Abstract
At the end of the 1960s, Knuth characterised the permutations that can be sorted using a stack in terms of forbidden patterns. He also showed that they are in bijection with Dyck paths and thus counted by the Catalan numbers. Subsequently, Even & Itai, Pratt and Tarjan studied permutations that can be sorted using two stacks in parallel. This problem is significantly harder. In particular, a sortable permutation can now be sorted by several distinct sequences of stack operations. Moreover, in order to be sortable, a permutation must avoid infinitely many patterns. The associated counting question has remained open for 40 years. We solve it by giving a pair of functional equations that characterise the generating function of permutations that can be sorted with two parallel stacks.The first component of this system describes the generating function Q(a,u) of square lattice loops confined to the positive quadrant, counted by the length and the number of North-West and East-South factors. Our analysis of the asymptotic number of sortable permutations relies at the moment on two intriguing conjectures dealing with the series Q(a,u). We prove that they hold for loops confined to the upper half plane, or not confined at all. They remain open for quarter plane loops. Given the recent activity on walks confined to cones, we believe them to be attractive per se.
Highlights
If we have a device whose only ability is to rearrange certain sequences of objects, it is natural to ask “What rearrangements can my device produce?” When the device is an abstract one that can operate on sequences of any size, Preprint submitted to Elsevier this becomes a combinatorial question
Such questions were apparently first considered by Knuth [19] who dealt with the case where the device was a stack, i.e. a storage mechanism operating in a last in, first out manner (Figure 1)
Using a stack it is clear that the input sequence abc cannot produce the output sequence cab as, in order for c to be the first element output, both a and b must be in the stack together but they will be output as ba and not as ab
Summary
Preprint submitted to Elsevier this becomes a combinatorial question. Such questions were apparently first considered by Knuth [19] who dealt with the case where the device was a stack, i.e. a storage mechanism operating in a last in, first out manner (Figure 1). The questions “How many permutations of length n can be produced by two stacks in series, or by two stacks in parallel?” have remained open for forty years. We determine the exponential growth of the numbers sn, modulo some conjectures that deal with square lattice walks confined to the quarter plane. These walks naturally encode the admissible sequences of stack operations, in the same way as Dyck paths do in the case of a single stack.
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