Year: 2026
Type: preprint
Abstract: We study the number of minimal transitive factorizations of the identity permutation in $S_n$ into transpositions supported on a quasi-threshold graph. We show that this number is always divisible by $(2n-2)!/n!$, which is the factorization count for a star graph, as shown by Irving and Rattan. To prove this, we give a combinatorial formula for the number of such factorizations as a weighted sum over a subset of the $n^{n-2}$ factorization trees, which are edge-weighted spanning trees satisfying certain flow constraints.
Source: arXiv (Cornell University)
Authors: Li, Cordelia Yuqiao, Liu, Ricky Ini
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214807143