In this article, we study polymatroids that are representable by means of linear restricted rank-metric codes, namely, by subspaces of the space of alternating, symmetric, or Hermitian square matrices endowed with the rank metric. More precisely, we characterize the rank function defining these polymatroids and establish sufficient conditions on the relevant parameters under which it is fully determined. We show that there are several differences in compared to the behaviour of $q$-polymatroids of unrestricted matrix codes.
Institute of Mathematics for Industry (IMI), Kyushu University（九州大学产业数学研究所）；Graduate School of Science and Technology, Kumamoto University（熊本大学科学技术研究生院）；Faculty of Advanced Science and Technology, Kumamoto University（熊本大学先进理工学部）
It is well known that linear rank-metric codes give rise to q-polymatroids. Analogously to matroid theory one may ask whether a given q-polymatroid is representable by a rank-metric code. We provide an answer by presenting an example of a q-matroid that is not representable by any linear rank-metric code and, via a relation to paving matroids, provide examples of various q-matroids that are not representable by F_{q^m}-linear rank-metric codes. We then go on and introduce deletion and contraction for q-polymatroids and show that they are mutually dual and correspond to puncturing and shortenin…
This paper contributes to the study of rank-metric codes from an algebraic and combinatorial point of view. We introduce $q$-polymatroids, the $q$-analogue of polymatroids, and develop their basic properties. We associate a pair of q-polymatroids to a rank-metric codes and show that several invariants and structural properties of the code, such as generalized weights, the property of being MRD or an optimal anticode, and duality, are captured by the associated combinatorial object.
We consider the notion of a $(q,m)$-polymatroid, due to Shiromoto, and the more general notion of $(q,m)$-demi-polymatroid, and show how generalized weights can be defined for them. Further, we establish a duality for these weights analogous to Wei duality for generalized Hamming weights of linear codes. The corresponding results of Ravagnani for Delsarte rank metric codes, and Martinez-Penas and Matsumoto for relative generalized rank weights are derived as a consequence.
This paper is devoted to the study of independent spaces of q-polymatroids. With the aid of an auxiliary q-matroid it is shown that the collection of independent spaces satisfies the same properties as for q-matroids. However, in contrast to q-matroids, the rank value of an independent space does not agree with its dimension. Nonetheless, the rank values of the independent spaces fully determine the q-polymatroid, and this fact can be exploited to derive a cryptomorphism of q-polymatroids. Finally, the notions of minimal spanning spaces, maximally strongly independent spaces, and bases will be…
The rank of a skew partition $λ/μ$, denoted $rank(λ/μ)$, is the smallest number $r$ such that $λ/μ$ is a disjoint union of $r$ border strips. Let $s_{λ/μ}(1^t)$ denote the skew Schur function $s_{λ/μ}$ evaluated at $x_1=...=x_t=1, x_i=0$ for $i>t$. The zrank of $λ/μ$, denoted $zrank(λ/μ)$, is the exponent of the largest power of $t$ dividing $s_{λ/μ}(1^t)$. Stanley conjectured that $rank(λ/μ)=zrank(λ/μ)$. We show the equivalence between the validity of the zrank conjecture and the nonsingularity of restricted Cauchy matrices. In support of Stanley's conjecture we give affirmative answers for s…
In this paper, we provide algorithms to rank, unrank, and randomly generate certain degree-restricted classes of Cayley trees. Specifically, we consider classes of trees that have a given degree sequence or a given multiset of degrees. If the underlying set of trees have n vertices, then the largest ranks involved in each case are of order n! so that it takes O(nlog(n)) bits just to write down the ranks. Our ranking and unranking algorithms for these degree-restricted classes are as efficient as can be expected since we show that they require O(n^2log(n)) bit operations if the underlying trees…
We derive an asymptotic formula for $A(n,j,r)$ the number of integer partitions of $n$ into at most $j$ parts each part $\le r$. We assume $j$ and $r$ are near their mean values. We also investigate the second largest part, the number of parts $\ge 2$, etc. We show that the fraction of the partitions of an even integer $n$ that are graphical, ie. whose parts form the degree sequence of a simple graph, is $O(\ln^{-1/2} n)$. Probabilistic results are used in our discussion of graphical partitions. The George Szekeres circle method is essential for our asymptotic results on partitions. We determi…
