Cayley's formula — interactive companion

Companion to An elaborate new proof of Cayley's formula. Pick a vertex-labelled tree on n vertices. The app draws its cyclic embedding, reads off the cyclic factorization of λn by walking clockwise from vertex n, and shows the maximal distinguished subword of λn corresponding to it. Email me if you find any errors or want some specific function added.

Pick a vertex-labelled tree on n vertices. The app draws its cyclic embedding (Definition 4.6), reads off the corresponding cyclic factorization of the translation λn ∈ S̃n by walking clockwise from vertex n along the marked edge (Theorem 4.8), and shows the maximal distinguished subword of λn = [s0, …, sn-1]n-1 that corresponds to it (Theorem 6.1).

● vertex n, the root · edges point towards n · the walk starts at n along the marked edge, to its smallest neighbour, and turns clockwise · the green numerals in the corners are the run-leaves.
Cyclic factorization rT
run-leaves
increasing sequence a0 < … < a2n-2 (Prop. 3.4)
m1 < … < m2n-2 (Prop. 4.4)
Maximal distinguished subword uT ∈ SUBn

A subword with exactly 2n−2 skips whose product of takes is the identity (Definition 5.1; the equivalence with the usual notion of distinguished is Cor. 5.15). Each skip j contributes the reflection u(j-1) sij u(j-1)−1 of Def. 5.3, shown in its cell. Drawn as the n × (n−1) array of Definition 5.1, whose rows are the n consecutive factors of length n−1 and whose columns are those rows vertically aligned. Click a cell to toggle it — as soon as the result is again a maximal distinguished subword, the tree and factorization above follow it.

Companion to E. Banaian, A. T. N. Hoang, E. Kelley, W. Miller, J. Stack, C. Stephen, N. Williams, An elaborate new proof of Cayley's formula, Algebraic Combinatorics 8 (2025), no. 4, 971–995, doi:10.5802/alco.429.