The Narayana numbers are $N(n,k) = {1 \over n}{n \choose k}{n \choose {k+1}}$. There are several natural statistics on Dyck paths with a distribution given by N(n,k). We show the equidistribution of Narayana statistics by computing the flag h-vector of $J({\bf 2} \times {\bf n})$ in different ways. In the process we discover new Narayana statistics and provide co-statistics for which the Narayana statistics in question have a distribution given by Fürlinger and Hofbauers q-Narayana numbers. We also interpret the h-vector in terms of semi-standard Young tableaux, which enables us to express the q-Narayana numbers in terms of Schur functions.
We consider Gaussian elliptic random matrices $X$ of a size $N \times N$ with parameter $ρ$, i.e., matrices whose pairs of entries $(X_{ij}, X_{ji})$ are mutually independent Gaussian vectors, $E X_{ij} = 0$, $E X^2_{ij} = 1$ and $E X_{ij} X_{ji} = ρ$. We are interested in the asymptotic distribution of eigenvalues of the matrix $W =\frac{1}{N^2} X^2 X^{*2}$. We have shown that this distribution is defined by its moments and we provide a recurrent relation for these moments. We have proven that the (symmetrized) asymptotic distribution is determined by its free cumulants, which are Narayana po…
Motzkin numbers have been widely studied since they count many different combinatorial objects. In this paper we present a new appearance of this remarkable sequence in the network coding setting through a particular case of multishot codes called flag codes. A flag code is a set of sequences of nested subspaces (flags) of a vector space over the finite field $\mathbb{F}_q$. If the list of dimensions is $(1, \dots, n-1)$, we speak about a full flag code. The flag distance is defined as the sum of the respective subspace distances and can be represented by means of the so-called distance vector…
We study the equivariant flag $f$-vector and equivariant flag $h$-vector of a balanced relative simplicial complex with respect to a group action. When the complex satisfies Serre's condition $(S_{\ell}),$ we show that the equivariant flag $h$-vector, the equivariant $h$-vector, and the equivariant $f$-vector satisfy several inequalities. We apply these results to the study of $P$-partitions of double posets, and weak colorings of mixed graphs.
We prove several relations on the $f$-vectors and Betti numbers of flag complexes. For every flag complex $Δ$, we show that there exists a balanced complex with the same $f$-vector as $Δ$, and whose top-dimensional Betti number is at least that of $Δ$, thereby extending a theorem of Frohmader by additionally taking homology into consideration. We obtain upper bounds on the top-dimensional Betti number of $Δ$ in terms of its face numbers. We also give a quantitative refinement of a theorem of Meshulam by establishing lower bounds on the $f$-vector of $Δ$, in terms of the top-dimensional Betti n…
We present a first example of a flag vector of a polyhedral sphere that is not the flag vector of any polytope. Namely, there is a unique 3-sphere with the parameters $(f_0,f_1,f_2,f_3;f_{02})=(12,40,40,12;120)$, but this sphere is not realizable by a convex 4-polytope. The 3-sphere, which is 2-simple and 2-simplicial, was found by Werner (2009); we present results of a computer enumeration which imply that the sphere with these parameters is unique. We prove that it is non-polytopal in two ways: First, we show that it has no oriented matroid, and thus it is not realizable; this proof was foun…
We prove several identities expressing polynomials counting permutations by various descent statistics in terms of Eulerian polynomials, extending results of Stembridge, Petersen, and Brändén. Additionally, we find $q$-exponential generating functions for $q$-analogues of these descent statistic polynomials that also keep track of the inversion number or inverse major index. We also present identities relating several of these descent statistic polynomials to refinements of type B Eulerian polynomials and flag descent polynomials by the number of negative letters of a signed permutation. Our m…
A colored complex of type $\mathbf{a} = (a_1, \dots, a_n)$ is a simplicial complex $Δ$ on a vertex set $V$, together with an ordered partition $(V_1, \dots, V_n)$ of $V$, such that every face $F$ of $Δ$ satisfies $|F \cap V_i| \leq a_i$. For each $\mathbf{b} = (b_1, \dots, b_n) \leq \mathbf{a}$, let $f_{\mathbf{b}}$ be the number of faces $F$ of $Δ$ such that $|F \cap V_i| = b_i$. The array of integers $\{f_{\mathbf{b}}\}_{\mathbf{b} \leq \mathbf{a}}$ is called the fine $f$-vector of $Δ$, and it is a refinement of the $f$-vector of $Δ$. In this paper, we generalize the notion of Macaulay repre…
We present some enumerative and structural results for flag homology spheres. For a flag homology sphere $Δ$, we show that its $γ$-vector $γ^Δ=(1,γ_1,γ_2,\ldots)$ satisfies: \begin{align*} γ_j=0,\text{ for all } j>γ_1, \quad γ_2\leq\binom{γ_1}{2}, \quad γ_{γ_1}\in\{0,1\}, \quad \text{ and }γ_{γ_1-1}\in\{0,1,2,γ_1\}, \end{align*} supporting a conjecture of Nevo and Petersen. Further we characterize the possible structures for $Δ$ in extremal cases. As an application, the techniques used produce infinitely many $f$-vectors of flag balanced simplicial complexes that are not $γ$-vectors of flag ho…
We determine the chromatic number of some graphs of flags in buildings of type $A_4$, namely of the Kneser graphs of flags of type $\{2,4\}$ in the vector spaces $GF(q)^5$ for $q\geq3$, and of the Kneser graph of flags of type $\{2,3\}$ in the vector spaces $GF(q)^5$ for large $q$.