Let $p$ be a prime and let $S$ be a non-empty subset of $\mathbb{F}_p$. Generalizing a result of Green and Tao on the equidistribution of high-rank polynomials over finite fields, we show that if $P: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ is a polynomial and its restriction to $S^n$ does not take each value with approximately the same frequency, then there exists a polynomial $P_0: \mathbb{F}_p^n \rightarrow \mathbb{F}_p$ that vanishes on $S^n$, such that the polynomial $P-P_0$ has bounded rank. Our argument uses two black boxes: that a tensor with high partition rank has high analytic rank …
We study the functions that count matrices of given rank over a finite field with specified positions equal to zero. We show that these matrices are $q$-analogues of permutations with certain restricted values. We obtain a simple closed formula for the number of invertible matrices with zero diagonal, a $q$-analogue of derangements, and a curious relationship between invertible skew-symmetric matrices and invertible symmetric matrices with zero diagonal. In addition, we provide recursions to enumerate matrices and symmetric matrices with zero diagonal by rank, and we frame some of our results …
Let $(G,+)$ be a finite abelian group. Then, $\so(G)$ and $η(G)$ denote the smallest integer $\ell$ such that each sequence over $G$ of length at least $\ell$ has a subsequence whose terms sum to $0$ and whose length is equal to and at most, resp., the exponent of the group. For groups of rank two, we study the inverse problems associated to these constants, i.e., we investigate the structure of sequences of length $\so(G)-1$ and $η(G)-1$ that do not have such a subsequence. On the one hand, we show that the structure of these sequences is in general richer than expected. On the other hand, as…
Matroid intersection is one of the most powerful frameworks of matroid theory that generalizes various problems in combinatorial optimization. Edmonds' fundamental theorem provides a min-max characterization for the unweighted setting, while Frank's weight-splitting theorem provides one for the weighted case. Several efficient algorithms were developed for these problems, all relying on the usage of one of the conventional oracles for both matroids. In the present paper, we consider the tractability of the matroid intersection problem under restricted oracles. In particular, we focus on the ra…
Four constructions for Ferrers diagram rank-metric (FDRM) codes are presented. The first one makes use of a characterization on generator matrices of a class of systematic maximum rank distance codes. By introducing restricted Gabidulin codes, the second construction is presented, which unifies many known constructions for FDRM codes. The third and fourth constructions are based on two different ways to represent elements of a finite field $\mathbb F_{q^m}$ (vector representation and matrix representation). Each of these constructions produces optimal codes with different diagrams and paramete…
Solomon showed that the Poincaré polynomial of a Coxeter group $W$ satisfies a product decomposition depending on the exponents of $W$. This polynomial coincides with the rank-generating function of the poset of regions of the underlying Coxeter arrangement. In this note we determine all instances when the analogous factorization property of the rank-generating function of the poset of regions holds for a restriction of a Coxeter arrangement. It turns out that this is always the case with the exception of some instances in type $E_8$.
We obtain a finite analogue of a recent generalization of an identity in Ramanujan's Notebooks. Differentiating it with respect to one of the parameters leads to a result whose limiting case gives a finite analogue of Andrews' famous identity for $\textup{spt}(n)$. The latter motivates us to extend the theory of the restricted partition function $p(n, N)$, namely, the number of partitions of $n$ with largest parts less than or equal to $N$, by obtaining the finite analogues of rank and crank for vector partitions as well as of the rank and crank moments. As an application of the identity for o…
Interaction between skew-representability, tensor products, extension properties, and rank inequalities
ELTE Eötvös Loránd University（罗兰大学）；HUN-REN Alfréd Rényi Institute of Mathematics（匈牙利科学院阿尔弗雷德·雷尼数学研究所）；Universitat Politècnica de Catalunya（加泰罗尼亚理工大学）；London School of Economics and Political Science（伦敦政治经济学院）
We classify primitive, rank 1, omega-categorical structures having polynomially many types over finite sets. For a fixed number of 4-types, we show that there are only finitely many such structures and that all are built out of finitely many linear orders interacting in a restricted number of ways. As an example of application, we deduce the classification of primitive structures homogeneous in a language consisting of n linear orders as well as all reducts of such structures.
