A simpler formula for the number of diagonal inversions of an <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si33.gif" display="inline" overflow="scroll"><mml:mrow><mml:mo>(</mml:mo><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math>-parking function and a returning fermionic formula

Hicks, Angela; Leven, Emily

doi:10.1016/j.disc.2014.10.016

Cited by 6 publications

(3 citation statements)

References 21 publications

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“…The following is a simpler description for the difference pdinv(π) − maxtdinv(π), given in [HL15]. Definition 3.16.…”

Section: Labelled Pathsmentioning

confidence: 99%

Rectangular analogues of the square paths conjecture and the univariate Delta conjecture

Iraci,

Pagaria,

Paolini

et al. 2023

Combinatorial Theory

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In this paper, we extend the rectangular side of the shuffle conjecture by stating a rectangular analogue of the square paths conjecture. In addition, we describe a set of combinatorial objects and one statistic that are a first step towards a rectangular extension of (the rise version of) the Delta conjecture, and of (the rise version of) the Delta square conjecture, corresponding to the case q = 1 of an expected general statement. We also prove our new rectangular paths conjecture in the special case when the sides of the rectangle are coprime.

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“…The following is a simpler description for the difference pdinv(π) − maxtdinv(π), given in [HL15]. Definition 3.16.…”

Section: Labelled Pathsmentioning

confidence: 99%

Rectangular analogues of the square paths conjecture and the univariate Delta conjecture

Iraci,

Pagaria,

Paolini

et al. 2023

Combinatorial Theory

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“…We shall use this definition of dinv in some combinatorial proofs in Section 3. However, our definition of dinv(PF) in Section 4 will follow the formulation by Leven and Hicks [14], who gave a simplified formula for dinv(PF) as the sum of two simpler statistics, tdinv and dinvcorr. Definition (dinvcorr).…”

Section: Combinatorial Sidementioning

confidence: 99%

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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“…The original formulation of Hikita [24] as modified by Gorsky-Mazin [17], [18] expressed the dinv statistic as a combination of tdinv and two other statistics. However, Hicks and Leven [23] showed that this can be simplified as follows. Let λ(P F ) be partitions whose english Ferrers diagram is formed by the cells above P F .…”

Section: 20mentioning

confidence: 99%

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

Garsia

Leven

Wallach

et al. 2017

Journal of Combinatorial Theory, Series A

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Our main result here is that the specialization at t = 1/q of the Q km,kn operators studied in [4] may be given a very simple plethystic form. This discovery yields elementary and direct derivations of several identities relating these operators at t = 1/q to the Rational Compositional Shuffle conjecture of [3].In particular we show that if m, n and k are positive integers and (m, n) is a coprime pair thenwhere as customarily, for any integer s ≥ 0 and indeterminate u we set [s] u = 1 + u + · · · + u s−1 . We also show that the symmetric polynomial on the right hand side is always Schur positive. Moreover, using the Rational Compositional Shuffle conjecture, we derive a precise formula expressing this polynomial in terms of Parking functions in the km × kn lattice rectangle.

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A simpler formula for the number of diagonal inversions of an (m,n)-parking function and a returning fermionic formula

Cited by 6 publications

References 21 publications

Rectangular analogues of the square paths conjecture and the univariate Delta conjecture

Rectangular analogues of the square paths conjecture and the univariate Delta conjecture

Schur Function Expansions and the Rational Shuffle Theorem

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

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