A joint absorption-Mahonian process (Q1347986)

An absorption ring is considered. The joint absorption-Mahonian process begins with a placement of a dot into a cell. A coin with probability \(q\) of landing tails up is then tossed until the dot escapes or until heads occur. Whenever the dot returns to the cell in which it was initially placed, it escapes from the ring with probability \(1-t\). If heads occur before the dot escapes, then the dot comes to rest and is said to be absorbed. The absorption set is the set of labels of the cells in which absorption has occured. The distribution of the cell labels in which absorption occured is derived by using the terms of the \(q\)-shifted factorial and \((t,q)\)-binomial coefficients. The properties of Mahonian statistics are discussed. The formula for the probability that the set \(A\subseteq C\) (\(C\) is the set of labels) is the absorption set is proved. The distributions of the absorption number and the Chu-Vandermonde type identities for the \((t,q)\)-binomial coefficients are also discussed.