Corrigendum to "Factoring, into edge transpositions of a tree, permutations fixing a terminal vertex" [J. Combin. Theory Ser. A 85 (1) (1999) 92-95]

The proof of Theorem 9 of [1] contains an error,1 and the assertion is in fact false, as is shown by the following example. (We use the same symbol for an edge and the associated transposition and multiply from right to left, e.g. (1,2)(1,3) = (1,3,2).) Let the vertices be {1,2,3,4,5,6,7,8} and the edges {a = (1,4),b1 = (3,4),b2 = (2,3), c1 = (4,5), c2 = (5,6), c3 = (6,7), c4 = (7,8)}. Let σ = (1)(2,7)(3,8)(4,6)(5) = ab1c1b2b1c2c1ac3c2c1b1c4c3c2c1b2b1a = b1b2c1b1c2c1b1b2b1c3c2c1b1c4c3c2c1b2b1 (both of length 19). Since, as a permutation of {2,3,4,5,6,7,8} σ has 19 inversions, no product for it involving only b’s and c’s can have smaller length. A bit of (Mathematica-aided) checking confirms that there is no shorter factorization of σ even allowing a, so the example is consistent with the weaker conjecture that there is always some factorization of minimal length not using the edge, i.e. an affirmative answer to the first part of Question 2 of [2] and with Conjecture 1. The example shows that the answer to the second part of Question 2 is (sometimes) affirmative.