Jobst Heitzig
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.
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We show that the sorting probability of the Catalan poset} $P_n$ satisfies $δ(P_n)= O \bigl(n^{-5/4}\bigr)$.
Poset Associahedra and Stack-sorting
本研究针对一类介于经典排列多面体与结合多面体之间的偏序集结合多面体，给出其$h$-向量的组合解释，关联栈排序理论并证明部分$h$-多项式的实根性。
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.
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…
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 $ω$.
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。
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…
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.
In this note we construct a poset map from a Boolean algebra to the Bruhat order which unveils an interesting connection between subword complexes, sorting orders, and certain totally nonnegative spaces. This relationship gives a new proof of Björner and Wachs' result \cite{BW} that the proper part of Bruhat order is homotopy equivalent to the proper part of a Boolean algebra --- that is, to a sphere. We also obtain a geometric interpretation for sorting orders. We conclude with two new results: that the intersection of all sorting orders is the weak order, and the union of sorting orders is t…
We analyze the quantum query complexity of sorting under partial information. In this problem, we are given a partially ordered set $P$ and are asked to identify a linear extension of $P$ using pairwise comparisons. For the standard sorting problem, in which $P$ is empty, it is known that the quantum query complexity is not asymptotically smaller than the classical information-theoretic lower bound. We prove that this holds for a wide class of partially ordered sets, thereby improving on a result from Yao (STOC'04).
Kahn and Kim (J. Comput. Sci., 1995) have shown that for a finite poset $P$, the entropy of the incomparability graph of $P$ (normalized by multiplying by the order of $P$) and the base-$2$ logarithm of the number of linear extensions of $P$ are within constant factors from each other. The tight constant for the upper bound was recently shown to be $2$ by Cardinal, Fiorini, Joret, Jungers and Munro (STOC 2010, Combinatorica). Here, we refine this last result in case $P$ has width $2$: we show that the constant can be replaced by $2-\varepsilon$ if one also takes into account the number of conn…
We are interested in maximizing the number of pairwise unrelated copies of a poset $P$ in the family of all subsets of $[n]$. We prove that for any $P$ the maximum number of unrelated copies of $P$ is asymptotic to a constant times the largest binomial coefficient. Moreover, the constant has the form $\frac{1}{c(P)}$, where $c(P)$ is the size of the smallest convex closure over all embeddings of $P$ into the Boolean lattice.
Extending the classical pop-stack sorting map on the lattice given by the right weak order on $S_n$, Defant defined, for any lattice $M$, a map $\mathsf{Pop}_{M}: M \to M$ that sends an element $x\in M$ to the meet of $x$ and the elements covered by $x$. In parallel with the line of studies on the image of the classical pop-stack sorting map, we study $\mathsf{Pop}_{M}(M)$ when $M$ is the weak order of type $B_n$, the Tamari lattice of type $B_n$, the lattice of order ideals of the root poset of type $A_n$, and the lattice of order ideals of the root poset of type $B_n$. In particular, we sett…
