88 research outputs found
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].Comment: 30 page
A proof of the Square Paths Conjecture
The modified Macdonald polynomials, introduced by Garsia and Haiman (1996),
have many astounding combinatorial properties. One such class of properties
involves applying the related operator of Bergeron and Garsia (1999)
to basic symmetric functions. The first discovery of this type was the
(recently proven) Shuffle Conjecture of Haglund, Haiman, Loehr, Remmel, and
Ulyanov (2005), which relates the expression to parking functions.
In (2007), Loehr and Warrington conjectured a similar expression for in terms of labeled square paths. In this paper, we extend Haglund and
Loehr's (2005) notion of schedules to labeled square paths and apply this
extension to prove the Square Paths Conjecture
A proof of the shuffle conjecture
We present a proof of the compositional shuffle conjecture, which generalizes
the famous shuffle conjecture for the character of the diagonal coinvariant
algebra. 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 over . We then extend
these operators to an action of an algebra acting on this space,
and interpret the right generalization of the using an involution of
the algebra which is antilinear with respect to the conjugation .Comment: some proofs are expanded. Accepted in JAM
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
A note on passing from a quasi-symmetric function expansion to a Schur function expansion of a symmetric function
Egge, Loehr and Warrington gave in \cite{ELW} a combinatorial formula that
permits to convert the expansion of a symmetric function, homogeneous of degree
, in terms of Gessel's fundamental quasisymmetric functions into an
expansion in terms of Schur functions. Surprisingly the Egge, Loehr and
Warrington result may be shown to be simply equivalent to replacing the Gessel
fundamental by a Schur function indexed by the same composition. In this paper
we give a direct proof of the validity of this replacement. This interpretation
of the result in \cite{ELW} has already been successfully applied to Schur
positivity problems
A new `dinv' arising from the two part case of the Shuffle Conjecture
In a recent paper J. Haglund showed that a certain symmetric function
expresion enumerates by t^{area} q^{dinv} of the parking functions whose
diagonal word is in the shuffle of 12...j and j+1...j+n with k of the cars
j+1,...,j+n in the main diagonal including car j+n in the cell (1,1). In view
of some recent conjectures of Haglund-Morse-Zabrocki it is natural to
conjecture that replacing E_{n,k} by the modified Hall-Littlewood functions
would yield a polynomial that enumerates the same collection of parking
functions but now restricted by the requirement that the Dyck path supporting
cars j+1,...,j+n hits the diagonal according to the composition
p=(p_1,p_2,...,p_k). We prove here this conjecture by deriving a recursion for
the symmetric function expression then using this recursion to construct a new
dinv statistic we will denote ndinv and show that this polynomial enumerates
the latter parking functions by t^{area} q^{ndinv}.Comment: 32 pages, 28 figure
The Delta Conjecture
We conjecture two combinatorial interpretations for the symmetric function
, where is an eigenoperator for the modified
Macdonald polynomials defined by Bergeron, Garsia, Haiman, and Tesler. Both
interpretations can be seen as generalizations of the Shuffle Conjecture of
Haglund, Haiman, Remmel, Loehr, and Ulyanov, which was proved recently by
Carlsson and Mellit. We show how previous work of the third author on Tesler
matrices and ordered set partitions can be used to verify several cases of our
conjectures. Furthermore, we use a reciprocity identity and LLT polynomials to
prove another case. Finally, we show how our conjectures inspire 4-variable
generalizations of the Catalan numbers, extending work of Garsia, Haiman, and
the first author
Rank complement of rational Dyck paths and conjugation of -core partitions
Given a coprime pair of positive integers, rational Catalan numbers
counts two combinatorial objects:rational
-Dyck paths are lattice paths in the rectangle that never go
below the diagonal; -cores are partitions with no hook length equal to
or .Anderson established a bijection between -Dyck paths and
-cores. We define a new transformation, called rank complement, on
rational Dyck paths. We show that rank complement corresponds to conjugation of
-cores under Anderson's bijection. This leads to: i) a new approach to
characterizing -cores; ii) a simple approach for counting the number of
self-conjugate -cores; iii) a proof of the equivalence of two
conjectured combinatorial sum formulas, one over rational -Dyck paths
and the other over -cores, for rational Catalan polynomials.Comment: Updated several references. 15 pages, 5 figure
A simpler formula for the number of diagonal inversions of an (m,n)-Parking Function and a returning Fermionic formula
Recent results have placed the classical shuffle conjecture of Haglund et al.
in a broader context of an infinite family of conjectures about parking
functions in any rectangular lattice. The combinatorial side of the new
conjectures has been defined using a complicated generalization of the dinv
statistic which is composed of three parts and which is not obviously
non-negative. Here we simplify the definition of dinv, prove that it is always
non-negative, and give a geometric description of the statistic in the style of
the classical case. We go on to show that in the n x (n+1) lattice, parking
functions satisfy a fermionic formula that is similar to the one given in the
classical case by Haglund and Loehr
-Positivity Results and Conjectures
In a 2016 ArXiv posting F. Bergeron listed a variety of symmetric functions
with the property that is -positive. A large subvariety
of his examples could be explained by the conjecture that the Dyck path LLT
polynomials exhibit the same phenomenon. In this paper we list the results of
computer explorations which suggest that other examples exhibit the same
phenomenon. We prove two of the resulting conjectures and propose algorithms
that would prove several of our conjectures. In writing this paper we have
learned that similar findings have been independently discovered by Per
Alexandersson.Comment: 19 pages, 15 figure
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