Passing through a stack \(k\) times

We consider the number of passes a permutation needs to take through a stack if we only pop the appropriate output values and start over with the remaining entries in their original order. We define a permutation \(\pi\) to be \(k\)-pass sortable if \(\pi\) is sortable using \(k\) passes through the stack. Permutations that are \(1\)-pass sortable are simply the stack sortable permutations as defined by Knuth. We define the permutation class of \(2\)-pass sortable permutations in terms of their basis. We also show all \(k\)-pass sortable classes have finite bases by giving bounds on the length of a basis element of the permutation class for any positive integer \(k\). Finally, we define the notion of tier of a permutation \(\pi\) to be the minimum number of passes after the first pass required to sort \(\pi\). We then give a bijection between the class of permutations of tier \(t\) and a collection of integer sequences studied by Parker. This gives an exact enumeration of tier \(t\) permutations of a given length and thus an exact enumeration for the class of \((t+1)\)-pass sortable permutations. Finally, we give a new derivation for the generating function in Parker's thesis and an explicit formula for the coefficients.