Sorting probability of Catalan posets

A Non-Messing-Up Phenomenon for Posets

We classify finite posets with a particular sorting property, generalizing a result for rectangular arrays. Each poset is covered by two sets of disjoint saturated chains such that, for any original labeling, after sorting the labels along both sets of chains, the labels of the chains in the first set remain sorted. We also characterize posets with more restrictive sorting properties.

A unifying poset perspective on alternating sign matrices, plane partitions, Catalan objects, tournaments, and tableaux

Alternating sign matrices (ASMs) are square matrices with entries 0, 1, or -1 whose rows and columns sum to 1 and whose nonzero entries alternate in sign. We present a unifying perspective on ASMs and other combinatorial objects by studying a certain tetrahedral poset and its subposets. We prove the order ideals of these subposets are in bijection with a variety of interesting combinatorial objects, including ASMs, totally symmetric self-complementary plane partitions (TSSCPPs), staircase shaped semistandard Young tableaux, Catalan objects, tournaments, and totally symmetric plane partitions. …

Sorting probability for large Young diagrams

For a finite poset $P=(X,\prec)$, let $\mathcal{L}_P$ denote the set of linear extensions of $P$. The sorting probability $δ(P)$ is defined as \[δ(P) \, := \, \min_{x,y\in X} \, \bigl| \mathbf{P} \, [L(x)\leq L(y) ] \ - \ \mathbf{P} \, [L(y)\leq L(x) ] \bigr|\,, \] where $L \in \mathcal{L}_P$ is a uniform linear extension of $P$. We give asymptotic upper bounds on sorting probabilities for posets associated with large Young diagrams and large skew Young diagrams, with bounded number of rows.

The asymptotic complexity of partial sorting -- How to learn large posets by pairwise comparisons

The expected number of pairwise comparisons needed to learn a partial order on n elements is shown to be at least n*n/4-o(n*n), and an algorithm is given that needs only n*n/4+o(n*n) comparisons on average. In addition, the optimal strategy for learning a poset with four elements is presented.

Promotion Sorting

Schützenberger's promotion operator is an extensively-studied bijection that permutes the linear extensions of a finite poset. We introduce a natural extension $\partial$ of this operator that acts on all labelings of a poset. We prove several properties of $\partial$; in particular, we show that for every labeling $L$ of an $n$-element poset $P$, the labeling $\partial^{n-1}(L)$ is a linear extension of $P$. Thus, we can view the dynamical system defined by $\partial$ as a sorting procedure that sorts labelings into linear extensions. For all $0\leq k\leq n-1$, we characterize the $n$-element…

Sorting and generating reduced words

We introduce a partial order on the set of all reduced words of a given permutation $ω$, called \emph{directed-braid poset} of $ω$. This poset enables us to produce two algorithms: One is a sorting algorithm applied on any reduced word of $ω$ and aims to obtained the natural word (lexicographically largest reduced word); the other one is a generation algorithm applied on the natural word and aims to obtained the set of all reduced words of $ω$.

提升、缠绕标记与排序生成函数

Promotion, Tangled Labelings, and Sorting Generating Functions

本文研究了Defant和Kravitz对有限偏序集任意标记的推广提升算子，提出了更严格的猜想，并引入了排序生成函数，证明了累积生成函数系数的对数凹性。

Sort-Invariant Non-Messing-Up

A poset has the non-messing-up property if it has two covering sets of disjoint saturated chains so that for any labeling of the poset, sorting the labels along one set of chains and then sorting the labels along the other set yields a linear extension of the poset. The linear extension yielded by thus twice sorting a labeled non-messing-up poset may be independent of which sort was performed first. Here we characterize such sort-invariant labelings for convex subposets of a cylinder. They are completely determined by avoidance of a particular subpattern: a diamond of four elements whose small…

Sweep maps: A continuous family of sorting algorithms

We define a family of maps on lattice paths, called sweep maps, that assign levels to each step in the path and sort steps according to their level. Surprisingly, although sweep maps act by sorting, they appear to be bijective in general. The sweep maps give concise combinatorial formulas for the q,t-Catalan numbers, the higher q,t-Catalan numbers, the q,t-square numbers, and many more general polynomials connected to the nabla operator and rational Catalan combinatorics. We prove that many algorithms that have appeared in the literature (including maps studied by Andrews, Egge, Gorsky, Haglun…

Narayana数的对称性与根偏序的行空化

Symmetry of Narayana numbers and rowvacuation of root posets

本文证明对于经典类型的根偏序，行空化算子实现了Panyushev提出的对合，从而展示了W-Narayana数的对称性。

The Catalan combinatorics of the hereditary artin algebras

This is a survey on the categorification of the poset of generalized non-crossing partitions, using the representation theory of a hereditary artin algebra H, looking at the set P of exceptional subcategories in mod H. This categorification is due to Ingalls and Thomas, and a subsequent paper by Igusa and Schiffler. Starting point is a refinement of the classical tilting theory for mod H, replacing torsion pairs by torsion triples, thus putting it into the realm of the stability theory of King. The torsion pairs in mod H correspond nicely to the perpendicular pairs of exceptional subcategories…