Related papers: n! matchings, n! posets
For a rank two root system and a pair of nonnegative integers, using only elementary combinatorics we construct two posets. The constructions are uniform across the root systems A1+A1, A2, C2, and G2. Examples appear in Figures 3.2 and 3.3.…
In 1989 Erd\H{o}s and Sz\'ekely showed that there is a bijection between (i) the set of rooted trees with $n+1$ vertices whose leaves are bijectively labeled with the elements of $[\ell]=\{1,2,\dots,\ell\}$ for some $\ell \leq n$, and (ii)…
We introduce a new concept of permutation avoidance pattern called hatted pattern, which is a natural generalization of the barred pattern. We show the growth rate of the class of permutations avoiding a hatted pattern in comparison to…
We resolve a question of Gillespie, Griffin, and Levinson that asks for a combinatorial bijection between two classes of trivalent trees, tournament trees and slide trees, that both naturally arise in the intersection theory of the moduli…
The confluent binomial posets with the atomic function A(n) = max (1,2^(n-2)) are classified. In particular, it is shown that in general there are non-isomorphic intervals of the same length.
Define an expansion poset to be the poset of monomials of a cluster variable attached to an arc in a polygon, where each monomial is represented by the corresponding combinatorial object from some fixed combinatorial cluster expansion…
In section 1 we consider a 3-tuple $S=(|S|,\preccurlyeq,E)$ where $|S|$ is a finite set, $\preccurlyeq$ a partial ordering on $|S|,$ and $E$ a set of unordered pairs of distinct members of $|S|,$ and study, as a function of $n\geq 0,$ the…
Given two posets $P,Q$ we say that $Q$ is $P$-free if $Q$ does not contain a copy of $P$. The size of the largest $P$-free family in $2^{[n]}$, denoted by $La(n,P)$, has been extensively studied since the 1980s. We consider several related…
A derangement is a permutation with no fixed point, and a nonderangement is a permutation with at least one fixed point. There is a one-term recurrence for the number of derangements of $n$ elements, and we describe a bijective proof of…
The class of ranked tree-child networks, tree-child networks arising from an evolution process with a fixed embedding into the plane, has recently been introduced by Bienvenu, Lambert, and Steel. These authors derived counting results for…
Let $B_n$ be the poset generated by the subsets of $[n]$ with the inclusion as relation and let $P$ be a finite poset. We want to embed $P$ into $B_n$ as many times as possible such that the subsets in different copies are incomparable. The…
The Interval poset of a permutation is an effective way of capturing all the intervals of the permutation and the inclusions between them and was introduced recently by Tenner. Thi paper explores the geometric interpretation of interval…
Matchings are frequently used to model RNA secondary structures; however, not all matchings can be realized as RNA motifs. One class of matchings, called the L $\&$ P matchings, is the most restrictive model for RNA secondary structures in…
We address a supersaturation problem in the context of forbidden subposets. A family $\mathcal{F}$ of sets is said to contain the poset $P$ if there is an injection $i:P \rightarrow \mathcal{F}$ such that $p \le_P q$ implies $i(p) \subset i…
In this short note, we prove that every twin-free graph on $n$ vertices contains a locating-dominating set of size at most $\lceil\frac{5}{8}n\rceil$. This improves the earlier bound of $\lfloor\frac{2}{3}n\rfloor$ due to Foucaud, Henning,…
The non-Abelian exponentiation theorem has recently been generalised to correlators of multiple Wilson line operators. The perturbative expansions of these correlators exponentiate in terms of sets of diagrams called webs, which together…
Given a set of $n$ red and $n$ blue points in the plane, we are interested in matching red points with blue points by straight line segments so that the segments do not cross. We develop a range of tools for dealing with the non-crossing…
We show that the well known {\em homotopy complementation formula} of Bj\"orner and Walker admits several closely related generalizations on different classes of topological posets (lattices). The utility of this technique is demonstrated…
We say a finite poset $P$ is a tree poset if its Hasse diagram is a tree. Let $k$ be the length of the largest chain contained in $P$. We show that when $P$ is a fixed tree poset, the number of $P$-free set systems in $2^{[n]}$ is…
The problem of genealogy of permutations has been solved partially by Stefan (odd order) and Acosta-Hum\'anez \& Bernhardt (power of two). It is well known that Sharkovskii's theorem shows the relationship between the cardinal of the set of…