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A versatile combinatorial approach of studying products of long cycles in symmetric groups
Ricky X. F. Chen
In symmetric groups, studies of permutation factorizations or triples of permutations satisfying certain conditions have a long history. One particular interesting case is when two…
Combinatorially refine a Zagier-Stanley result on products of permutations
Ricky X. F. Chen
In this paper, we enumerate the pairs of permutations that are long cycles and whose product has a given cycle-type. Our main result is a simple relation concerning the desired num…
Separation probabilities and analogues of a Zagier-Stanley formula
Ricky X. F. Chen
In this paper, we first obtain some analogues of a formula of Zagier (1995) and Stanley (2011). For instance, we prove that the number of pairs of -cycles whose product has …
A refinement of the formula for -ary trees and the Gould-Vandermonde's convolution
Ricky X. F. Chen
In this paper, we present an involution on some kind of colored -ary trees which provides a combinatorial proof of a combinatorial sum involving the generalized Catalan numbers…
Lcm-lattice, Taylor Bases and Minimal Free Resolutions of a Monomial ideal
Ri-Xiang Chen
We use the lcm-lattice of a monomial ideal to study its minimal free resolutions. A new concept called a Taylor basis of a minimal free resolution is introduced and then used throu…