We introduce two polynomials (in $q$) associated with a finite poset $P$ that encode some information on the covering relation in $P$. If $P$ is a distributive lattice, and hence $P$ is isomorphic to the poset of dual order ideals in a poset $L$, then these polynomials coincide and the coefficient of $q$ equals the number of $k$-element antichains in $L$. In general, these two covering polynomials are different, and we introduce a deviation polynomial of $P$, which measures the difference between these two. We then compute all these polynomials in the case, where $P$ is one of the posets associated with an irreducible root system. These are 1) the posets of positive roots, 2) the poset of ad-nilpotent ideals, and 3) the poset of Abelian ideals.
Let $B$ be an $n \times n$ nilpotent matrix with entries in an infinite field $\k$. Assume that $B$ is in Jordan canonical form with the associated Jordan block partition $P$. In this paper, we study a poset $\mathcal{D}_P$ associated to the nilpotent commutator of $B$ and a certain partition of $n$, denoted by $λ_U(P)$, defined in terms of the lengths of unions of special chains in $\mathcal{D}_P$. Polona Oblak associated to a given partition $P$ another partition $Ob(P)$ resulting from a recursive process. She conjectured that $Ob(P)$ is the same as the Jordan partition $Q(P)$ of a generic e…
In this manuscript we develop the theory of poset pinball, a combinatorial game recently introduced by Harada and Tymoczko for the study of the equivariant cohomology rings of GKM-compatible subspaces of GKM spaces. Harada and Tymoczko also prove that in certain circumstances, a successful outcome of Betti poset pinball yields a module basis for the equivariant cohomology ring of the GKM-compatible subspace. Our main contributions are twofold. First we construct an algorithm (which we call the dimension pair algorithm) which yields the result of a successful outcome of Betti poset pinball for …
The chain covering number $\Cov(P)$ of a poset $P$ is the least number of chains needed to cover $P$. For a cardinal $ν$, we give a list of posets of cardinality and covering number $ν$ such that for every poset $P$ with no infinite antichain, $\Cov(P)\geq ν$ if and only if $P$ embeds a member of the list. This list has two elements if $ν$ is a successor cardinal, namely $[ν]^2$ and its dual, and four elements if $ν$ is a limit cardinal with $\cf(ν)$ weakly compact. For $ν= \aleph_1$, a list was given by the first author; his construction was extended by F. Dorais to every infinite successor c…
We introduce the notion of a coloring of a poset, which consists of a labeling of the edges in its Hasse diagram by elements in a given group $G$. We use $G$-colorings to describe the covering maps of posets and present a new method based on colorings to obtain concrete and simple presentations of the fundamental group of polyhedra.
The set of weights of a finite-dimensional representation of a reductive Lie algebra has a natural poset structure ("weight poset"). Studying certain combinatorial problems related to antichains in weight posets, we realised that the best setting is provided by the representations associated with $\mathbb Z$-gradings of simple Lie algebras (arXiv: math.CO 1411.7683). If $\mathfrak g$ is a simple Lie algebra, then a $\mathbb Z$-grading of $\mathfrak g$ induces a $\mathbb Z$-grading of the corresponding root system $Δ$. In this article, we elaborate on a general theory of lower ideals (or antich…
Let $Δ$ be a root system with a subset of positive roots, $Δ^+$. We consider edges of the Hasse diagrams of some posets associated with $Δ^+$. For each edge one naturally defines its type, and we study the partition of the set of edges into types. For $Δ^+$, the type is a simple root, and for the posets of ad-nilpotent and Abelian ideals the type is an affine simple roots. We give several descriptions of the set of edges of given type and uniform expressions for the number of edges. By a result of Peterson, the number of Abelian ideals is $2^n$, where $n$ is the rank of $Δ$. We prove that the …
We prove a general lemma (inspired by a lemma of Holroyd and Talbot) about the connection of the largest cardinalities (or weight) of structures satisfying some hereditary property and substructures satisfying the same hereditary property. We use it to show how results concerning forbidden subposet problems in the Boolean poset imply analogous results in the poset of subspaces of a finite vector space. We also study generalized forbidden subposet problems in the poset of subspaces.
