Abstract
This paper continues the analysis of the pattern-avoiding sorting machines recently introduced by Cerbai, Claesson and Ferrari (2020). These devices consist of two stacks, through which a permutation is passed in order to sort it, where the content of each stack must at all times avoid a certain pattern. Here we characterize and enumerate the set of permutations that can be sorted when the first stack is $132$-avoiding, solving one of the open problems proposed by the above mentioned authors. To that end we present several connections with other well known combinatorial objects, such as lattice paths and restricted growth functions (which encode set partitions). We also provide new proofs for the enumeration of some sets of pattern-avoiding restricted growth functions and we expect that the tools introduced can be fruitfully employed to get further similar results.
Highlights
Pattern-avoiding sorting machines were introduced in a recent paper by Cerbai, Claesson and Ferrari [CCF]
There is a well known algorithm, called Stacksort, that sorts every sortable permutation. It has two key properties: 1. the stack is increasing, meaning that the elements inside the stack are maintained in increasing order; 2. the algorithm is right greedy, meaning that it always chooses to perform a push operation as long as the stack remains increasing in the above sense; here the expression “right greedy” refers to the usual pictorial representation of this problem, in which the input permutation is on the right, the stack is in the middle and the output permutation is on the left
Given two permutations σ ∈ Sk and π = π1 · · · πn ∈ Sn, with k n, we say that σ is a pattern of π when there exist indices 1 i1 < i2 < · · · < ik n such that πi1πi2 . . . πik is isomorphic to σ, that is, πi1, πi2, . . . , πik are in the same relative order of size as the elements of σ, in which case we write σ πi1πi2 . . . πik
Summary
Pattern-avoiding sorting machines were introduced in a recent paper by Cerbai, Claesson and Ferrari [CCF]. Given that describing the set of sortable permutations is rather manageable in the classical case, one would think that similar results can be derived by considering a slightly more general version of the problem, where a second stack is connected in series to the first one. Amongst the six permutations of length three, Sort(321) = Av(123, 132) as a consequence of the previous result, but so far the only other solved pattern is 123: 123-sortable permutations are shown to be enumerated by the partial sums of partial sums of the Catalan numbers (sequence A294790 in [Sl]) via a bijection with Schroder paths avoiding the pattern UHD [CF]. Some of the results lead to an independent proof of the enumeration of Sort(132)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
