2015

Rank complement of rational Dyck paths and conjugation of $(m,n)$-core partitions

Given a coprime pair $(m,n)$ of positive integers, rational Catalan numbers $\frac{1}{m+n} \binom{m+n}{m,n}$ counts two combinatorial objects:rational $(m,n)$-Dyck paths are lattice paths in the $m\times n$ rectangle that never go below the diagonal; $(m,n)$-cores are partitions with no hook length equal to $m$ or $n$.Anderson established a bijection between $(m,n)$-Dyck paths and $(m,n)$-cores. We define a new transformation, called rank complement, on rational Dyck paths. We show that rank complement corresponds to conjugation of $(m,n)$-cores under Anderson's bijection. This leads to: i) a new approach to characterizing $n$-cores; ii) a simple approach for counting the number of self-conjugate $(m,n)$-cores; iii) a proof of the equivalence of two conjectured combinatorial sum formulas, one over rational $(m,n)$-Dyck paths and the other over $(m,n)$-cores, for rational Catalan polynomials.

math.CO↗
arXiv2015

A Euclid style algorithm for MacMahon's partition analysis

Solutions to a linear Diophantine system, or lattice points in a rational convex polytope, are important concepts in algebraic combinatorics and computational geometry. The enumeration problem is fundamental and has been well studied, because it has many applications in various fields of mathematics. In algebraic combinatorics, MacMahon's partition analysis has become a general approach for linear Diophantine system related problems. Many algorithms have been developed, but "bottlenecks" always arise when dealing with complex problems. While in computational geometry, Barvinok's important result asserts the existence of a polynomial time algorithm when the dimension is fixed. However, the implementation by the LattE package of De Loera et. al. does not perform well in many situations. By combining excellent ideas in the two fields, we generalize Barvinok's result by giving a polynomial time algorithm for MacMahon's partition analysis in a suitable condition. We also present an elementary Euclid style algorithm, which might not be polynomial but is easy to implement and performs well. As applications, we contribute the generating series for magic squares of order 6.

math.CO↗
arXiv2015

Hankel determinant solutions to several discrete integrable systems and the Laurent Property

Many discrete integrable systems exhibit the Laurent phenomenon. In this paper, we investigate three integrable systems: the Somos-4 recurrence, the Somos-5 recurrence and a system related to so-called $A_1$ $Q$-system, whose general solutions are derived in terms of Hankel determinant. As a result, we directly confirm that they satisfy the Laurent property. Additionally, it is shown that the Somos-5 recurrence can be viewed as a specified Bäcklund transformation of the Somos-4 recurrence. The related topics about Somos polynomials are also studied.

math-ph↗
arXiv2014

Compositional (km,kn)-Shuffle Conjectures

In 2008, Haglund, Morse and Zabrocki formulated a Compositional form of the Shuffle Conjecture of Haglund et al. In very recent work, Gorsky and Negut by combining their discoveries with the work of Schiffmann-Vasserot on the symmetric function side and the work of Hikita and Gorsky-Mazin on the combinatorial side, were led to formulate an infinite family of conjectures that extend the original Shuffle Conjecture of Haglund et al. In fact, they formulated one conjecture for each pair (m,n) of coprime integers. This work of Gorsky-Negut leads naturally to the question as to where the Compositional Shuffle Conjecture of Haglund-Morse-Zabrocki fits into these recent developments. Our discovery here is that there is a compositional extension of the Gorsky-Negut Shuffle Conjecture for each pair (km,kn), with (m,n) co-prime and k > 1.

math.CO↗
arXiv2014

Some remarkable new Plethystic Operators in the Theory of Macdonald Polynomials

In the 90's a collection of Plethystic operators were introduced in [3], [7] and [8] to solve some Representation Theoretical problems arising from the Theory of Macdonald polynomials. This collection was enriched in the research that led to the results which appeared in [5], [6] and [9]. However since some of the identities resulting from these efforts were eventually not needed, this additional work remained unpublished. As a consequence of very recent publications [4], [11], [19], [20], [21], a truly remarkable expansion of this theory has taken place. However most of this work has appeared in a language that is virtually inaccessible to practitioners of Algebraic Combinatorics. Yet, these developments have led to a variety of new conjectures in [2] in the Combinatorics and Symmetric function Theory of Macdonald Polynomials. The present work results from an effort to obtain in an elementary and accessible manner all the background necessary to construct the symmetric function side of some of these new conjectures. It turns out that the above mentioned unpublished results provide precisely the tools needed to carry out this project to its completion.

math.CO↗
arXiv2013

A three shuffle case of the compositional parking function conjecture

We prove here that the polynomial q, t-enumerates, by the statistics dinv and area, the parking functions whose supporting Dyck path touches the main diagonal according to the composition p of size a + b + c and have a reading word which is a shuffle of one decreasing word and two increasing words of respective sizes a, b, c. Here Cp(1) is a rescaled Hall-Littlewood polynomial and "nabla" is the Macdonald eigenoperator introduced in [1]. This is our latest progress in a continued effort to settle the decade old shuffle conjecture of [14]. It includes as special cases all previous results connected with this conjecture such as the q, t-Catalan [3] and the Schroder and h, h results of Haglund in [12] as well as their compositional refinements recently obtained in [9] and [10]. It also confirms the possibility that the approach adopted in [9] and [10] has the potential to yield a resolution of the shuffle parking function conjecture as well as its compositional refinement more recently proposed by Haglund, Morse and Zabrocki in [15].

math.CO↗
arXiv2013

Hermite Reduction and Creative Telescoping for Hyperexponential Functions

We present a reduction algorithm that simultaneously extends Hermite's reduction for rational functions and the Hermite-like reduction for hyperexponential functions. It yields a unique additive decomposition and allows to decide hyperexponential integrability. Based on this reduction algorithm, we design a new method to compute minimal telescopers for bivariate hyperexponential functions. One of its main features is that it can avoid the costly computation of certificates. Its implementation outperforms Maple's function DEtools[Zeilberger]. Moreover, we derive an order bound on minimal telescopers, which is more general and tighter than the known one.

cs.SC↗
arXiv2011

On Zeilberger's Constant Term for Andrews' TSSCPP Theorem

This paper studies Zeilberger's two prized constant term identities. For one of the identities, Zeilberger asked for a simple proof that may give rise to a simple proof of Andrews theorem for the number of totally symmetric self complementary plane partitions. We obtain an identity reducing a constant term in $2k$ variables to a constant term in $k$ variables. As applications, Zeilberger's constant terms are converted to single determinants. The result extends for two classes of matrices, the sum of all of whose full rank minors is converted to a single determinant. One of the prized constant term problems is solved, and we give a seemingly new approach to Macdonald's constant term for root system of type BC.

math.CO↗
arXiv2008

Enumeration of bilaterally symmetric 3-noncrossing partitions

Schutzenberger's theorem for the ordinary RSK correspondence naturally extends to Chen et. al's correspondence for matchings and partitions. Thus the counting of bilaterally symmetric $k$-noncrossing partitions naturally arises as an analogue for involutions. In obtaining the analogous result for 3-noncrossing partitions, we use a different technique to develop a Maple package for 2-dimensional vacillating lattice walk enumeration problems. The package also applies to the hesitating case. As applications, we find several interesting relations for some special bilaterally symmetric partitions.

math.CO↗