Graded Poset Research Articles

The Endomorphism Conjecture for Graded Posets with Whitney Numbers at most 4

We prove the endomorphism conjecture for graded posets with largest Whitney number at most 4.

Some Computational Study of the Root Distribution of Ehrhart Polynomials

In this paper, we investigate the root distribution of the Ehrhart polynomials of lattice polytopes. When the lattice polytope is reflexive, the roots of the Ehrhart polynomial distribute symmetrically with respect to the line [Formula: see text]. A special case of this distribution is when all the roots lie on this line. Our main concern is to find out which reflexive polytopes satisfy this special condition. Such lattice polytopes are called CL-polytopes. Another special case opposite to this is when all the roots are real. Such lattice polytopes are called real polytopes. The first topic of this paper is the Ehrhart polynomials of equatorial spheres, which are related to graded posets. We discuss the CL-ness of the equatorial Ehrhart polynomials for complete graded posets and zig-zag posets. The second topic is to investigate the relation of the root distribution of the Ehrhart polynomials of a reflexive polytope [Formula: see text] and its dual [Formula: see text]. We discuss the CL-ness and realness of [Formula: see text] and [Formula: see text] in pair, of dimensions up to 4. Throughout this paper, we investigate these problems by computer calculation.

The birational Lalanne–Kreweras involution

The Lalanne–Kreweras involution is an involution on the set of Dyck paths which combinatorially exhibits the symmetry of the number of valleys and major index statistics. We define piecewise-linear and birational extensions of the Lalanne–Kreweras involution. Actually, we show that the Lalanne–Kreweras involution is a special case of a more general operator, called rowvacuation, which acts on the antichains of any graded poset. Rowvacuation, like the closely related and more studied rowmotion operator, is a composition of toggles. We obtain the piecewise-linear and birational lifts of the Lalanne–Kreweras involution by using the piecewise-linear and birational toggles of Einstein and Propp. We show that the symmetry properties of the Lalanne–Kreweras involution extend to these piecewise-linear and birational lifts.

Special idempotents and projections

We define, for any special matching of a finite graded poset, an idempotent, regressive and order preserving function. We consider the monoid generated by such functions. We call special idempotent any idempotent element of this monoid. They are interval retracts. Some of them realize a kind of parabolic map and are called special projections. We prove that, in Eulerian posets, the image of a special projection, and its complement, are graded induced subposets. In a finite Coxeter group, all projections on right and left parabolic quotients are special projections, and some projections on double quotients too. We extend our results to special partial matchings.

Symmetric Chain Decompositions of Products of Posets with Long Chains

We ask if there exists a symmetric chain decomposition of the cuboid $Q_k \times n$ such that no chain is taut, i.e. no chain has a subchain of the form $(a_1,\ldots, a_k,0)\prec \cdots\prec (a_1,\ldots,a_k,n-1)$. In this paper, we show this is true precisely when $k \ge 5$ and $n\ge 3$. This question arises naturally when considering products of symmetric chain decompositions which induce orthogonal chain decompositions — the existence of the decompositions provided in this paper unexpectedly resolves the most difficult case of previous work by the second author on almost orthogonal symmetric chain decompositions (2017), making progress on a conjecture of Shearer and Kleitman (1979). In general, we show that for a finite graded poset $P$, there exists a canonical bijection between symmetric chain decompositions of $P \times m$ and $P \times n$ for $m, n\ge rk(P) + 1$, that preserves the existence of taut chains. If $P$ has a unique maximal and minimal element, then we also produce a canonical $(rk(P) +1)$ to $1$ surjection from symmetric chain decompositions of $P \times (rk(P) + 1)$ to symmetric chain decompositions of $P \times rk(P)$ which sends decompositions with taut chains to decompositions with taut chains.

Families of Subsets Without a Given Poset in Double Chains and Boolean Lattices

Given a finite poset P, the intensively studied quantity La(n, P) denotes the largest size of a family of subsets of [n] not containing P as a weak subposet. Burcsi and Nagy (J. Graph Theory Appl. 1, 40–49 2013) proposed a double-chain method to get an upper bound \({\mathrm La}(n,P)\le \frac {1}{2}(|P|+h-2)\left (\begin {array}{c}n \\ \lfloor {n/2}\rfloor \end {array}\right )\) for any finite poset P of height h. This paper elaborates their double-chain method to obtain a new upper bound $${\mathrm La}(n,P)\le \left( \frac{|P|+h-\alpha(G_{P})-2}{2}\right)\left( \begin{array}{c}n \\ \lfloor{\frac{n}{2}}\rfloor\end{array}\right) $$ for any graded poset P, where α(G P ) denotes the independence number of an auxiliary graph defined in terms of P. This result enables us to find more posets which verify an important conjecture by Griggs and Lu (J. Comb. Theory (Ser. A) 119, 310–322, 2012).

