Journal of Combinatorial Theory Series A最新文献

Let p be a prime and let G be a finite group which is generated by the set $G_p$ of its p-elements. We show that if G is solvable and not a p-group, then the minimal number ${\sigma }_p\left(G\right)$ of proper subgroups of G whose union contains $G_p$ is equal to 1 less than the minimal number of proper subgroups of G whose union is G. For p-solvable groups G, we always have ${\sigma }_p\left(G\right)\ge p+1$. We study the case of alternating and symmetric groups G in detail.

In 2012, Andrews and Merca investigated the truncated version of the Euler pentagonal number theorem. Their work has opened up a new study on truncated theta series and has inspired several mathematicians to work on the topic. In 2019, Merca studied the Rogers-Ramanujan functions and posed three groups of conjectures on truncated series involving the Rogers-Ramanujan functions. In this paper, we present a uniform method to prove the three groups of conjectures given by Merca based on a result due to Pólya and Szegö.

We study the proportion of metric matroids whose Jacobians have nontrivial p-torsion. We establish a correspondence between these Jacobians and the $F_p$-rational points on configuration hypersurfaces, thereby relating their proportions. By counting points over finite fields, we prove that the proportion of these Jacobians is asymptotically equivalent to $1/p$.

Constant weight codes (CWCs) and constant composition codes (CCCs) are two important classes of codes that have been studied extensively in both combinatorics and coding theory for nearly sixty years. In this paper we show that for all fixed odd distances, there exist near-optimal CWCs and CCCs asymptotically achieving the classic Johnson-type upper bounds.

Let $A_q(n,d,w)$ denote the maximum size of q-ary CWCs of length n with constant weight w and minimum distance d. One of our main results shows that for all fixed $q,w$ and odd d, one has ${\lim }_{n\to \infty }\frac{A_q(n,d,w)}{\left(\begin{array}{c}n\\ t\end{array}\right)}=\frac{{(q-1)}^t}{\left(\begin{array}{c}w\\ t\end{array}\right)}$, where $t=\frac{2w-d+1}{2}$. This implies the existence of near-optimal generalized Steiner systems originally introduced by Etzion, and can be viewed as a counterpart of a celebrated result of Rödl on the existence of near-optimal Steiner systems. Note that prior to our work, very little is known about $A_q(n,d,w)$ for $q\ge 3$. A similar result is proved for the maximum size of CCCs.

We provide different proofs for our two main results, based on two strengthenings of the well-known Frankl-Rödl-Pippenger theorem on the existence of near-optimal matchings in hypergraphs: the first proof follows by Kahn's linear programming variation of the above theorem, and the second follows by the recent independent work of Delcourt-Postle, and Glock-Joos-Kim-Kühn-Lichev on the existence of near-optimal matchings avoiding certain forbidden configurations.

We also present several intriguing open questions for future research.

For a positive integer m, a group G is said to admit a tournament m-semiregular representation (TmSR for short) if there exists a tournament Γ such that the automorphism group of Γ is isomorphic to G and acts semiregularly on the vertex set of Γ with m orbits. It is easy to see that every finite group of even order does not admit a TmSR for any positive integer m. The T1SR is the well-known tournament regular representation (TRR for short). In 1970s, Babai and Imrich proved that every finite group of odd order admits a TRR except for $Z_3^2$, and every group (finite or infinite) without element of order 2 having an independent generating set admits a T2SR in (1979) [3]. Later, Godsil correct the result by showing that the only finite groups of odd order without a TRR are $Z_3^2$ and $Z_3^3$ by a probabilistic approach in (1986) [11]. In this note, it is shown that every finite group of odd order has a TmSR for every $m\ge 2$.

The exact value of the separating Noether number of an arbitrary finite abelian group of rank two is determined. This is done by a detailed study of the monoid of zero-sum sequences over the group.

The tableau reconstruction problem, posed by Monks (2009), asks the following. Starting with a standard Young tableau T, a 1-minor of T is a tableau obtained by first deleting any cell of T, and then performing jeu de taquin slides to fill the resulting gap. This can be iterated to arrive at the set of k-minors of T. The problem is this: given k, what are the values of n such that every tableau of size n can be reconstructed from its set of k-minors? For $k=1$, the problem was recently solved by Cain and Lehtonen. In this paper, we solve the problem for $k=2$, proving the sharp lower bound $n\ge 8$. In the case of multisets of k-minors, we also give a lower bound for arbitrary k, as a first step toward a sharp bound in the general multiset case.

In this paper, we establish some general expansion formulas for q-series. Three of Liu's identities motivate us to search and find such type of formulas. These expansion formulas include as special cases or limiting cases many q-identities including the q-Gauss summation formula, the q-Pfaff-Saalschütz summation formula, three of Jackson's transformation formulas and Sears' terminating ${}_4\varphi _3$ transformation formula. As applications, we provide a new proof of the orthogonality relation for continuous dual q-Hahn polynomials, establish some generating functions for special values of the Dirichlet L-functions and the Hurwitz zeta functions, give extensions of three of Liu's identities, establish an extension of Dilcher's identity, and deduce various double Rogers-Ramanujan type identities.

Let $r\text{des}$ and $r\text{exc}$ denote the permutation statistics r-descent number and r-excedance number, respectively. We prove that the pairs of permutation statistics $(r\text{des},r\text{maj})$ and $(r\text{exc},r\text{den})$ are equidistributed, where $r\text{maj}$ denotes the r-major index defined by Don Rawlings and $r\text{den}$ denotes the r-Denert's statistic defined by Guo-Niu Han. When $r=1$, this result reduces to the equidistribution of $(\text{des},\text{maj})$ and $(\text{exc},\text{den})$, which was conjectured by Denert in 1990 and proved that same year by Foata and Zeilberger. We call a pair of permutation statistics that is equidistributed with $(r\text{des},r\text{maj})$ and $(r\text{exc},r\text{den})$ an r-Euler-Mahonian statistic, which reduces to the classical Euler-Mahonian statistic when $r=1$.

We then introduce the notions of r-level descent number, r-level excedance number, r-level major index, and r-level Denert's statistic, denoted by ${\text{des}}_r,{\text{exc}}_r,{\text{maj}}_r$, and ${\text{den}}_r$, respectively. We prove that $({\text{des}}_r,{\text{maj}}_r)$ is r-Euler-Mahonian and conjecture that $({\text{exc}}_r,{\text{den}}_r)$ is r-Euler-Mahonian. Furthermore, we give an extension of the above result and conjecture.