Permutations sortable by n-4 passes through a stack

We characterise and enumerate permutations that are sortable by n-4 passes through a stack. We conjecture the number of permutations sortable by n-5 passes, and also the form of a formula for the general case n-k, which involves a polynomial expression.Comment: 6 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

Deterministic stack-sorting for set partitions

A sock sequence is a sequence of elements, which we will refer to as socks, from a finite alphabet. A sock sequence is sorted if all occurrences of a sock appear consecutively. We define equivalence classes of sock sequences called sock patterns, which are in bijection with set partitions. The notion of stack-sorting for set partitions was originally introduced by Defant and Kravitz. In this paper, we define a new deterministic stack-sorting map $\phi_{\sigma}$ for sock sequences that uses a $\sigma$-avoiding stack, where pattern containment need not be consecutive. When $\sigma = aba$, we show that our stack-sorting map sorts any sock sequence with $n$ distinct socks in at most $n$ iterations, and that this bound is tight for $n \geq 3$. We obtain a fine-grained enumeration of the number of sock patterns of length $n$ on $r$ distinct socks that are $1$-stack-sortable under $\phi_{aba}$, and we also obtain asymptotics for the number of sock patterns of length $n$ that are $1$-stack-sortable under $\phi_{aba}$. Finally, we show that for all unsorted sock patterns $\sigma \neq a\cdots a b a \cdots a$, the map $\phi_{\sigma}$ cannot eventually sort all sock sequences on any multiset $M$ unless every sock sequence on $M$ is already sorted