Journal of Algebraic Combinatorics最新文献

Ariki and Mathas (Math Z 233(3):601–623, 2000) showed that the simple modules of the Ariki–Koike algebras (mathcal {H}_{mathbb {C},v;Q_1,ldots , Q_m}big (G(m, 1, n)big )) (when the parameters are roots of unity and (v ne 1)) are labeled by the so-called Kleshchev multipartitions. This together with Ariki’s categorification theorem enabled Ariki and Mathas to obtain the generating function for the number of Kleshchev multipartitions by making use of the Weyl–Kac character formula. In this paper, we revisit their generating function relation for the (v=-1) case. In particular, this (v=-1) scenario is of special interest as the corresponding Kleshchev multipartitions are closely tied with generalized Rogers–Ramanujan type partitions when (Q_1=cdots =Q_a=-1) and (Q_{a+1}=cdots =Q_m =1). Based on this connection, we provide an analytic proof of the result of Ariki and Mathas for the above choice of parameters. Our second objective is to investigate simple modules of the Ariki–Koike algebra in a fixed block, which are, as widely known, labeled by the Kleshchev multiparitions with a fixed partition residue statistic. This partition statistic is also studied in the works of Berkovich, Garvan, and Uncu. Employing their results, we provide two bivariate generating function identities for the (m=2) scenario.

A good range of problems on trees can be described by the following general setting: Given a bilinear map (*:mathbb {R}^dtimes mathbb {R}^drightarrow mathbb {R}^d) and a vector (sin mathbb {R}^d), we need to estimate the largest possible absolute value g(n) of an entry over all vectors obtained from applying (n-1) applications of (*) to n instances of s. When the coefficients of (*) are nonnegative and the entries of s are positive, the value g(n) is known to follow a growth rate (lambda =lim _{nrightarrow infty } root n of {g(n)}). In this article, we prove that for such (*) and s there exist nonnegative numbers (r,r') and positive numbers (a,a') so that for every n,

While proving the upper bound, we actually also provide another approach in proving the limit (lambda ) itself. The lower bound is proved by showing a certain form of submultiplicativity for g(n). Corollaries include a lower bound and an upper bound for (lambda ), which are followed by a good estimation of (lambda ) when we have the value of g(n) for an n large enough.

Recently, H. Dao and R. Nair gave a combinatorial description of simplicial complexes (Delta ) such that the squarefree reduction of the Stanley–Reisner ideal of (Delta ) has the WLP in degree 1 and characteristic zero. In this paper, we apply the connections between analytic spread of equigenerated monomial ideals, mixed multiplicities and birational monomial maps to give a sufficient and necessary condition for the squarefree reduction (A(Delta )) to satisfy the WLP in degree i and characteristic zero in terms of mixed multiplicities of monomial ideals that contain combinatorial information of (Delta ), we call them incidence ideals. As a consequence, we give an upper bound to the possible failures of the WLP of (A(Delta )) in degree i in positive characteristics in terms of mixed multiplicities. Moreover, we extend Dao and Nair’s criterion to arbitrary monomial ideals in positive odd characteristics.

We propose versions of higher Bruhat orders for types B and C. This is based on a theory of higher Bruhat orders of type A and their geometric interpretations (due to Manin–Shekhtman, Voevodskii–Kapranov, and Ziegler), and on our study of the so-called symmetric cubillages of cyclic zonotopes.

We prove that every regular graph of valency at least four whose automorphism group contains a nilpotent subgroup acting transitively on the vertex set admits a nowhere-zero 3-flow.

The original motivation of this paper was to find the context-free grammar for the joint distribution of peaks and valleys on permutations. Although such attempt was unsuccessful, we can obtain noncommutative symmetric function identities for the joint distributions of several descent-based statistics, including peaks, valleys and even/odd descents, on permutations via Zhuang’s generalized run theorem. Our results extend in a unified way several generating function formulas exist in the literature, including formulas of Carlitz and Scoville (Discrete Math 5:45–59, 1973; J Reine Angew Math 265:110–137, 1974), J. Combin. Theory Ser. A, 20: 336-356 (1976), Zhuang (Adv Appl Math 90:86–144, 2017), Pan and Zeng (Adv Appl Math 104:85–99, 2019; Discrete Math 346:113575, 2023). As applications of these generating function formulas, Wachs’ involution and Foata–Strehl action on permutations, we also investigate the signed counting of even and odd descents, and of descents and peaks, which provide two generalizations of Désarménien and Foata’s classical signed Eulerian identity.

In this paper we develop the theory of cyclic flats of q-matroids. We show that the cyclic flats, together with their ranks, uniquely determine a q-matroid and hence derive a new q-cryptomorphism. We introduce the notion of (mathbb {F}_{q^m})-independence of an (mathbb {F}_q)-subspace of (mathbb {F}_q^n) and we show that q-matroids generalize this concept, in the same way that matroids generalize the notion of linear independence of vectors over a given field.

We continue the study of intersection bodies of polytopes, focusing on the behavior of IP under translations of P. We introduce an affine hyperplane arrangement and show that the polynomials describing the boundary of (I(P+t)) can be extended to polynomials in variables (tin mathbb {R}^d) within each region of the arrangement. In dimension 2, we give a full characterization of those polygons such that their intersection body is convex. We give a partial characterization for general dimensions.