Sorting and preimages of pattern classes

We introduce an algorithm to determine when a sorting operation, such as stack-sort or bubble-sort, outputs a given pattern. The algorithm provides a new proof of the description of West-2-stack-sortable permutations, that is permutations that are completely sorted when passed twice through a stack, in terms of patterns. We also solve the long-standing problem of describing West-3-stack-sortable permutations. This requires a new type of generalized permutation pattern we call a decorated pattern.Comment: 13 pages, 5 figures, to appear at FPSAC 201

Which School Systems Sort Weaker Students into Smaller Classes? International Evidence

We examine whether the sorting of differently achieving students into differently sized classes results in a regressive or compensatory pattern of class sizes for a sample of national school systems. Sorting effects are identified by subtracting the causal effect of class size on performance from their total correlation. Our empirical results indicate substantial compensatory sorting within and especially between schools in many countries. Only the United States, a country with decentralized education finance and considerable residential mobility, exhibits regressive between-school sorting. Between-school sorting is more compensatory in systems with ability tracking. Within-school sorting is more compensatory when administrators rather than teachers assign students to classrooms.student sorting, class size, educational achievement, education

Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations

Defant, Engen, and Miller defined a permutation to be uniquely sorted if it has exactly one preimage under West's stack-sorting map. We enumerate classes of uniquely sorted permutations that avoid a pattern of length three and a pattern of length four by establishing bijections between these classes and various lattice paths. This allows us to prove nine conjectures of Defant.Comment: 18 pages, 16 figures, new version with updated abstract and reference

2-stack pushall sortable permutations

In the 60's, Knuth introduced stack-sorting and serial compositions of stacks. In particular, one significant question arise out of the work of Knuth: how to decide efficiently if a given permutation is sortable with 2 stacks in series? Whether this problem is polynomial or NP-complete is still unanswered yet. In this article we introduce 2-stack pushall permutations which form a subclass of 2-stack sortable permutations and show that these two classes are closely related. Moreover, we give an optimal O(n^2) algorithm to decide if a given permutation of size n is 2-stack pushall sortable and describe all its sortings. This result is a step to the solve the general 2-stack sorting problem in polynomial time.Comment: 41 page

Revstack sort, zigzag patterns, descent polynomials of tt-revstack sortable permutations, and Steingr\'imsson's sorting conjecture

In this paper we examine the sorting operator $T(LnR)=T(R)T(L)n$. Applying this operator to a permutation is equivalent to passing the permutation reversed through a stack. We prove theorems that characterise $t$-revstack sortability in terms of patterns in a permutation that we call $zigzag$ patterns. Using these theorems we characterise those permutations of length $n$ which are sorted by $t$ applications of $T$ for $t=0,1,2,n-3,n-2,n-1$. We derive expressions for the descent polynomials of these six classes of permutations and use this information to prove Steingr\'imsson's sorting conjecture for those six values of $t$. Symmetry and unimodality of the descent polynomials for general $t$-revstack sortable permutations is also proven and three conjectures are given