Journal of Combinatorial Theory Series A最新文献

Zagier provided eleven conjectural rank two examples for Nahm's problem. All of them have been proved in the literature except for the fifth example, and there is no q-series proof for the tenth example. We prove that the fifth and the tenth examples are in fact equivalent. Then we give a q-series proof for the fifth example, which confirms a recent conjecture of Wang. This also serves as the first q-series proof for the tenth example, whose explicit form was conjectured by Vlasenko and Zwegers in 2011 and whose modularity was proved by Cherednik and Feigin in 2013 via nilpotent double affine Hecke algebras.

A family $F$ of sets is union-closed if the union of any two sets from $F$ belongs to $F$. The union-closed sets conjecture states that if $F$ is a finite union-closed family of finite sets, then there is an element that belongs to at least half of the sets in $F$. The conjecture has several equivalent formulations in terms of other combinatorial structures such as lattices and graphs. In its whole generality the conjecture remains wide open, but it was verified for various important classes of lattices, such as lower semimodular lattices, and graphs, such as chordal bipartite graphs. In the present paper we develop a Boolean approach to the conjecture and verify it for several classes of Boolean functions, such as submodular functions and double Horn functions.

Recently Lazar and Wachs proved two new permutation models, called D-permutations and E-permutations, for Genocchi and median Genocchi numbers. In a follow-up, Eu et al. studied the even-odd descent permutations, which are in bijection with E-permutations. We generalize Eu et al.'s descent polynomials with eight statistics and obtain an explicit J-fraction formula for their ordinary generaing function. The J-fraction permits us to confirm two conjectures of Lazar-Wachs about cycles of D and E permutations and obtain a $(p,q)$-analogue of Eu et al.'s gamma-formula. Moreover, the $(p,q)$ gamma-coefficients have the same factorization flavor as the gamma-coefficients of Brändén's $(p,q)$-Eulerian polynomials.

Constructing bent functions by composing a Boolean function with a permutation was introduced by Hou and Langevin in 1997. The approach appears simple but heavily depends on the construction of desirable permutations. In this paper, we further study this approach by investigating the exponential sums of certain monomials and permutations. We propose several classes of bent functions from quadratic permutations and permutations with (generalized) Niho exponents, and also a class of bent functions from a generalization of the Maiorana-McFarland class. The relations among the proposed bent functions and the known families of bent function are studied. Numerical results show that our constructions include bent functions that are not contained in the completed Maiorana-McFarland class $M^{\#}$, the class ${\mathrm{PS}}_{ap}$ or the class $H$.

For a set of positive integers $A\subseteq \left[n\right]$, an r-coloring of A is rainbow sum-free if it contains no rainbow Schur triple. In this paper we initiate the study of the rainbow Erdős-Rothschild problem in the context of sum-free sets, which asks for the subsets of $\left[n\right]$ with the maximum number of rainbow sum-free r-colorings. We show that for $r=3$, the interval $\left[n\right]$ is optimal, while for $r\ge 8$, the set $[\lfloor n/2\rfloor ,n]$ is optimal. We also prove a stability theorem for $r\ge 4$. The proofs rely on the hypergraph container method, and some ad-hoc stability analysis.

A graph $G=(V,E)$ is called fully regular if for every independent set $I\subset V$, the number of vertices in $V\setminus I$ that are not connected to any element of I depends only on the size of I. A linear ordering of the vertices of G is called successive if for every i, the first i vertices induce a connected subgraph of G. We give an explicit formula for the number of successive vertex orderings of a fully regular graph.

As an application of our results, we give alternative proofs of two theorems of Stanley and Gao & Peng, determining the number of linear edge orderings of complete graphs and complete bipartite graphs, respectively, with the property that the first i edges induce a connected subgraph.

As another application, we give a simple product formula for the number of linear orderings of the hyperedges of a complete 3-partite 3-uniform hypergraph such that, for every i, the first i hyperedges induce a connected subgraph. We found similar formulas for complete (non-partite) 3-uniform hypergraphs and in another closely related case, but we managed to verify them only when the number of vertices is small.

The development of the theory of the second-order Eulerian polynomials began with the works of Buckholtz and Carlitz in their studies of an asymptotic expansion. Gessel-Stanley introduced Stirling permutations and provided combinatorial interpretations for the second-order Eulerian polynomials in terms of Stirling permutations. The Stirling permutations have been extensively studied by many researchers. The motivation of this paper is to develop a general method for finding equidistributed statistics on Stirling permutations. Firstly, we show that the up-down-pair statistic is equidistributed with the ascent-plateau statistic, and that the exterior up-down-pair statistic is equidistributed with the left ascent-plateau statistic. Secondly, we introduce the Stirling permutation code (called SP-code). A large number of equidistribution results follow from simple applications of the SP-codes. In particular, we find that six bivariable set-valued statistics are equidistributed on the set of Stirling permutations, and we generalize a classical result on trivariate version of the second-order Eulerian polynomial, which was independently established by Dumont and Bóna. Thirdly, we explore the bijections among Stirling permutation codes, perfect matchings and trapezoidal words. We then show the e-positivity of the enumerators of Stirling permutations by left ascent-plateaux, exterior up-down-pairs and right plateau-descents. In the final part, the e-positivity of the multivariate k-th order Eulerian polynomials is established, which improves a classical result of Janson-Kuba-Panholzer and generalizes a recent result of Chen-Fu. These e-positive expansions are derived from the combinatorial theory of context-free grammars.

In 1968, Golomb and Welch conjectured that there is no perfect Lee codes with radius $r\ge 2$ and dimension $n\ge 3$. A diameter perfect code is a natural generalization of the perfect code. In 2011, Etzion (2011) [5] proposed the following problem: Are there diameter perfect Lee (DPL, for short) codes with distance greater than four besides the $DPL(3,6)$ code? Later, Horak and AlBdaiwi (2012) [12] conjectured that there are no $DPL(n,d)$ codes for dimension $n\ge 3$ and distance $d>4$ except for $(n,d)=(3,6)$. In this paper, we give a counterexample to this conjecture. Moreover, we prove that for $n\ge 3$, there is a linear $DPL(n,6)$ code if and only if $n=3,11$.

The problem of covering the ground set of two matroids by a minimum number of common independent sets is notoriously hard even in very restricted settings, i.e. when the goal is to decide if two common independent sets suffice or not. Nevertheless, as the problem generalizes several long-standing open questions, identifying tractable cases is of particular interest. Strongly base orderable matroids form a class for which a basis-exchange condition that is much stronger than the standard axiom is met. As a result, several problems that are open for arbitrary matroids can be solved for this class. In particular, Davies and McDiarmid showed that if both matroids are strongly base orderable, then the covering number of their intersection coincides with the maximum of their covering numbers.

Motivated by their result, we propose relaxations of strongly base orderability in two directions. First we weaken the basis-exchange condition, which leads to the definition of a new, complete class of matroids with distinguished algorithmic properties. Second, we introduce the notion of covering the circuits of a matroid by a graph, and consider the cases when the graph is (A) 2-regular, or (B) a path. We give an extensive list of results explaining how the proposed relaxations compare to existing conjectures and theorems on coverings by common independent sets.