A proof of the Square Paths Conjecture

Sergel, Emily

doi:10.1016/j.jcta.2017.06.013

“…In [6] the authors conjecture a combinatorial formula for [n−k] t [n] t ∆ h m ∆ e n−k ω(p n ) in terms of risedecorated partially labelled square paths that we call generalised Delta square conjecture (rise version). This conjecture extends the square conjecture of Loehr and Warrington [18] (now a theorem [23]), i.e. it reduces to that one for m = k = 0.…”

Section: Introductionsupporting

confidence: 76%

“…The special case •, e n of this conjecture, known as the q, t-square, was proved by Can and Loehr in [3]. Recently Sergel [23] proved the full square conjecture, by showing that the shuffle theorem by Carlsson and Mellit [4] implies the square conjecture (now square theorem).…”

Section: Introductionmentioning

confidence: 99%

A Valley Version of the Delta Square Conjecture

Iraci

¹

,

Wyngaerd

²

2021

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Inspired by [Qiu, Wilson 2020] and [D'Adderio, Iraci, Vanden Wyngaerd 2019], we formulate a generalised Delta square conjecture (valley version). Furthermore, we use similar techniques as in [Haglund, Sergel 2019] to obtain a schedule formula for the combinatorics of our conjecture. We then use this formula to prove that the (generalised) valley version of the Delta conjecture implies our (generalised) valley version of the Delta square conjecture. This implication broadens the argument in [Sergel 2016], relying on the formulation of the touching version in terms of the Θ f operators introduced in [D'Adderio, Iraci, Vanden Wyngaerd 2020-Theta Operators].

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“…

We conjecture a formula for the symmetric function [n−k]t [n]t ∆ hm ∆e n−k ω(pn) in terms of decorated partially labelled square paths. This can be seen as a generalization of the square conjecture of Loehr and Warrington [20], recently proved by Sergel [25] after the breakthrough of Carlsson and Mellit [4]. Moreover, it extends to the square case the combinatorics of the generalized Delta conjecture of Haglund, Remmel and Wilson [14], answering one of their questions.

…”

mentioning

confidence: 72%

“…The special case ·, e n of this conjecture, known as q, t-square, has been proved earlier by Can and Loehr in [3]. Recently the full square conjecture has been proved by Sergel in [25] after the breakthrough of Carlsson and Mellit in [4].…”

Section: Introductionmentioning

confidence: 95%

The Delta Square Conjecture

D'Adderio

¹

,

Iraci

²

,

Wyngaerd

³

2019

International Mathematics Research Notices

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We conjecture a formula for the symmetric function [n−k]t [n]t ∆ hm ∆e n−k ω(pn) in terms of decorated partially labelled square paths. This can be seen as a generalization of the square conjecture of Loehr and Warrington [20], recently proved by Sergel [25] after the breakthrough of Carlsson and Mellit [4]. Moreover, it extends to the square case the combinatorics of the generalized Delta conjecture of Haglund, Remmel and Wilson [14], answering one of their questions. We support our conjecture by proving the specialization m = q = 0, reducing it to the same case of the Delta conjecture, and the Schröder case, i.e. the case ·, e n−d h d . The latter provides a broad generalization of the q, t-square theorem of Can and Loehr [3]. We give also a combinatorial involution, which allows to establish a linear relation among our conjectures (as well as the generalized Delta conjectures) with fixed m and n. Finally, in the appendix, we give a new proof of the Delta conjecture at q = 0.Partially labelled Dyck paths differ from labelled Dyck paths only in that 0 is allowed as a label in the former and not in the latter.Definition 2.2. We define for each D ∈ PLD(m, n) a monomial in the variables x 1 , x 2 , . . . : we setwhere l i (D) is the label of the i-th vertical step of D (the first being at the bottom). Notice that x 0 does not appear, which explains the word partially.

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“…But the sum is exactly the one given in the definition of maj, so the two statistics match. 16. We define the set SOP(m, n) k of standardized ordered multiset partitions to be the set OP(m, (1, 1, .…”

Section: Ordered Set Partitionsmentioning

confidence: 99%