A new plethystic symmetric function operator and the rational compositional shuffle conjecture at t= 1/q

Garsia, Adriano M.; Leven, Emily; Wallach, Nolan R.; Xin, Guoce

doi:10.1016/j.jcta.2016.07.001

Cited by 6 publications

(7 citation statements)

References 38 publications

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“…Garsia, Leven, Wallach, and Xin [6] and Gorsky and Negut [9] conjectured the following theorem which was proved by Mellit [19]. Theorem 6 (Mellit).…”

Section: The Rational Shuffle Conjecturementioning

confidence: 93%

“…In Figure 2(c), tdinv(PF) = 7 since the pairs of cars contributing to tdinv are (1,3), (1,4), (3,5), (3,6), (4,6), (5,7) and (6,7). Then, the statistic max dinv of a path is defined as the maximum of temporary dinvs of parking functions on the path.…”

Section: Combinatorial Sidementioning

confidence: 99%

“…Given m, n coprime and k ≥ 1, we defined the return of a (km, kn)-parking function PF, ret(PF) to be the smallest positive integer i such that the supporting path of PF goes through the point (im, in). Then following the formulation of Garsia, Leven, Wallach, and Xin [6], we define the extended Hikita polynomial to be H km,kn [X; q, t] = PF∈PF km,kn…”

Section: Combinatorial Sidementioning

confidence: 99%

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Schur Function Expansions and the Rational Shuffle Theorem

Qiu¹,

Remmel²

2018

Preprint

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Gorsky and Negut [9] introduced operators Q m,n on symmetric functions and conjectured that, in the case where m and n are relatively prime, the expansion of Q m,n (−1) n in terms of the fundamental quasi-symmetric functions are given by polynomials introduced by Hikita [15]. Later, Bergeron, Garsia, Leven and Xin [3] extended and refined the conjectures of Gorsky and Negut to give a combinatorial interpretation of the coefficients that arise in the expansion of Q m,n (−1) n in terms of the fundamental quasi-symmetric functions for arbitrary m and n which we will call the Rational Shuffle Conjecture. In the special case Q n+1,n (−1) n , the Rational Shuffle Conjecture becomes the Shuffle Conjecture of Haglund, Haiman, Loehr, Remmel, and Ulyanov [12]. The Shuffle Conjecture was proved in 2015 by Carlsson and Mellit [4] and full Rational Shuffle Conjecture was later proved by Mellit [19]. The main goal of this paper is to study the combinatorics of the coefficients that arise in the Schur function expansion of Q m,n (−1) n in certain special cases. Leven gave a combinatorial proof of the Schur function expansion of Q 2,2n+1 (−1) 2n+1 and Q 2n+1,2 (−1) 2 in [17]. In this paper, we explore several symmetries in the combinatorics of the coefficients that arise in the Schur function expansion of Q m,n (−1) n . Especially, we study the hook-shaped Schur function coefficients, and the Schur function expansion of Q m,n (−1) n in the case where m or n equals 3.

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“…Garsia, Leven, Wallach, and Xin [6] and Gorsky and Negut [9] conjectured the following theorem which was proved by Mellit [19]. Theorem 6 (Mellit).…”

Section: The Rational Shuffle Conjecturementioning

confidence: 93%

Section: Combinatorial Sidementioning

confidence: 99%

Section: Combinatorial Sidementioning

confidence: 99%

See 1 more Smart Citation

Schur Function Expansions and the Rational Shuffle Theorem

Qiu¹,

Remmel²

2018

Preprint

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“…The fundamental problem we are concerned here is to compute the constant term of in a set of variables of a formal series in the field of iterated Laurent series K = Q((x n )) · · · ((x 1 )), which is called the working field. The reader is referred to [9] for the original development of the field of iterated Laurent series. Here we only recall that K defines a total ordering 0 < x 1 < x 2 < · · · < x n < 1 on the variables (more precisely, a total group order on its monomials), which can be formally treated as 0 < < x 1 < < x 2 < < · · · < < x n < < 1.…”

Section: 2mentioning

confidence: 99%

On Parity Unimodality of $q$-Catalan Polynomials

Xin

Zhong

2020

Electron. J. Combin.

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A polynomial A(q) = n i=0 a i q i is said to be unimodal if a 0 ≤ a 1 ≤ · · · ≤ a k ≥ a k+1 ≥ · · · ≥ a n . We investigate the unimodality of rational q-Catalan polynomials, which is defined to be C m,n (q) = [n+m]m+n n q for a coprime pair of positive integers (m, n). We conjecture that they are unimodal with respect to parity, or equivalently, (1 + q)C m+n (q) is unimodal. By using generating functions and the constant term method, we verify our conjecture for m ≤ 5 in a straightforward way.Mathematics Subject Classifications: 05A15, 05A20, 05E05

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“…This polynomial also appears to be connected to certain modules arising in the theory of rational Cherednik algebras. Recently, it is shown in [11] that the modified version [gcd(m, n)] q Cat m,n (q) for arbitrary m and n is also a polynomial in q with N coefficients.…”

Section: Introductionmentioning

confidence: 99%

A note on rank complement of rational Dyck paths and conjugation of $(m,n)$-core partitions

Xin

2017

Journal of Combinatorics

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Given a coprime pair (m, n) of positive integers, rational Catalan numbers 1 m+n m+n m,n counts two combinatorial objects: rational (m, n)-Dyck paths are lattice paths in the m × n rectangle that never go below the diagonal; (m, n)-cores are partitions with no hook length equal to m or n. Anderson established a bijection between (m, n)-Dyck paths and (m, n)-cores. We define a new transformation, called rank complement, on rational Dyck paths. We show that rank complement corresponds to conjugation of (m, n)-cores under Anderson's bijection. This leads to: i) a new approach to characterizing n-cores; ii) a simple approach for counting the number of self-conjugate (m, n)-cores; iii) a proof of the equivalence of two conjectured combinatorial sum formulas, one over rational (m, n)-Dyck paths and the other over (m, n)-cores, for rational Catalan polynomials.Mathematics Subject Classifications: 05A17, 05A19, 05E05

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A new plethystic symmetric function operator and the rational compositional shuffle conjecture at t= 1/q

Cited by 6 publications

References 38 publications

Schur Function Expansions and the Rational Shuffle Theorem

Schur Function Expansions and the Rational Shuffle Theorem

On Parity Unimodality of $q$-Catalan Polynomials

A note on rank complement of rational Dyck paths and conjugation of $(m,n)$-core partitions

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