Two Parking Function Bijections: A Sharpening of the q,t-Catalan and Shröder Theorems

In a recent work Jim Haglund, Jennifer Morse and Mike Zabrocki proved a variety of identities involving Hall-Littlewood symmetric functions indexed by compositions. When they applied ∇ to these symmetric functions the resulting identities and computer data led them to some truly remarkable refinements of the shuffle conjecture. We prove here the symmetric function side of a recursion which combined with a recent parking function recursion of Angela Hicks [18] settles some some special cases of the Haglund-Morse-Zabrocki conjectures. Our main result of a compositional q, t-Catalan and Schröder theorem yields as a consequence surprisingly simple new proofs of the original q, t-Catalan and Schröder results.

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“…These identities combined with two beautiful parking function bijections proved by Angela Hicks in [18] yield the following combinatorial result.…”

Section: More Preciselymentioning

confidence: 85%

Hall–Littlewood Operators in the Theory of Parking Functions and Diagonal Harmonics

Garsia

Xin

Zabrocki

2011

International Mathematics Research Notices

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“…Denote v = (1 + u) m+1 u −m , and let the series u i be defined as above. Denote A i = A(u i ), where A(u) is given by (25). Then…”

Section: Proposition 12mentioning

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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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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“…A desirable approach to proving (1.2) would be to determine a recursive formula for D α pX; q, tq, and interpret the result in terms of some commutation relations for ∇. Indeed, this approach has been applied in some important special cases, see [GH02,GXZ12,Hic12]. In [GXZ12], for instance, the authors devise a recursive formula (Proposition 3.12) to prove the Catalan case of the compositional conjecture, extending the results of [GH02].…”

Section: Introductionmentioning

confidence: 99%

A proof of the shuffle conjecture

Carlsson

Mellit

2017

J. Amer. Math. Soc.

142

230

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We present a proof of the compositional shuffle conjecture [HMZ12], which generalizes the famous shuffle conjecture for the character of the diagonal coinvariant algebra [HHL`05b]. We first formulate the combinatorial side of the conjecture in terms of certain operators on a graded vector space V˚whose degree zero part is the ring of symmetric functions SymrXs over Qpq, tq. We then extend these operators to an action of an algebraÃ 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 pq, tq Þ Ñ pq´1, t´1q.2010 Mathematics Subject Classification. 05E05, 05E10, 05A30. 1 (ii) The involution N commutes with d´, and maps y α to N α . (iii) The restriction of N to V 0 " SymrXs agrees with ∇ composed with a conjugation map which essentially exchanges the B α rX; qs and C α rX; qs. It then becomes clear that these properties imply (1.2).While the compositional shuffle conjecture is clearly our main application, the shuffle conjecture has been further generalized in several remarkable directions such as the rational compositional shuffle conjecture, and relationships to knot invariants, double affine Hecke algebras, and the cohomology of the affine Springer fibers, see [BGLX14,GORS14,GN15,Neg13,Hik14,SV11,SV13]. We hope that future applications to these fascinating topics will be forthcoming.1.1. Acknowledgments. The authors would like to thank François Bergeron, Adriano Garsia, Mark Haiman, Jim Haglund, Fernando Rodriguez-Villegas and Guoce Xin for many valuable discussions on this and related topics. The authors also acknowledge the International Center for Theoretical Physics, Trieste, Italy, at which most of the research for this paper was performed. Erik Carlsson was also supported by the Center for Mathematical Sciences and Applications at Harvard University during some of this period, which he gratefully acknowledges.2. The Compositional shuffle conjecture 2.1. Plethystic operators. A λ-ring is a ring R with a family of ring endomorphisms pp i q iPZ ą0 satisfying p 1 rxs " x, p m rp n rxss " p mn rxs, px P R, m, n P Z ą0 q.

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Two Parking Function Bijections: A Sharpening of the q,t-Catalan and Shröder Theorems

Cited by 10 publications

References 7 publications

Hall–Littlewood Operators in the Theory of Parking Functions and Diagonal Harmonics

Hall–Littlewood Operators in the Theory of Parking Functions and Diagonal Harmonics

The representation of the symmetric group on m -Tamari intervals

A proof of the shuffle conjecture

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