Connectivity of reduced bumpless pipe dreams under local 2×2 flips

Prove that for every n, the graph on reduced bumpless pipe dreams of size n, with edges given by a single 2×2 flip that preserves reducedness, is connected.

Background

In the six-vertex/ASM setting, 2×2 flips connect the full (unrestricted) state space. For reduced bumpless pipe dreams (RBPDs), flips that would create a double crossing must be rejected, and connectivity is no longer immediate.

The authors verified connectivity by computation for n ≤ 8 and observed “traps” requiring non-monotone paths, motivating a formal conjecture of connectivity under flips that preserve reducedness.

References

Restricting to the subset of {\em reduced} BPDs, we conjecture that the same $2\times 2$ flips suffice to connect the state space: For every $n$, the set of reduced bumpless pipe dreams of size~$n$ is connected under the $2 \times 2$ flips (\Cref{fig:local_flips}) that preserve reducedness.

— Computation and sampling for Schubert specializations  (2603.20104 - Anderson et al., 20 Mar 2026) in Section 4.2