Pipe Dreams in a Rectangle     permutations

Displays a canvas with colored pipes crossing through an N-by-M rectangular grid, where crosses and bumps form a pipe dream. Shows the resulting permutation as a matrix plot below. Adjust rows, columns, crossing probability, and Demazure reduction, then zoom and pan to explore the pipe configuration.

This simulation generates pipe dreams (also known as RC-graphs) on an $N \times M$ grid.

Pipe dream construction:

• Pipes enter from the left edge (labeled $1, 2, \ldots, N$ from top to bottom) and the bottom edge (labeled $N+1, N+2, \ldots, N+M$ from left to right)
• Each cell is randomly assigned as a cross ($+$, with probability $p$) or a bump (elbow, with probability $1-p$)
• At a cross, pipes pass straight through; at a bump, pipes turn 90 degrees

Demazure reduction: When enabled, if two pipes that have already crossed try to cross again (i.e., the left pipe has a larger label than the bottom pipe), the cross is forced to become a bump. This ensures each pair of pipes crosses at most once, producing a reduced pipe dream.

The output permutation is read from the top edge (left to right) followed by the right edge (top to bottom).

Skip to simulation canvas

code

• Link to code (This simulation is interactive, written in JavaScript/WASM)
• Link to code (C++ source code compiled to WebAssembly)

references

1. Colin Defant. Permutons from Demazure Products • https://arxiv.org/abs/2505.15630 (opens in new tab)

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