A combinatorial bijection between k-edge colored trees and colored Prüfer codes for labelled trees is established. This bijection gives a simple combinatorial proof for the number k(n − 2)! nk−n n−2 of k-edge colored trees with n vertices.
We give a new decomposition of derangements, which gives a direct interpretation of a formula for their generating function. This decomposition also works for counting derangements by number of excedances.
We prove a formula for the linearization coefficients of the general Sheffer polynomials, which unifies all the special known results for Hermite, Charlier, Laguerre, Meixner and Meixner-Pollaczek polynomials. Furthermore, we give a new and explicit real version of the corresponding formula for Meixner-Pollaczek polynomials. Our proof is based on some… (More)
For a permutation π = π 1 π 2 · · · π n ∈ S n and a positive integer i ≤ n, we can view π 1 π 2 · · · π i as an element of S i by order-preserving relabeling. The j-set of π is the set of i's such that π 1 π 2 · · · π i is an involution in S i. We prove a characterization theorem for j-sets, give a generating function for the number of different j-sets of… (More)
By considering a family of orthogonal polynomials generalizing the Tchebycheff polynomials of the second kind we refine the corresponding results of De Sainte-Catherine and Viennot on Tchebycheff polynomials of the second kind
Minimal transitive cycle factorizations and parking functions are related very closely. Using the correspondence between them, we find a bijection between minimal transitive factorizations of permutations of type (1, n − 1) and prime parking functions of length n.