2011/01/19 by Alejandro H. Morales, Morales, Alejandro H., Ekaterina A. Vassilieva +1
Mathematics · Engineering · Computer Science · #Advanced Combinatorial Mathematics #graph theory and CDMA systems #Cellular Automata and Applications
paper · pdf · doi:10.48550/arxiv.1101.3614
We evaluate combinatorially certain connection coefficients of the symmetric\ngroup that count the number of factorizations of a long cycle as a product of\nthree permutations. Such factorizations admit an important topological\ninterpretation in terms of unicellular constellations on orientable surfaces.\nAlgebraic computation of these coefficients was first done by Jackson using\nirreducible characters of the symmetric group. However, bijective computations\nof these coefficients are so far limited to very special cases. Thanks to a new\nbijection that refines the work of Schaeffer and Vassilieva, and Vassilieva, we\ngive an explicit closed form evaluation of the generating series for these\ncoefficients. The main ingredient in the bijection is a modified oriented\ntricolored tree tractable to enumerate. Finally, reducing this bijection to\nfactorizations of a long cycle into two permutations, we get the analogue\nformula for the corresponding generating series.\n