Given $\mathbb{F}_q$ the finite field with $q$ elements and an integer $n\geq 2$, a flag is a sequence of nested subspaces of $\mathbb{F}_q^n$ and a flag code is a nonempty set of flags. In this context, the distance between flags is the sum of the corresponding subspace distances. Hence, a given flag distance value might be obtained by many different combinations. To capture such a variability, in the paper at hand, we introduce the notion of distance vector as an algebraic object intrinsically associated to a flag code that encloses much more information than the distance parameter itself. O…
The $n^{th}$ partial flag incidence algebra of a poset $P$ is the set of functions from $P^n$ to some ring which are zero on non-partial flag vectors. These partial flag incidence algebras for $n>2$ are not commutative, not unitary, and not associative. However, partial flag incidence algebras contain generalized zeta, delta, and Möbius functions which are finer and more delicate invariants than their classical analogues. We also study some generalized characteristic polynomials of posets which are not evaluations of Tutte polynomials and compute them for Boolean lattices. Motivation for this …
In this paper, we study flag codes on the vector space $\mathbb{F}_q^n$, being $q$ a prime power and $\mathbb{F}_q$ the finite field of $q$ elements. More precisely, we focus on flag codes that attain the maximum possible distance (optimum distance flag codes) and can be obtained from a spread of $\mathbb{F}_q^n$. We characterize the set of admissible type vectors for this family of flag codes and also provide a construction of them based on well-known results about perfect matchings in graphs. This construction attains both the maximum distance for its type vector and the largest possible car…
Grünbaum, Barnette, and Reay in 1974 completed the characterization of the pairs $(f_i,f_j)$ of face numbers of $4$-dimensional polytopes. Here we obtain a complete characterization of the pairs of flag numbers $(f_0,f_{03})$ for $4$-polytopes. Furthermore, we describe the pairs of face numbers $(f_0,f_{d-1})$ for $d$-polytopes; this description is complete for even $d\ge6$ except for finitely many exceptional pairs that are "small" in a well-defined sense, while for odd $d$ we show that there are also "large" exceptional pairs. Our proofs rely on the insight that "small" pairs need to be defi…
The flag vector contains all the face incidence data of a polytope, and in the poset setting, the chain enumerative data. It is a classical result due to Bayer and Klapper that for face lattices of polytopes, and more generally, Eulerian graded posets, the flag vector can be written as a cd-index, a non-commutative polynomial which removes all the linear redundancies among the flag vector entries. This result holds for regular CW complexes. We relax the regularity condition to show the cd-index exists for Whitney stratified manifolds by extending the notion of a graded poset to that of a quasi…
We determine the chromatic number of the Kneser graph qΓ_{7,{3,4}} of flags of vectorial type {3, 4} of a rank 7 vector space over the finite field GF(q) for large q and describe the colorings that attain the bound. This result relies heavily, not only on the independence number, but also on the structure of all large independent sets. Furthermore, our proof is more general in the following sense: it provides the chromatic number of the Kneser graphs qΓ_{2d+1,{d,d+1}} of flags of vectorial type {d, d+1} of a rank 2d+1 vector space over GF(q) for large q as long as the large independent sets of…
The main results of this paper are accessible with only basic linear algebra. Given an increasing sequence of dimensions, a flag in a vector space is an increasing sequence of subspaces with those dimensions. The set of all such flags (the flag manifold) can be projectively coordinatized using products of minors of a matrix. These products are indexed by tableaux on a Young diagram. A basis of "standard monomials" for the vector space generated by such projective coordinates over the entire flag manifold has long been known. A Schubert variety is a subset of flags specified by a permutation. L…
We use equivariant localization and divided difference operators to determine formulas for the torus-equivariant fundamental cohomology classes of $K$-orbit closures on the flag variety $G/B$ for various symmetric pairs $(G,K)$. In type $A$, we realize the closures of $K=GL(p,\C) \times GL(q,\C)$-orbits on $GL(p+q,\C)/B$ as universal degeneracy loci for a vector bundle over a variety which is equipped with a single flag of subbundles and which splits as a direct sum of subbundles of ranks $p$ and $q$. The precise description of such a degeneracy locus relies upon knowing a set-theoretic descri…
