In this note we consider the question how the set of inversions of a permutation $π\in S_n$ can be partitioned into two subset, such that those are itself inversion sets of permutations. This is archived by exploiting a connection to a graph theoretical result. For this we establish a correspondence between the substitution decomposition of $π$ and the modular decomposition of its inversion graph. We also consider the special case of multiplicative decompositions.
If $α\in S_n$ is a permutation of $\{1, 2, \ldots, n\}$, the inversion set of $α$ is $Φ(α) = \{(i, j) \, | \, 1 \leq i < j \leq n, α(i) > α(j)\}$. We describe all $r$-tuples $α_1, α_2, \ldots, α_r \in S_n$ such that $Δ_n^+ = \{(i, j) \, | \, 1 \leq i < j \leq n\}$ is the disjoint union of $Φ(α_1), Φ(α_2), \ldots, Φ(α_r)$. Using this description we prove that certain faces of the Littlewood-Richardson cone are simplicial and provide an algorithm for writing down their sets of generating rays. We also discuss analogous problems for the Weyl groups of root systems of types $B$, $C$ and $D$ provid…
We introduce ballot matrices, a signed combinatorial structure whose definition naturally follows from the generating function for labeled interval orders. A sign reversing involution on ballot matrices is defined. We show that matrices fixed under this involution are in bijection with labeled interval orders and that they decompose to a pair consisting of a permutation and an inversion table. To fully classify such pairs, results pertaining to the enumeration of permutations having a given set of ascent bottoms are given. This allows for a new formula for the number of labeled interval orders…
We solve two related extremal problems in the theory of permutations. A set $Q$ of permutations of the integers 1 to $n$ is inversion-complete (resp., pair-complete) if for every inversion $(j,i)$, where $1 \le i \textless{} j \le n$, (resp., for every pair $(i,j)$, where $i\not= j$) there exists a permutation in~$Q$ where $j$ is before~$i$. It is minimally inversion-complete if in addition no proper subset of~$Q$ is inversion-complete; and similarly for pair-completeness. The problems we consider are to determine the maximum cardinality of a minimal inversion-complete set of permutations, and…
We show that there exist $k$-colorable matroids that are not $(b,c)$-decomposable when $b$ and $c$ are constants. A matroid is $(b,c)$-decomposable, if its ground set of elements can be partitioned into sets $X_1, X_2, \ldots, X_l$ with the following two properties. Each set $X_i$ has size at most $ck$. Moreover, for all sets $Y$ such that $|Y \cap X_i| \leq 1$ it is the case that $Y$ is $b$-colorable. A $(b,c)$-decomposition is a strict generalization of a partition decomposition and, thus, our result refutes a conjecture from arXiv:1911.10485v2 .
In this extended abstract we announce a proof that, in a Coxeter group of rank 3, low elements are in bijection with small inversion sets. This gives a partial confirmation of Conjecture 2 in [Dyer, Hohlweg '16]. That same article provides the main ingredient: the bipodality of the set of small roots is used to propagate information on the vertices of inversion polytopes.
In this article, we investigate the existence of joins in the weak order of an infinite Coxeter group W. We give a geometric characterization of the existence of a join for a subset X in W in terms of the inversion sets of its elements and their position relative to the imaginary cone. Finally, we discuss inversion sets of infinite reduced words and the notions of biconvex and biclosed sets of positive roots.
Let $\mathfrak g$ be a simple Lie algebra, $\mathfrak b$ a fixed Borel subalgebra, and $W$ the Weyl group of $\mathfrak g$. In this note, we study a relationship between the maximal abelian ideals of $\mathfrak b$ and the minimal inversion complete sets of $W$. The latter have been recently defined by Malvenuto et al. (J. Algebra, 424 (2015), 330-356.)
We compute for reflection groups of type $A,B,D,F_4,H_3$ and for dihedral groups a statistic counting the maximal cardinality of a set of elements in the group whose generalized inversions yield the full set of inversions and which are minimal with respect to this property. We also provide lower bounds for the $E$ types that we conjecture to be the exact value of our statistic.
The problem of finding necessary and sufficient conditions to decompose a complete tripartite graph $K_{r,s,t}$ into 5-cycles was first considered by E.S. Mahmoodian and Maryam Mirzakhani (1995). They stated some necessary conditions and conjectured that those conditions are also sufficient. Since then, many cases of the problem have been solved by various authors; however, the case when the partite sets $r\leq s\leq t$ have odd and distinct sizes remained open. We show the conjecture is true when $r$, $s$ and $t$ are all multiples of 5, $t+90 \leq \frac{4rs}{r+s}$, and $t \neq s+10$.
Beijing International Center for Mathematical Research, Peking University（北京大学北京国际数学研究中心）；Qiuzhen College, Tsinghua University（清华大学邱耀学院）；School of Mathematical Sciences, Peking University（北京大学数学科学学院）
We define and study odd analogues of classical geometric and combinatorial objects associated to permutations, namely odd Schubert varieties, odd diagrams, and odd inversion sets. We show that there is a bijection between odd inversion sets of permutations and acyclic orientations of the Turán graph, that the dimension of the odd Schubert variety associated to a permutation is the odd length of the permutation, and give several necessary conditions for a subset of $[n] \times [n]$ to be the odd diagram of a permutation. We also study the sign-twisted generating function of the odd length over …
Given a sufficiently large and sufficiently dense bipartite graph $G=(A, B; E),$ we present a novel method for decomposing the majority of the edges of $G$ into quasirandom graphs so that the vertex sets of these quasirandom graphs partition the majority of $A.$ The method works for relatively small or sparse graphs, and can be used to substitute the Regularity lemma of Szemerédi in some graph embedding problems.
Recently, Hong and Li launched a systematic study of length-four pattern avoidance in inversion sequences, and in particular, they conjectured that the number of $0021$-avoiding inversion sequences can be enumerated by the OEIS entry A218225. Meanwhile, Burstein suggested that the same sequence might also count three sets of pattern restricted permutations. The objective of this paper is not only a confirmation of Hong and Li's conjecture and Burstein's first conjecture, but also two more delicate generating function identities with the $\mathsf{ides}$ statistic concerned in the restricted per…
Decomposing a digraph into subdigraphs with a fixed structure or property is a classical problem in graph theory and a useful tool in a number of applications of networks and communication. A digraph is strongly connected if it contains a directed path from each vertex to all others. In this paper we consider multipartite tournaments, and we study the existence of a partition of a multipartite tournament with $c$ partite sets into strongly connected $c$-tournaments. This is a continuation of the study started in 1999 by Volkmann of the existence of strongly connected subtournaments in multipar…
We consider decomposing a 3-connected planar graph $G$ using laminar separators of size three. We show how to find a maximal set of laminar 3-separators in such a graph in linear time. We also discuss how to find maximal laminar set of 3-separators from special families. For example we discuss non-trivial cuts, ie. cuts which split $G$ into two components of size at least two. For any vertex $v$, we also show how to find a maximal set of 3-separators disjoint from $v$ which are laminar and satisfy: every vertex in a separator $X$ has two neighbours not in the unique component of $G-X$ containi…
