Pattern bounds for principal specializations of $\beta $-Grothendieck polynomials
Algebraic Combinatorics, Volume 8 (2025) no. 3, pp. 745-763.

There has been recent interest in lower bounds for the principal specializations of Schubert polynomials $\nu _w := \mathfrak{S}_w(1,\dots ,1)$. We prove a conjecture of Yibo Gao in the setting of $1243$-avoiding permutations that gives a lower bound for $\nu _w$ in terms of the permutation patterns contained in $w$. We extend this result to principal specializations of $\beta $-Grothendieck polynomials $\nu ^{(\beta )}_w := \mathfrak{G}^{(\beta )}_w(1,\dots ,1)$ by restricting to the class of vexillary $1243$-avoiding permutations. Our methods are bijective, offering a combinatorial interpretation of the coefficients $c_w$ and $c^{(\beta )}_w$ appearing in these conjectures.

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DOI: 10.5802/alco.422
Classification: 05A05, 05E05, 05E15, 14N15
Keywords: permutation pattern, Schubert polynomial, Grothendieck polynomial, bumpless pipe dream

Dennin, Hugh 1

1 The Ohio State University Department of Mathematics 231 West 18th Avenue Columbus OH 43210-1174 (USA)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Dennin, Hugh. Pattern bounds for principal specializations of $\beta $-Grothendieck polynomials. Algebraic Combinatorics, Volume 8 (2025) no. 3, pp. 745-763. doi : 10.5802/alco.422. https://alco.centre-mersenne.org/articles/10.5802/alco.422/

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