We prove that in every cover of a Young diagram with $\binom{2k}{k}$ steps with generalized rectangles there is a row or a column in the diagram that is used by at least $k+1$ rectangles. We show that this is best-possible by partitioning any Young diagram with $\binom{2k}{k}-1$ steps into actual rectangles, each row and each column used by at most $k$ rectangles. This answers two questions by Kim et al. (2018). Our results can be rephrased in terms of local covering numbers of difference graphs with complete bipartite graphs, which has applications in the recent notion of local dimension of p…
We study covering numbers and local covering numbers with respect to difference graphs and complete bipartite graphs. In particular we show that in every cover of a Young diagram with $\binom{2k}{k}$ steps with generalized rectangles there is a row or a column in the diagram that is used by at least $k+1$ rectangles, and prove that this is best-possible. This answers two questions by Kim, Martin, Masa{ř}{\'ı}k, Shull, Smith, Uzzell, and Wang (Europ. J. Comb. 2020), namely: - What is the local complete bipartite cover number of a difference graph? - Is there a sequence of graphs with constant l…
We study type $A$ nilpotent Hessenberg varieties equipped with a natural $S^1$-action using techniques introduced by Tymoczko, Harada-Tymoczko, and Bayegan-Harada, with a particular emphasis on a special class of nilpotent Springer varieties corresponding to the partition $λ= (n-2,2)$ for $n \geq 4$. First we define the adjacent-pair matrix corresponding to any filling of a Young diagram with $n$ boxes with the alphabet $\{1,2,\ldots,n\}$. Using the adjacent-pair matrix we make more explicit and also extend some statements concerning highest forms of linear operators in previous work of Tymocz…
Let $\be$ be a Borel subalgebra of a complex simple Lie algebra $\g$. An ideal of $\be$ is called ad-nilpotent, if it is contained in $[\be,\be]$. The generators of an ad-nilpotent ideal give rise to an antichain in the poset of positive roots, and the whole theory can be expressed in a combinatorial fashion, in terms of antichains. The aim of this paper is to present a refinement of the enumerative theory of ad-nilpotent ideals for the case in which $\g$ has roots of different length. An antichain is called short, if it consists of short roots. We obtain, for short antichains, analogues of al…
Let $k$ be an infinite field. Fix a Jordan nilpotent $n$ by $n$ matrix $B = J_P$ with entries in $k$ and associated Jordan type $P$. Let $Q(P)$ be the Jordan type of a generic nilpotent matrix commuting with $B$. In this paper, we use the combinatorics of a poset associated to the partition $P$, to give an explicit formula for the smallest part of $Q(P)$, which is independent of the characteristic of $k$. This, in particular, leads to a complete description of $Q(P)$ when it has at most three parts.
Let $(\mathcal{P},\leqslant)$ be a finite poset. Define the numbers $a_1,a_2,\ldots$ (respectively, $c_1,c_2,\ldots$) so that $a_1+\ldots+a_k$ (respectively, $c_1+\ldots+c_k$) is the maximal number of elements of $\mathcal{P}$ which may be covered by $k$ antichains (respectively, $k$ chains.) Then the number $e(\mathcal{P})$ of linear extensions of poset $\mathcal{P}$ is not less than $\prod a_i!$ and not more than $n!/\prod c_i!$. A corollary: if $\mathcal{P}$ is partitioned onto disjoint antichains of size $b_1,b_2, \ldots$, then $e(\mathcal{P})\geqslant \prod b_i!$.
We study a min-max relation conjectured by Saks and West: For any two posets $P$ and $Q$ the size of a maximum semiantichain and the size of a minimum unichain covering in the product $P\times Q$ are equal. For positive we state conditions on $P$ and $Q$ that imply the min-max relation. Based on these conditions we identify some new families of posets where the conjecture holds and get easy proofs for several instances where the conjecture had been verified before. However, we also have examples showing that in general the min-max relation is false, i.e., we disprove the Saks-West conjecture.
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…
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 covering number of a nontrivial finite group $G$, denoted $σ(G)$, is the smallest number of proper subgroups of $G$ whose set-theoretic union equals $G$. In this article, we focus on a dual problem to that of covering numbers of groups, which involves maximal subgroups of finite groups. For a nontrivial finite group $G$, we define the intersection number of $G$, denoted $ι(G)$, to be the minimum number of maximal subgroups whose intersection equals the Frattini subgroup of $G$. We elucidate some basic properties of this invariant, and give an exact formula for $ι(G)$ when $G$ is a nontrivi…
Let alpha = (a,b,...) be a composition. Consider the associated poset F(alpha), called a fence, whose covering relations are x_1 < x_2 < ... < x_{a+1} > x_{a+2} > ... > x_{a+b+1} < x_{a+b+2} < ... . We study the associated distributive lattice L(alpha) consisting of all lower order ideals of F(alpha). These lattices are important in the theory of cluster algebras and their rank generating functions can be used to define q-analogues of rational numbers. In particular, we make progress on a recent conjecture of Morier-Genoud and Ovsienko that L(alpha) is rank unimodal. We show that if one of the…
Associated with any composition beta=(a,b,...) is a corresponding fence poset F(beta) whose covering relations are x_1 < x_2 < ... < x_{a+1} > x_{a+2} > ... > x_{a+b+1} < x_{a+b+2} < ... The distributive lattice L(beta) of all lower order ideals of F(beta) is important in the theory of cluster algebras. In addition, its rank generating function r(q;beta) is used to define q-analogues of rational numbers. Oguz and Ravichandran recently showed that its coefficients satisfy an interlacing condition, proving a conjecture of McConville, Smyth and Sagan, which in turn implies a previous conjecture o…
The k-Young lattice Y^k is a weak subposet of the Young lattice containing partitions whose first part is bounded by an integer k>0. The Y^k poset was introduced in connection with generalized Schur functions and later shown to be isomorphic to the weak order on the quotient of the affine symmetric group by a maximal parabolic subgroup. We prove a number of properties for $Y^k$ including that the covering relation is preserved when elements are translated by rectangular partitions with hook-length $k$. We highlight the order ideal generated by an $m\times n$ rectangular shape. This order ideal…
