We extend the idea of interval pattern avoidance defined by Yong and the author for $S_n$ to arbitrary Weyl groups using the definition of pattern avoidance due to Billey and Braden, and Billey and Postnikov. We show that, as previously shown by Yong and the author for $GL_n$, interval pattern avoidance is a universal tool for characterizing which Schubert varieties have certain local properties, and where these local properties hold.
We show that the hyperplane arrangement of a coconvex set in a finite root system is free if and only if it is free in corank 4. As a consequence, we show that the inversion arrangement of a Weyl group element w is free if and only if w avoids a finite list of root system patterns. As a key part of the proof, we use a recent theorem of Abe and Yoshinaga to show that if the root system does not contain any factors of type C or F, then Peterson translation of coconvex sets preserves freeness. This also allows us to give a Kostant-Shapiro-Steinberg rule for the coexponents of a free inversion arr…
Following Anders and Archer, we say that an unordered rooted labeled forest avoids the pattern $σ\in\mathcal{S}_k$ if in each tree, each sequence of labels along the shortest path from the root to a vertex does not contain a subsequence with the same relative order as $σ$. For each permutation $σ\in\mathcal{S}_{k-2}$, we construct a bijection between $n$-vertex forests avoiding $(σ)(k-1)k:=σ(1)\cdotsσ(k-2)(k-1)k$ and $n$-vertex forests avoiding $(σ)k(k-1):=σ(1)\cdotsσ(k-2)k(k-1)$, giving a common generalization of results of West on permutations and Anders--Archer on forests. We further define…
Positroids are certain representable matroids originally studied by Postnikov in connection with the totally nonnegative Grassmannian and now used widely in algebraic combinatorics. The positroids give rise to determinantal equations defining positroid varieties as subvarieties of the Grassmannian variety. Rietsch, Knutson-Lam-Speyer and Pawlowski studied geometric and cohomological properties of these varieties. In this paper, we continue the study of the geometric properties of positroid varieties by establishing several equivalent conditions characterizing smooth positroid varieties using a…
We present an infinite family of 3-connected non-bipartite graphs with chromatic roots in the interval (1,2) thus resolving a conjecture of Jackson's in the negative. In addition, we briefly consider other graph classes that are conjectured to have no chromatic roots in (1,2).
The odd diagram of a permutation is a subset of the classical diagram with additional parity conditions. In this paper, we study classes of permutations with the same odd diagram, which we call odd diagram classes. First, we prove a conjecture relating odd diagram classes and 213- and 312-avoiding permutations. Secondly, we show that each odd diagram class is a Bruhat interval. Instrumental to our proofs is an explicit description of the Bruhat edges that link permutations in a class.
This paper considers the enumeration of ternary trees (i.e. rooted ordered trees in which each vertex has 0 or 3 children) avoiding a contiguous ternary tree pattern. We begin by finding recurrence relations for several simple tree patterns; then, for more complex trees, we compute generating functions by extending a known algorithm for pattern-avoiding binary trees. Next, we present an alternate one-dimensional notation for trees which we use to find bijections that explain why certain pairs of tree patterns yield the same avoidance generating function. Finally, we compare our bijections to k…
We develop interval pattern avoidance and Mars-Springer ideals to study singularities of symmetric orbit closures in a flag variety. This paper focuses on the case of the Levi subgroup GL_p x GL_q acting on the classical flag variety. We prove that all reasonable singularity properties can be classified in terms of interval patterns of clans.
Jim Propp recently introduced a variant of chip-firing on a line where the chips are given distinct integer labels. Hopkins, McConville, and Propp showed that this process is confluent from some (but not all) initial configurations of chips. We recast their set-up in terms of root systems: labeled chip-firing can be seen as a root-firing process which allows the moves $λ\to λ+ α$ for $α\in Φ^{+}$ whenever $\langleλ,α^\vee\rangle = 0$, where $Φ^{+}$ is the set of positive roots of a root system of Type A and $λ$ is a weight of this root system. We are thus motivated to study the exact same root…
Hultman, Linusson, Shareshian, and Sjöstrand gave a pattern avoidance characterization of the permutations for which the number of chambers of its associated inversion arrangement is the same as the size of its lower interval in Bruhat order. Hultman later gave a characterization, valid for an arbitrary finite reflection group, in terms of distances in the Bruhat graph. On the other hand, the pattern avoidance criterion for permutations had earlier appeared in independent work of Sjöstrand and of Gasharov and Reiner. We give characterizations of the elements of the hyperoctahedral groups satis…
Babson and Steingr\'ımsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We consider n-permutations that avoid the generalized pattern 1-32 and whose k rightmost letters form an increasing subword. The number of such permutations is a linear combination of Bell numbers. We find a bijection between these permutations and all partitions of an $(n-1)$-element set with one subset marked that satisfy certain additional conditions. Also we find the e.g.f. for the number of permutations that avoid a g…
It is proved that if $G$ is a graph containing a spanning tree with at most three leaves, then the chromatic polynomial of $G$ has no roots in the interval $(1,t_1]$, where $t_1 \approx 1.2904$ is the smallest real root of the polynomial $(t-2)^6 +4(t-1)^2(t-2)^3 -(t-1)^4$. We also construct a family of graphs containing such spanning trees with chromatic roots converging to $t_1$ from above. We employ the Whitney $2$-switch operation to manage the analysis of an infinite class of chromatic polynomials.