In 2022, Defant and Kravitz introduced extended promotion (denoted $\partial$), a map that acts on the set of labelings of a poset. Extended promotion is a generalization of Schützenberger's promotion operator, a well-studied map that permutes the set of linear extensions of a poset. It is known that if $L$ is a labeling of an $n$-element poset $P$, then $\partial^{n-1}(L)$ is a linear extension. This allows us to regard $\partial$ as a sorting operator on the set of all labelings of $P$, where we think of the linear extensions of $P$ as the labelings which have been sorted. The labelings requ…
A poset $(P',\le_{P'})$ contains a copy of some other poset $(P,\le_P)$ if there is an injection $f\colon P'\to P$ where for every $X,Y\in P$, $X\le_P Y$ if and only if $f(X)\le_{P'} f(Y)$. For any posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the smallest integer $N$ such that any blue/red coloring of a Boolean lattice of dimension $N$ contains either a copy of $P$ with all elements blue or a copy of $Q$ with all elements red. We denote by $K_{t_1,\dots,t_\ell}$ a complete $\ell$-partite poset, i.e.\ a poset consisting of $\ell$ pairwise disjoint sets $A^i$ of size $t_i$, $1\le i\le…
Variance vs. range for linear extensions, and balancing extensions in posets of bounded width
本研究证明Kahn-Saks关于大宽度偏序集存在平衡元素对的猜想，可推广至仅存在大量与某元素不可比元素的情况，还借此简化证明了Young图排序概率相关结果及其高维推广。
In order theory, a rank function measures the vertical "level" of a poset element. It is an integer-valued function on a poset which increments with the covering relation, and is only available on a graded poset. Defining a vertical measure to an arbitrary finite poset can be accomplished by extending a rank function to be interval-valued. This establishes an order homomorphism from a base poset to a poset over real intervals, and a standard (canonical) specific interval rank function is available as an extreme case. Various ordering relations are available over intervals, and we begin in this…
The Order Dimension of the Poset of Regions in a Hyperplane Arrangement
研究了Coxeter群弱序的序维数等于其秩，并为其他有限类型给出了序维数的界限，通过统一方法简化了A和B类型已知结果。
Pop, Crackle, Snap (and Pow): Some Facets of Shards
本文将超平面构型中的碎片解释为补集基本群的生成元，并构造纯碎片幺半群，进而通过Pop算子与嵌入Crackle、Snap建立碎片交序与纯碎片幺半群及正辫子幺半群之间的联系。
Poset Ramsey number $R(P,Q_n)$. III. Chain Compositions and Antichains
本文研究偏序集 Ramsey 数 $R(P,Q_n)$，为链的平行复合和细分 $Q_2$ 两类偏序集给出上界，从而完成至多4元素偏序集的结果确定，并证明反链情形 $R(A_t,Q_n)=n+3$ 对 $3\le t\le \log\log n$ 成立。
On the Gap of Finite Posets
该研究针对有限偏序集，证明了宽度与期望秩间隙的猜想关系，构造了宽度2偏序集使间隙任意大，还构建了满足熵与间隙特定关系的偏序集，核心思想来自ChatGPT 5.6 Sol。
On the Average Sizes of Ideals and Antichains in Some Infinite Families of Posets
研究有根无标号树生成的偏序集族中理想与反链的平均大小，利用生成函数和奇点分析证明二次根分解族中理想平均大小为 n/3，并分析反链常数及分支限制的影响。
Sperner theorems for unrelated copies of some partially ordered sets in a powerset lattice and minimum generating sets of powers of distributive lattices
该研究确定了含0、1元的有限偏序集在幂集格中的最大无关副本数Sp(U,n)并给出部分偏序集的估计，证明其与分配格幂的最小生成集函数互为左伴随，可计算多种分配格幂的最小生成集，为密码认证协议提供了适用的格结构。
On the order sequence of a group
本文引入有限群的阶序列及其逐元支配偏序，刻画循环群的极大性与阶乘积、和的界，并建立 Abel p-群与分拆偏序集及互素直积的对应关系。
Minimum-sized generating sets of the direct powers of free distributive lattices
该研究针对自由分配格直幂的最小生成元数问题，通过估计子集格中全段偏序集无关副本的最大数量，给出了多组参数下的精确或高精度结果，并提及该研究与格论及密码学相关。
Sorting algorithms have attracted a great deal of attention and study, as they have numerous applications to Mathematics, Computer Science and related fields. In this thesis, we first deal with the mathematical analysis of the Quicksort algorithm and its variants. Specifically, we study the time complexity of the algorithm and we provide a complete demonstration of the variance of the number of comparisons required, a known result but one whose detailed proof is not easy to read out of the literature. We also examine variants of Quicksort, where multiple pivots are chosen for the partitioning …
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