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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Keywords: permutation pattern, Schubert polynomial, Grothendieck polynomial, bumpless pipe dream
Dennin, Hugh 1

@article{ALCO_2025__8_3_745_0, author = {Dennin, Hugh}, title = {Pattern bounds for principal specializations of $\beta ${-Grothendieck} polynomials}, journal = {Algebraic Combinatorics}, pages = {745--763}, publisher = {The Combinatorics Consortium}, volume = {8}, number = {3}, year = {2025}, doi = {10.5802/alco.422}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.422/} }
TY - JOUR AU - Dennin, Hugh TI - Pattern bounds for principal specializations of $\beta $-Grothendieck polynomials JO - Algebraic Combinatorics PY - 2025 SP - 745 EP - 763 VL - 8 IS - 3 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.422/ DO - 10.5802/alco.422 LA - en ID - ALCO_2025__8_3_745_0 ER -
%0 Journal Article %A Dennin, Hugh %T Pattern bounds for principal specializations of $\beta $-Grothendieck polynomials %J Algebraic Combinatorics %D 2025 %P 745-763 %V 8 %N 3 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.422/ %R 10.5802/alco.422 %G en %F ALCO_2025__8_3_745_0
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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