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1. A Pipe Dream Perspective on Totally Symmetric Self-Complementary Plane Partitions

   https://doi.org/10.1017/fms.2023.131  

   Huang, Daoji; Striker, Jessica (January 2024, Forum of Mathematics, Sigma)

   Abstract We characterize totally symmetric self-complementary plane partitions (TSSCPP) as bounded compatible sequences satisfying a Yamanouchi-like condition. As such, they are in bijection with certain pipe dreams. Using this characterization and the recent bijection of Gao–Huang between reduced pipe dreams and reduced bumpless pipe dreams, we give a bijection between alternating sign matrices and TSSCPP in the reduced, 1432-avoiding case. We also give a different bijection in the 1432- and 2143-avoiding case that preserves natural poset structures on the associated pipe dreams and bumpless pipe dreams. 

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2. The Canonical Bijection between Pipe Dreams and Bumpless Pipe Dreams

   https://doi.org/10.1093/imrn/rnad083  

   Gao, Yibo; Huang, Daoji (April 2023, International Mathematics Research Notices)

   Abstract We present a direct bijection between reduced pipe dreams and reduced bumpless pipe dreams (BPDs) by interpreting reduced compatible sequences on BPDs and show that this is the unique bijection preserving bijective realizations of Monk’s formula, establishing its canonical nature. 

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3. Bijective Proofs of Monk's rule for Schubert and Double Schubert Polynomials with Bumpless Pipe Dreams

   https://doi.org/10.37236/11824  

   Huang, Daoji (July 2023, The Electronic Journal of Combinatorics)

   We give bijective proofs of Monk's rule for Schubert and double Schubert polynomials computed with bumpless pipe dreams. In particular, they specialize to bijective proofs of transition and cotransition formulas of Schubert and double Schubert polynomials, which can be used to establish bijections with ordinary pipe dreams. 

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4. Bijective Proofs of Monk’s rule for Schubert and Double Schubert Polynomials with Bumpless Pipe Dreams

   Huang, Daoji (January 2021, Séminaire lotharingien de combinatoire)

   null (Ed.)

   We give bijective proofs of Monk’s rule for Schubert and double Schubert polynomials computed with bumpless pipe dreams. In particular, they specialize to bijective proofs of transition and cotransition formulas of Schubert and double Schubert polynomials, which can be used to establish bijections with ordinary pipe dreams. 

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5. Grothendieck Shenanigans: Permutons from pipe dreams via integrable probability

   Morales, Alejandro H; Panova, Greta; Petrov, Leonid; Yeliussizov, Damir (July 2024, arXiv)

   We study random permutations arising from reduced pipe dreams. Our main model is motivated by Grothendieck polynomials with parameter $$\beta=1$$ arising in K-theory of the flag variety. The probability weight of a permutation is proportional to the principal specialization (setting all variables to 1) of the corresponding Grothendieck polynomial. By mapping this random permutation to a version of TASEP (Totally Asymmetric Simple Exclusion Process), we describe the limiting permuton and fluctuations around it as the order $$n$$ of the permutation grows to infinity. The fluctuations are of order $$n^{\frac13}$$ and have the Tracy-Widom GUE distribution, which places this algebraic (K-theoretic) model into the Kardar-Parisi-Zhang universality class. We also investigate non-reduced pipe dreams and make progress on a recent open problem on the asymptotic number of inversions of the resulting permutation. Inspired by Stanley's question for the maximal value of principal specializations of Schubert polynomials, we resolve the analogous question for $$\beta=1$$ Grothendieck polynomials, and provide bounds for general $$\beta$$. 

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