A Bijective Proof of Loehr-Warrington's Formulas for the Statistics $${\rm{ctot}_\frac{q}{p}}$$ ctot q p and $${\rm{mid}_\frac{q}{p}}$$ mid q p

Mazin, Mikhail

doi:10.1007/s00026-014-0241-x

Cited by 6 publications

(5 citation statements)

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“…One can use [31,Thm. 16] and [34] along with the relationships among varations on the sweep map found in [4] to prove the formula h + b,a (µ) = (a − 1)(b − 1)/2 − area(sweep(µ)). This equality implies that Conjectures 6 and 21 are equivalent.…”

Section: Rational Q T-catalan Numbersmentioning

confidence: 99%

Rational Parking Functions and Catalan Numbers

2015

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The "classical" parking functions, counted by the Cayley number (n + 1) n−1 , carry a natural permutation representation of the symmetric group Sn in which the number of orbits is the Catalan number 1 n+1 2n n . In this paper, we will generalize this setup to "rational" parking functions indexed by a pair (a, b) of coprime positive integers. We show that these parking functions, which are counted by b a−1 , carry a permutation representation of Sa in which the number of orbits is the "rational" Catalan number 1 a+b a+b a . We compute the Frobenius characteristic of the Sa-module of (a, b)-parking functions. Next we study q-analogues of the rational Catalan numbers, proposing a combinatorial formula for 1[a+b]q a+b a q and relating this formula to a new combinatorial model for q-binomial coefficients. Finally, we discuss q, t-analogues of rational Catalan numbers and parking functions (generalizing the shuffle conjecture for the classical case) and present several conjectures.

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Section: Rational Q T-catalan Numbersmentioning

confidence: 99%

Rational Parking Functions and Catalan Numbers

2015

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“…Proof. This result essentially follows from Corollary 1 on page 8 in [23]. With the terminology above, arm( ) is the number of boxes strictly between the box ∈ R N,M and the horizontal step h of the boundary of D, whereas its leg( ) is the number of boxes strictly between the box and v .…”

Section: Relatively Prime Casementioning

confidence: 86%

Rational Dyck Paths in the Non Relatively Prime Case

Gorsky

Mazin

Vazirani

2017

Electron. J. Combin.

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We study the relationship between rational slope Dyck paths and invariant subsets of Z, extending the work of the first two authors in the relatively prime case. We also find a bijection between (dn, dm)-Dyck paths and d-tuples of (n, m)-Dyck paths endowed with certain gluing data. These are the first steps towards understanding the relationship between rational slope Catalan combinatorics and the geometry of affine Springer fibers and knot invariants in the non relatively prime case. Y n,m M n,m Y n,m Core n,m D G ζ A FIGURE 1. Rational Catalan maps in the coprime case

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“…The dinv sweeps to area result was also known for (m, n) rational Dyck paths by [11,7,12,5]. Our work for k-Dyck paths are inspired by Garsia-Xin's visual proof in [5].…”

Section: Discussionmentioning

confidence: 99%

Dinv, Area, and Bounce for $\vec{k}$-Dyck paths

Xin¹,

Zhang²

2020

Preprint

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The well-known q, t-Catalan sequence has two combinatorial interpretations as weighted sums of ordinary Dyck paths: one is Haglund's area-bounce formula, and the other is Haiman's dinv-area formula. The zeta map was constructed to connect these two formulas: it is a bijection from ordinary Dyck paths to themselves, and it takes dinv to area, and area to bounce. Such a result was extended for k-Dyck paths by Loehr. The zeta map was extended by Armstrong-Loehr-Warrington for a very general class of paths.In this paper, We extend the dinv-area-bounce result for k-Dyck paths by: i) giving a geometric construction for the bounce statistic of a k-Dyck path, which includes the k-Dyck paths and ordinary Dyck paths as special cases; ii) giving a geometric interpretation of the dinv statistic of a k-Dyck path. Our bounce construction is inspired by Loehr's construction and Xin-Zhang's linear algorithm for inverting the sweep map on k-Dyck paths. Our dinv interpretation is inspired by Garsia-Xin's visual proof of dinv-to-area result on rational Dyck paths.

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A Bijective Proof of Loehr-Warrington's Formulas for the Statistics $${\rm{ctot}_\frac{q}{p}}$$ ctot q p and $${\rm{mid}_\frac{q}{p}}$$ mid q p

Cited by 6 publications

References 6 publications

Rational Parking Functions and Catalan Numbers

Rational Parking Functions and Catalan Numbers

Rational Dyck Paths in the Non Relatively Prime Case

Dinv, Area, and Bounce for $\vec{k}$-Dyck paths

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