Hook Formulas for Skew Shapes IV. Increasing Tableaux and Factorial Grothendieck Polynomials

Morales, Alejandro H.; Pak, Igor; Panova, Greta

doi:10.1007/s10958-022-05777-0

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2022

2024

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Cited by 3 publications

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“…w have previously been studied for w vexillary, where a determinantal upper bound was established in terms of Delannoy paths on Young tableaux [15]. There is a q-analog of ν (β) w given by G (β) w (1, q, .…”

Section: Minimal and Reducedmentioning

confidence: 99%

Pattern bounds for principal specializations of $β$-Grothendieck Polynomials

Dennin¹

2022

Preprint

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There has been recent interest in lower bounds for the principal specializations of Schubert polynomials νw := Sw(1, . . . , 1). We prove a conjecture of Yibo Gao in the setting of 1243-avoiding permutations that gives a lower bound for νw in terms of the permutation patterns contained in w. We extended this result to principal specializations of β-Grothendieck polynomials ν (β) w := G (β) w (1, . . . , 1) by restricting to the class of vexillary 1243-avoiding permutations. Our methods are bijective, offering a combinatorial interpretation of the coefficients cw and c (β) w appearing in these conjectures.cwhere this sum is taken over all permutation patterns u properly contained in w (so u = w).Conjecture 1.1 (Gao 1 [5]). For all permutations w, c w ≥ 0.Note that if Conjecture 1.1 were true, then we would get a lower bound ν w ≥ u

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Section: Minimal and Reducedmentioning

confidence: 99%