The case $k=2$ of the Shuffle Conjecture

Garsia, Adriano M.; Hicks, Angela; Stout, Andrew R.

doi:10.4310/joc.2011.v2.n2.a2

Cited by 5 publications

(5 citation statements)

References 7 publications

(11 reference statements)

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“…In the remainder of this section, we recall some of the conjectured connections between Tamari intervals and trivariate diagonal coinvariant spaces. They seem to parallel the (now largely proved) connections between ballot paths and bivariate diagonal coinvariant spaces, which have attracted considerable attention in the past 20 years [14,17,18,21,24,23,29] and are still a very active area of research today [1,2,13,16,22,19,25,28]. Let X = (x i,j )1≤i≤k 1≤j≤n be a matrix of variables.…”

Section: Introduction and Main Resultsmentioning

confidence: 75%

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The representation of the symmetric group on m -Tamari intervals

Bousquet-Mélou

Chapuy²,

Préville-Ratelle³

2013

Advances in Mathematics

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An m-ballot path of size n is a path on the square grid consisting of north and east unit steps, starting at (0, 0), ending at (mn, n), and never going below the line {x = my}. The set of these paths can be equipped with a lattice structure, called the m-Tamari lattice and denoted by T (m) n , which generalizes the usual Tamari lattice T n obtained when m = 1. This lattice was introduced by F. Bergeron in connection with the study of diagonal coinvariant spaces in three sets of n variables. The representation of the symmetric group S n on these spaces is conjectured to be closely related to the natural representation of S n on (labeled) intervals of the m-Tamari lattice, which we study in this paper.Anis labeled if the north steps of Q are labeled from 1 to n in such a way the labels increase along any sequence of consecutive north steps. The symmetric group S n acts on labeled intervals of T (m) n by permutation of the labels. We prove an explicit formula, conjectured by F. Bergeron and the third author, for the character of the associated representation of S n . In particular, the dimension of the representation, that is, the number of labeled m-Tamari intervals of size n, is found to be ✩ G. 310M. Bousquet-Mélou et al. / Advances in Mathematics 247 (2013) (m + 1) n (mn + 1) n−2 .These results are new, even when m = 1.The form of these numbers suggests a connection with parking functions, but our proof is not bijective. The starting point is a recursive description of m-Tamari intervals. It yields an equation for an associated generating function, which is a refined version of the Frobenius series of the representation. This equation involves two additional variables x and y, a derivative with respect to y and iterated divided differences with respect to x. The hardest part of the proof consists in solving it, and we develop original techniques to do so, partly inspired by previous work on polynomial equations with "catalytic" variables.

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Section: Introduction and Main Resultsmentioning

confidence: 75%

“…This is the second functional equation satisfied by F (x, y) given in Proposition 5. The third one, (13), follows by differentiating with respect to y.…”

Section: From the Construction To The Functional Equationmentioning

confidence: 99%

The representation of the symmetric group on m -Tamari intervals

Bousquet-Mélou

Chapuy²,

Préville-Ratelle³

2013

Advances in Mathematics

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“…a,b originally discovered by Haglund [2004]. Below we reproduce a simple surjective proof of the latter which was given by the first author and Stout in Garsia et al [2011].…”

Section: Introductionmentioning

confidence: 67%

“…Below we reproduce a simple surjective proof of the latter which was given by the first author and Stout in Garsia et al [2011]. In point of fact, Haglund [2004] proved, by a highly non-trivial sequence of manipulations, that Parkqt…”

Section: Introductionmentioning

confidence: 99%

A refinement of the Shuffle Conjecture with cars of two sizes and $t=1/q$

Hicks

Leven

2014

Journal of Combinatorics

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The original Shuffle Conjecture of Haglund et al. has a symmetric function side and a combinatorial side. The symmetric function side may be simply expressed as ∇e n , h µ where ∇ is the Macdonald polynomial eigen-operator of Bergeron and Garsia and h µ is the homogeneous basis indexed by µ = (µ 1 , µ 2 , . . . , µ k ) n. The combinatorial side q,tenumerates a family of Parking Functions whose reading word is a shuffle of k successive segments of 123 · · · n of respective lengths µ 1 , µ 2 , . . . , µ k . It can be shown that for t = 1/q the symmetric function side reduces to a product of q-binomial coefficients and powers of q. This reduction suggests a surprising combinatorial refinement of the general Shuffle Conjecture. Here we prove this refinement for k = 2 and t = 1/q. The resulting formula gives a q-analogue of the well studied Narayana numbers.

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“…which is the q, t-Catalan result of [3]. Setting c = 0 or a = 0 gives the following two identities proved by Haglund in [11], Namely the Schröder result (1.15) ∇e n , e k h n−k = P F ∈PF n t area(P F ) q dinv(P F ) χ σ(P F ) ∈ ↓A ∪∪B and the shuffle of two segments result [11] (see also [12] and [7])…”

Section: Introductionmentioning

confidence: 99%

A three shuffle case of the compositional parking function conjecture

Garsia¹,

Xin²,

Zabrocki³

2014

Journal of Combinatorial Theory, Series A

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We prove here that the polynomial ∇Cp 1 , eah b hc 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 |= 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 ∇ is the Macdonald eigen-operator introduced by Bergeron and Garsia [1]. This is our latest progress in a continued effort to settle the decade old shuffle conjecture of Haglund et. al. [13]. This result includes as special cases all previous results connected with the shuffle conjecture such as the q, t-Catalan [3] and the Schröder and h, h results of Haglund in [11] as well as their compositional refinements recently obtained by the authors in [8] and [9]. It also confirms the possibility that the approach adopted in [8] and [9] 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 [14].

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The case $k=2$ of the Shuffle Conjecture

Cited by 5 publications

References 7 publications

The representation of the symmetric group on m -Tamari intervals

The representation of the symmetric group on m -Tamari intervals

A refinement of the Shuffle Conjecture with cars of two sizes and $t=1/q$

A three shuffle case of the compositional parking function conjecture

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