In 1993, Fomin and Kirillov [4] introduced a new set of diagrams called RC graphs or pipe dreams that aid in computing Schubert polynomials. Nowadays, there are many studies on different kinds of pipe dreams and their bijections with other combinatorial objects. Serrano and Stump [20] gave bijections between pipe dreams, certain north-east fillings, and k-flagged tableaux. Hamaker, Marberg, and Pawlowski [8] studied a new class of involution pipe dreams. Lam, Lee, and Shimozono [13] more recently introduced bumpless pipe dreams and Gao and Huang [6] have provided a bijection between bumpless pipe dreams and ordinary pipe dreams.

As featured in the work of Knutson and Miller [12], there is a close connection between pipe dreams and the geometry of Schubert varieties. Work of Marberg and...[ Read more ]

In 1993, Fomin and Kirillov [4] introduced a new set of diagrams called RC graphs or pipe dreams that aid in computing Schubert polynomials. Nowadays, there are many studies on different kinds of pipe dreams and their bijections with other combinatorial objects. Serrano and Stump [20] gave bijections between pipe dreams, certain north-east fillings, and k-flagged tableaux. Hamaker, Marberg, and Pawlowski [8] studied a new class of involution pipe dreams. Lam, Lee, and Shimozono [13] more recently introduced bumpless pipe dreams and Gao and Huang [6] have provided a bijection between bumpless pipe dreams and ordinary pipe dreams.

As featured in the work of Knutson and Miller [12], there is a close connection between pipe dreams and the geometry of Schubert varieties. Work of Marberg and Pawlowski [16] extends this connection to involution pipe dreams. Apart from these geometric motivations, pipe dreams are also of interest on their own in combinatorics.

This thesis first studies bijections between pipe dreams and a variety of other objects: for example, certain north-east chains, north-east fillings, bumpless pipe dreams, and reverse plane partitions. This expository material surveys the main results from [6] and [20]. Then we study symplectic variations of these constructions, proving some new results. Specifically, we investigate bijections between involution pipe dreams, certain reflected north-east chains and fillings, and shifted reverse plane partitions. We also study a generalization of bumpless pipe dreams for involutions. We give a conjectural bijection between these objects and involution pipe dreams, extending the correspondence of [6].