Markov Chains on Graded Posets

We consider two types of discrete-time Markov chains where the state space is a graded poset and the transitions are taken along the covering relations in the poset. The first type of Markov chain goes only in one direction, either up or down in the poset (an up chain or down chain). The second type toggles between two adjacent rank levels (an up-and-down chain). We introduce two compatibility concepts between the up-directed transition probabilities (an up rule) and the down-directed (a down rule), and we relate these to compatibility between up-and-down chains. This framework is used to prove a conjecture about a limit shape for a process on Young’s lattice. Finally, we settle the questions whether the reverse of an up chain is a down chain for some down rule and whether there exists an up or down chain at all if the rank function is not bounded.

Peckness of Edge Posets

For any graded poset P, we define a new graded poset, 𝓔(P), whose elements are the edges in the Hasse diagram of P. For any group G acting on the boolean algebra Bn in a rank-preserving fashion we conjecture that 𝓔(Bn/G) is Peck. We prove that the conjecture holds for “common cover transitive” actions. We give some infinite families of common cover transitive actions and show that the common cover transitive actions are closed under direct and semidirect products.

Further Properties and Applications of Koszul Pairs

Koszul pairs were introduced in [arXiv:1011.4243] as an instrument for the study of Koszul rings. In this paper, we continue the enquiry of such pairs, focusing on the description of the second component, as a follow-up of the study in [arXiv:1605.05458]. As such, we introduce Koszul corings and prove several equivalent characterizations for them. As applications, in the case of locally finite $R$-rings, we show that a graded $R$-ring is Koszul if and only if its left (or right) graded dual coring is Koszul. Finally, for finite graded posets, we obtain that the respective incidence ring is Koszul if and only if the incidence coring is so.

On Koszulity of finite graded posets

In this paper, we continue our research on Koszul rings, started in [P. Jara, J. López-Peña and D. Ştefan, Koszul pairs and applications, to appear in J. Noncommut. Geom., http://arxiv.org/pdf/1011.4243.pdf ]. In Theorem 1.9, we prove in a unifying way several equivalent descriptions of Koszul rings, some of which being well known in the literature. Most of them are stated in terms of coring theoretical properties of [Formula: see text]. As an application of these characterizations, we investigate the Koszulity of the incidence rings for finite graded posets, see Theorems 2.8 and 2.9. Based on these results, we describe an algorithm to produce new classes of Koszul posets (i.e. graded posets, whose incidence rings are Koszul). Specific examples of Koszul posets are included.

Normalized Matching Property of Posets Generated by Orbits of Subspaces Under Finite Symplectic Groups

Let be the 2ν-dimensional symplectic space over a finite field 𝔽 q , and let ℳ be a given nontrivial orbit of subspaces in under the symplectic group Sp 2ν(𝔽 q ). Denote by 𝒫 the set of subspaces which are intersections of subspaces in ℳ. By ordering 𝒫 by inclusion, 𝒫 is a finite graded poset. In this article we show that 𝒫 has the normalized matching (NM) property, which implies that 𝒫 has the strong Sperner property and the Lubell-Yamamoto-Meschalkin (LYM) property.

An equivalence relation on the symmetric group and multiplicity-free flag $h$-vectors

We consider the equivalence relation ∼ on the symmetric group Sn generated by the interchange of any two adjacent elements ai and ai+1 of w = a1 · · · an ∈ Sn such that |ai − ai+1| = 1. We count the number of equivalence classes and the sizes of the equivalence classes. The results are generalized to permutations of multisets. In the original problem, the equivalence class containing the identity permutation is the set of linear extensions of a certain poset. Further investigation yields a characterization of all finite graded posets whose flag h-vector takes on only the values 0,±1.

Centralizer of an idempotent in a reductive monoid

Abstract Let M be a reductive monoid. If e is an idempotent in M, we prove that the centralizer M(e) of e in M is a regular monoid with a finite graded poset of 𝒥-classes. We compute this poset explicitly when M is of canonical or dual canonical type and e is the relevant pivotal idempotent.

The Rees product of posets

We determine how the flag f-vector of any graded poset changes under the Rees product with the chain, and more generally, any t-ary tree. As a corollary, the M\"obius function of the Rees product of any graded poset with the chain, and more generally, the t-ary tree, is exactly the same as the Rees product of its dual with the chain, respectively, t-ary chain. We then study enumerative and homological properties of the Rees product of the cubical lattice with the chain. We give a bijective proof that the M\"obius function of this poset can be expressed as n times a signed derangement number. From this we derive a new bijective proof of Jonsson's result that the M\"obius function of the Rees product of the Boolean algebra with the chain is given by a derangement number. Using poset homology techniques we find an explicit basis for the reduced homology and determine a representation for the reduced homology of the order complex of the Rees product of the cubical lattice with the chain over the symmetric group.

