Results and conjectures on simultaneous core partitions

Armstrong, Drew; Hanusa, Christopher R. H.; Jones, Brant

doi:10.1016/j.ejc.2014.04.007

Cited by 78 publications

(128 citation statements)

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“…In this section we present four combinatorial descriptions for computing the zeta and eta maps, starting with an interpretation involving core partitions implicit in [1], followed by with an equivalent description via the sweep maps considered in [3]. Our main contributions are two new combinatorial descriptions of the zeta map involving interval intersections and a laser filling, along with the study of the eta map in all four contexts.…”

Section: The Zeta Map (And Eta)mentioning

confidence: 98%

“…See, for example, [1][2][3]12], with equivalence of many definitions given in [3]. The precise description of zeta depends on making some choices; in our experience, these choices always resolve into one of two distinct maps, which we call zeta and eta.…”

Section: The Zeta Map (And Eta)mentioning

confidence: 99%

“…The reverse reading permutation of P , denoted τ (P ), encodes the relative order of the values in M (P ). In our running example, the one-line notation for σ(P ) and τ (P ) are σ(P ) = [1,3,7,12,9,13,11,8,5,10,6,4,2] and τ (P ) = [1,2,4,6,10,5,8,11,13,9,12,7,3].…”

Section: Dictionary Of Notationmentioning

confidence: 99%

“…Section 2 sets the stage by introducing key combinatorial concepts and statistics associated to (a, b)-Dyck paths. In Section 3 we investigate the skew length statistic sl(P ), originally defined in the context of (a, b)-cores in [1]. The original definition of skew length seems to depend on the ordering of a and b; we show that skew length is in fact independent of this choice.…”

Section: Introductionmentioning

confidence: 99%

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Combinatorics of the zeta map on rational Dyck paths

Ceballos

Denton

Hanusa

2016

Journal of Combinatorial Theory, Series A

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An (a, b)-Dyck path P is a lattice path from (0, 0) to (b, a) that stays above the line y = a b x. The zeta map is a curious rule that maps the set of (a, b)-Dyck paths into itself; it is conjecturally bijective, and we provide progress towards proof of bijectivity in this paper, by showing that knowing zeta of P and zeta of P conjugate is enough to recover P . Our method begets an area-preserving involution χ on the set of (a, b)-Dyck paths when ζ is a bijection, as well as a new method for calculating ζ −1 on classical Dyck paths. For certain nice (a, b)-Dyck paths we give an explicit formula for ζ −1 and χ and for additional (a, b)-Dyck paths we discuss how to compute ζ −1 and χ inductively. We also explore Armstrong's skew length statistic and present two new combinatorial methods for calculating the zeta map involving lasers and interval intersections. We provide a combinatorial statistic δ that can be used to recursively ✩ compute ζ −1 and show that δ is computable from ζ(P ) in the Fuss-Catalan case.

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Section: The Zeta Map (And Eta)mentioning

confidence: 98%

Section: The Zeta Map (And Eta)mentioning

confidence: 99%

Section: Dictionary Of Notationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Combinatorics of the zeta map on rational Dyck paths

Ceballos

Denton

Hanusa

2016

Journal of Combinatorial Theory, Series A

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“…For example, if π is the path on the left in Figure 23 This problem has been studied by Gorsky, Mazin, and Vazirani [GMV14] and Armstrong, Loehr, and Warrington [ALW14]. See also [AHJ14]. In [GMV14] it is shown that the sweep map is a bijection whenever m = kn + 1 or m = kn − 1 for some positive integer k. We note that in the case m = kn + 1 Loehr [Loe03], [Loe05b] (see also [Hag08][pp.…”

Section: Tesler Matrices and The Superpolynomialmentioning

confidence: 99%

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Haglund¹

2015

Handbook of Enumerative Combinatorics

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This chapter contains an account of a two-parameter version of the Catalan numbers, and corresponding two-parameter versions of related objects such as parking functions and Schröder paths, which have become important in algebraic combinatorics and other areas of mathematics as well. Although the original motivation for the definition of these objects was the study of Macdonald polynomials and the representation theory of diagonal harmonics, in this account we focus only on the combinatorics associated to their description in terms of lattice paths. Hence this chapter can be read by anyone with a modest background in combinatorics. In Section 1 we include basic facts involving q-analogues, permutation statistics, and symmetric functions which we need in later sections. Sections 2, 3, and 4 contain the results on the q, t-versions of the Catalan numbers, parking functions, and Schröder paths, respectively. Section 5 contains a brief account of the recent exciting extensions of these objects which have arisen in the study of string theory, knot invariants, and the Hilbert scheme from algebraic geometry.

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The combinatorics of knot invariants arising from the study of Macdonald polynomials

Haglund

2016

The IMA Volumes in Mathematics and Its Applications

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This chapter gives an expository account of some unexpected connections which have arisen over the last few years between Macdonald polynomials, invariants of torus knots, and lattice path combinatorics. The study of polynomial knot invariants is a well-known branch of topology which originated in the 1920's with the one-parameter Alexander polynomial [Ale28]. In the early 1980's Jones [Jon85] introduced a different one-parameter polynomial invariant, with important connections to physics. Shortly thereafter a number of authors more or less simultaneously discovered the HOMFLY polynomial, a two-parameter invariant which includes both the Alexander and Jones polynomials as special cases. The HOMFLY polynomial can be calculated recursively through skein relations. In the late 1980's Witten showed that the Jones polynomial and related invariants have an interpretation in terms of Chern-Simons theory, which is central to string theory. In 2006 Dunfield, Gukov, and Rasmussen [DGR06] hypothesized the existence of a threeparameter knot invariant, now known as the "superpolynomial knot invariant" of a knot K, denoted P K (a, q, t), which includes the HOMFLY polynomial as a special case. Since then various authors proposed different possible definitions of the superpolynomial, which are conjecturally all equivalent. These definitions typically involve homology though, and are difficult to compute. In the case of torus knots an accepted definition of the superpolynomial has recently emerged from work of Aganagic and Shakirov [AS12], [AS11] (using refined Chern-Simons theory) and Cherednik [Che13] (using the double affine Hecke algebra). Gorsky and Negut [GN13] showed that these two different constructions yield the same three parameter knot invariant which is now accepted as the definition of the superpolynomial for torus knots. These constructions involve symmetric functions in a set of variables X known as Macdonald polynomials, which also depend on two extra parameters q, t. These symmetric functions are important in algebraic combinatorics and other areas, and play a central role in various character formulas for S n modules connected to the Hilbert scheme from algebraic geometry. In particular, Haiman's formula for the bigraded character of the space DH n of diagonal harmonics under the diagonal action of the symmetric group is expressed in terms of Macdonald polynomials. E. Gorsky [Gor12], [ORS12, Appendix] noticed that the coefficient of a j in the superpolynomial of the (n + 1, n) torus knot equals the bigraded multiplicity of a certain hook shape in the character of DH n. This polynomial is known as the (q, t)-Schröder polynomial since the author showed it can be expressed as a weighted sum over Schröder lattice paths in the n × n + 1 rectangle. Gorsky and Negut have shown that the coefficient of a j in the superpolynomial of the (m, n) torus knot can be viewed as the coefficient of a certain hook Schur function in a symmetric function expression involving Macdonald polynomials, and they

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Results and conjectures on simultaneous core partitions

Cited by 78 publications

References 22 publications

Combinatorics of the zeta map on rational Dyck paths

Combinatorics of the zeta map on rational Dyck paths

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The combinatorics of knot invariants arising from the study of Macdonald polynomials

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