Related papers: Marked Bumpless Pipedreams and Compatible Pairs
We present a direct bijection between reduced pipe dreams and reduced bumpless pipe dreams by interpreting reduced compatible sequences on bumpless pipe dreams and show that this bijection preserves Monk's formula, establishing its…
Pipedreams and bumpless pipedreams are two combinatorial models that compute double Grothendieck polynomials. While studying matrix Schubert varieties, Pechenik, Speyer, and Weigandt defined a permutation statistic$\mathsf{rajcode}(\cdot)$…
In their work on the infinite flag variety, Lam, Lee, and Shimozono (2018) introduced objects called bumpless pipe dreams and used them to give a formula for double Schubert polynomials. We extend this formula to the setting of K-theory,…
Pipe dreams and bumpless pipe dreams for vexillary permutations are each known to be in bijection with certain semistandard tableaux via maps due to Lenart and Weigandt, respectively. Recently, Gao and Huang have defined a bijection between…
Lascoux and Sch\"utzenberger introduced Schubert and Grothendieck polynomials to study the cohomology and K-theory of the complete flag variety. We present explicit combinatorial rules for expressing Grothendieck polynomials in the basis of…
In their study of infinite flag varieties, Lam, Lee, and Shimozono (2021) introduced bumpless pipe dreams in a new combinatorial formula for double Schubert polynomials. These polynomials are the TxT-equivariant cohomology classes of matrix…
Bumpless pipe dreams (BPDs) are combinatorial objects used in the study of Schubert and Grothendieck polynomials. Weigandt recently introduced a co-BPD object associated to each BPD and used them to give an analogue to the change of bases…
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…
In [KU23] were introduced hybrid pipe dreams interpolating between classic and bumpless pipe dreams, each hybridization giving a different formula for double Schubert polynomials. A bijective proof was given (following [GH23]) of the…
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…
Schubert polynomials are distinguished representatives of Schubert cycles in the cohomology of the flag variety. In the spirit of Bergeron and Sottile, we use the Bruhat order to give $(n-1)!$ different combinatorial formulas for the…
Knutson and Zinn-Justin recently found a puzzle rule for the expansion of the product $\mathfrak{G}_{u}(x,t)\cdot \mathfrak{G}_{v}(x,t)$ of two double Grothendieck polynomials indexed by permutations with separated descents. We establish…
In this note, a new puzzle is introduced where the pipe dream and bumpless pipe dream can be played simultaneously. Using these, a combinatorial proof of the (ordinary) Schubert polynomials in terms of bumpless pipe dream is given. The main…
Pipedreams are combinatorial objects that compute Grothendieck polynomials. We introduce a new combinatorial object that naturally recast the pipedream formula. From this, we obtain the first direct combinatorial formula for the top degree…
Recent work of Pechenik, Speyer, and Weigandt proved a formula for the degree of any Grothendieck polynomial. A distinct formula for the degree of vexillary Grothendieck polynomials was proven by Rajchgot, Robichaux, and Weigandt. We give a…
Weighted enumeration of reduced pipe dreams (or rc-graphs) results in a combinatorial expression for Schubert polynomials. The duality between the set of reduced pipe dreams and certain antidiagonals has important geometric implications [A.…
In this paper we show that the pipe dream complex associated to the permutation 1n(n-1)...2 can be geometrically realized as a triangulation of the vertex figure of a root polytope. Leading up to this result we show that the Grothendieck…
In this note a bijection is constructed between the set of partitions of n simultaneously s-regular and t-distinct, and those simultaneously t-regular and s-distinct. Some implications of the map are discussed. As a generalized version of…
Schubert polynomials are polynomial representatives of Schubert classes in the cohomology of the complete flag variety and have a combinatorial formulation in terms of bumpless pipe dreams. Quantum double Schubert polynomials are polynomial…
The Cauchy identity gives a recipe for decomposing a double Grothendieck polynomial $\mathfrak{G}^{(\beta)}_w(x;y)$ as a sum of products $\mathfrak{G}^{(\beta)}_v(x)\mathfrak{G}^{(\beta)}_u(y)$ of single Grothendieck polynomials.…