Weighted Derivations and the cd-Index

Weighted derivations W 1 and W 2 allowed R. Ehrenborg and M. Readdy (Discrete Comput. Geom. 21:389–403, [1999]) to give a recursive description of the cd-indices of the lattices of the regions of the arrangements \(\mathcal{A}_{n}\) and ℬ n . In part motivated by this, we describe a new basis for the subspace spanned by ab-indices of all simplicial graded posets and determine the action of certain linear maps (associated with weighted derivations W k ) on this basis. Extending the “pyramid” and “prism” operations, we define operations Σ k on graded posets and show that relations between the ab-indices (or cd-indices, for an Eulerian poset P) of barycentric subdivisions of Σ k (P) and P can be described by using the linear maps associated by weighted derivations W k . Finally, in response to a question from Ehrenborg and Readdy (Discrete Comput. Geom. 21:389–403, [1999]), we obtain the formulae that express the cd-index of the lattice \(R_{\mathcal{D}_{n}}\) of regions of the arrangement \(\mathcal{D}_{n}\) in terms of the cd-indices of \(R_{\mathcal{B}_{n}}\) and \(R_{\mathcal{A}_{n-2}}\) .

Enumeration of totally positive Grassmann cells

Postnikov (Webs in totally positive Grassmann cells, in preparation) has given a combinatorially explicit cell decomposition of the totally nonnegative part of a Grassmannian, denoted Gr k, n +, and showed that this set of cells is isomorphic as a graded poset to many other interesting graded posets. The main result of our work is an explicit generating function which enumerates the cells in Gr k, n + according to their dimension. As a corollary, we give a new proof that the Euler characteristic of Gr k, n + is 1. Additionally, we use our result to produce a new q-analog of the Eulerian numbers, which interpolates between the Eulerian numbers, the Narayana numbers, and the binomial coefficients.

An upper bound for the size of the largest antichain in the poset of partitions of an integer

Let Pi n be the poset of partitions of an integer n, ordered by refinement. Let b( Pi n ) be the largest size of a level and d( Pi n ) be the largest size of an antichain of Pi n . We prove that d(Pi n) b(Pi n) ⩽ e+ o(1) as n→∞. The denominator is determined asymptotically. In addition, we show that the incidence matrices in the lower half of Pi n have full rank, and we prove a tight upper bound for the ratio from above if Pi n is replaced by any graded poset P.

Hopf Algebras and Edge-Labeled Posets

Given a finite graded poset with labeled Hasse diagram, we construct a quasi-symmetric generating function for chains whose labels have fixed descents. This is a common generalization of a generating function for the flag f-vector defined by Ehrenborg and of a symmetric function associated with certain edge-labeled posets that arose in the theory of Schubert polynomials. We show that this construction gives a Hopf morphism from an incidence Hopf algebra of edge-labeled posets to the Hopf algebra of quasi-symmetric functions.

Flag-symmetric and Locally Rank-symmetric Partially Ordered Sets

For every finite graded poset $P$ with $\hat{0}$ and $\hat{1}$ we associate a certain formal power series $F_P(x) = F_P(x_1,x_2,\dots)$ which encodes the flag $f$-vector (or flag $h$-vector) of $P$. A relative version $F_{P/\Gamma}$ is also defined, where $\Gamma$ is a subcomplex of the order complex of $P$. We are interested in the situation where $F_P$ or $F_{P/\Gamma}$ is a symmetric function of $x_1,x_2,\dots$. When $F_P$ or $F_{P/\Gamma}$ is symmetric we consider its expansion in terms of various symmetric function bases, especially the Schur functions. For a class of lattices called $q$-primary lattices the Schur function coefficients are just values of Kostka polynomials at the prime power $q$, thus giving in effect a simple new definition of Kostka polynomials in terms of symmetric functions. We extend the theory of lexicographically shellable posets to the relative case in order to show that some examples $(P,\Gamma)$ are relative Cohen-Macaulay complexes. Some connections with the representation theory of the symmetric group and its Hecke algebra are also discussed.

A shellable poset that is not lexicographically shellable

It is known that a lexicographically shellable poset is shellable, and it has been asked whether the two concepts are equivalent. We provide a counterexample, a shellable graded poset that is not lexicographically shellable.