From a permutation to its mosaic and its bipolar orientation

Rooms show the value π(k); positions switches to k. The wall graph turns each vertical wall into a vertex and each room into a directed edge.

Each vertex sits at its literal BBF coordinate wa = (a+½, ℓa+½). Every edge points strictly north-east by construction.

π (●) is completed to length 2n−1 by inserting n−1 connector points (○). The default loop form has a straight blue thread and alternating, noncrossing red semicircles; monotone morphs both into the corresponding curves. Both forms ℓ1 and ℓ2 thread through all 2n−1 points and cross exactly that many times.

The two binary search trees encode their missing children as canopy marks: empty left child, empty right child. A permutation is Baxter exactly when the canopies are complementary.

Room k is the BBF point (k, π(k)); rows are in position order and identify the corresponding graph edge.

Every catalogued prime (simple Baxter) permutation up to n=13, one tier per order. An edge is the SHORTEST deletion witness a child has -- 2-point deletions are shown only when no 1-point parent exists -- so it proves every prime has some short-deletion prime ancestor, not that this lists every one. Selecting a node draws it in every other panel above.