2011/12/21 by Olivier Bernardi, Alejandro H. Morales, Bernardi, Olivier +1 · 1 citation
Engineering · Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems
paper · doi:10.48550/arxiv.1112.4970
openalex publication_date 2011/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the factorizations of the permutation (1,2,...,n) into k factors of given cycle types. Using representation theory, Jackson obtained for each k an elegant formula for counting these factorizations according to the number of cycles of each factor. In the cases k=2,3 Schaeffer and Vassilieva gave a combinatorial proof of Jackson's formula, and Morales and Vassilieva obtained more refined formulas exhibiting a surprising symmetry property. These counting results are indicative of a rich combinatorial theory which has remained elusive to this point, and it is the goal of this article to establish a series of bijections which unveil some of the combinatorial properties of the factorizations of (1,2,...,n) into k factors for all k. We thereby obtain refinements of Jackson's formulas which extend the cases k=2,3 treated by Morales and Vassilieva. Our bijections are described in terms of "constellations", which are graphs embedded in surfaces encoding the transitive factorizations of permutations.