Positroids are certain representable matroids originally studied by Postnikov in connection with the totally nonnegative Grassmannian and now used widely in algebraic combinatorics. The positroids give rise to determinantal equations defining positroid varieties as subvarieties of the Grassmannian variety. Rietsch, Knutson-Lam-Speyer, and Pawlowski studied geometric and cohomological properties of these varieties. In this paper, we continue the study of the geometric properties of positroid varieties by establishing several equivalent conditions characterizing smooth positroid varieties using …
For a tree $T$, the subtree polynomial of $T$ is the generating polynomial for the number of subtrees of $T$. We show that the complex roots of the subtree polynomial are contained in the disk $\left\{z\in\mathbb{C}\colon\ |z|\leq 1+\sqrt[3]{3}\right\}$, and that $K_{1,3}$ is the only tree whose subtree polynomial has a root on the boundary. We also prove that the closure of the collection of all real roots of subtree polynomials contains the interval $[-2,-1]$, while the intervals $(\infty,-1-\sqrt[3]{3})$, $[-1,0)$, and $(0,\infty)$ are root-free.
This article studies real roots of the flow polynomial $F(G,λ)$ of a bridgeless graph $G$. For any integer $k\ge 0$, let $ξ_k$ be the supremum in $(1,2]$ such that $F(G,λ)$ has no real roots in $(1,ξ_k)$ for all graphs $G$ with $|W(G)|\le k$, where $W(G)$ is the set of vertices in $G$ of degrees larger than $3$. We prove that $ξ_k$ can be determined by considering a finite set of graphs and show that $ξ_k=2$ for $k\le 2$, $ξ_3=1.430\cdots$, $ξ_4=1.361\cdots$ and $ξ_5=1.317\cdots$. We also prove that for any bridgeless graph $G=(V,E)$, if all roots of $F(G,λ)$ are real but some of these roots a…
The $σ$-polynomial is given by $σ(G,x) = \sum_{i=χ(G)}^{n} a_{i}(G)\, x^{i}$, where $a_{i}(G)$ is the number of partitions of the vertices of $G$ into $i$ nonempty independent sets. These polynomials are closely related to chromatic polynomials, as the chromatic polynomial of $G$ is given by $\sum_{i=χ(G)}^{n} a_{i}(G)\, x(x-1) \cdots (x-(i-1))$. It is known that the closure of the real roots of chromatic polynomials is precisely $\{0,~1\} \bigcup [32/27,\infty)$, with $(-\infty,0)$, $(0,1)$ and $(1,32/27)$ being maximal zero-free intervals for roots of chromatic polynomials. We ask here wheth…
A syntax tree is a planar rooted tree where internal nodes are labeled on a graded set of generators. There is a natural notion of occurrence of contiguous pattern in such trees. We describe a way, given a set of generators $\mathfrak{G}$ and a set of patterns $\mathcal{P}$, to enumerate the trees constructed on $\mathfrak{G}$ and avoiding $\mathcal{P}$. The method is built around inclusion-exclusion formulas forming a system of equations on formal power series of trees, and composition operations of trees. This does not require particular conditions on the set of patterns to avoid. We connect…
Björner-Ekedahl prove that general intervals $[e,w]$ in Bruhat order are "top-heavy", with at least as many elements in the $i$-th corank as the $i$-th rank. Well-known results of Carrell and of Lakshmibai-Sandhya give the equality case: $[e,w]$ is rank-symmetric if and only if the permutation $w$ avoids the patterns $3412$ and $4231$ and these are exactly those $w$ such that the Schubert variety $X_w$ is smooth. In this paper we study the finer structure of rank-symmetric intervals $[e,w]$, beyond their rank functions. In particular, we show that these intervals are still "top-heavy" if one c…
Given arbitrary integers $k$ and $d$ with $0 \leq 2k \leq d$, we construct a Gorenstein Fano polytope $\Pc \subset \RR^d$ of dimension $d$ such that (i) its Ehrhart polynomial $i(\Pc, n)$ possesses $d$ distinct roots; (ii) $i(\Pc, n)$ possesses exactly $2k$ imaginary roots; (iii) $i(\Pc, n)$ possesses exactly $d - 2k$ real roots; (iv) the real part of each of the imaginary roots is equal to $- 1 / 2$; (v) all of the real roots belong to the open interval $(-1, 0)$.
We present a combinatorial and computational commutative algebra methodology for studying singularities of Schubert varieties of flag manifolds. We define the combinatorial notion of *interval pattern avoidance*. For "reasonable" invariants P of singularities, we geometrically prove that this governs (1) the P-locus of a Schubert variety, and (2) which Schubert varieties are globally not P. The prototypical case is P="singular"; classical pattern avoidance applies admirably for this choice [Lakshmibai-Sandhya'90], but is insufficient in general. Our approach is analyzed for some common invaria…
