Journal of Combinatorial Theory Series A最新文献

An orthogonal array of index unity, order v, degree 5 and strength 3, or an OA$(3,5,v)$ in short, is a $5\times v^3$ array on v symbols and in every $3\times v^3$ subarray, each 3-tuple column vector occurs exactly once. The existence of an OA$(3,5,4n+2)$ is still open except for few known infinite classes of n. In this paper, we introduce a new combinatorial structure called three dimensions orthogonal complete large sets of disjoint incomplete Latin squares and use it to obtain many new infinite classes of OA$(3,5,4n+2)$s.

q-Supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial are very rare in the literature. In this paper, we establish some q-supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial in terms of Jackson's ${}_8\varphi _7$ summation, Watson's ${}_8\varphi _7$ transformation, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials. More concretely, we give a q-analogue of a nice formula due to Long and Ramakrishna [Adv. Math. 290 (2016), 773–808] and two q-supercongruences involving double series.

Tutte's 3-flow conjecture asserts that every 4-edge-connected graph admits a nowhere-zero 3-flow. We prove that this conjecture is true for every Cayley graph of valency at least four on any supersolvable group with a noncyclic Sylow 2-subgroup and every Cayley graph of valency at least four on any group whose derived subgroup is of square-free order.

A code C is a subset of the vertex set of a graph and C is s-neighbour-transitive if its automorphism group $\mathrm{Aut}\left(C\right)$ acts transitively on each of the first $s+1$ parts $C_0,C_1,\dots ,C_s$ of the distance partition $\{C=C_0,C_1,\dots ,C_{\rho }\}$, where ρ is the covering radius of C. While codes have traditionally been studied in the Hamming and Johnson graphs, we consider here codes in the Kneser graphs. Let Ω be the underlying set on which the Kneser graph $K(n,k)$ is defined. Our first main result says that if C is a 2-neighbour-transitive code in $K(n,k)$ such that C has minimum distance at least 5, then $n=2k+1$ (i.e., C is a code in an odd graph) and C lies in a particular infinite family or is one particular sporadic example. We then prove several results when C is a neighbour-transitive code in the Kneser graph $K(n,k)$. First, if $\mathrm{Aut}\left(C\right)$ acts intransitively on Ω we characterise C in terms of certain parameters. We then assume that $\mathrm{Aut}\left(C\right)$ acts transitively on Ω, first proving that if C has minimum distance at least 3 then either $K(n,k)$ is an odd graph or $\mathrm{Aut}\left(C\right)$ has a 2-homogeneous (and hence primitive) action on Ω. We then assume that C is a code in an odd graph and $\mathrm{Aut}\left(C\right)$ acts imprimitively on Ω and characterise C in terms of certain parameters. We give examples in each of these cases and pose several open problems.

In recent years, monotone double Hurwitz numbers were introduced as a naturally combinatorial modification of double Hurwitz numbers. Monotone double Hurwitz numbers share many structural properties with their classical counterparts, such as piecewise polynomiality, while the quantitative properties of these two numbers are quite different. We consider real analogues of monotone double Hurwitz numbers and study the asymptotics for these real analogues. The key ingredient is an interpretation of real tropical covers with arbitrary splittings as factorizations in the symmetric group which generalizes the result from Guay-Paquet et al. (2016) [18]. By using the above interpretation, we consider three types of real analogues of monotone double Hurwitz numbers: real monotone double Hurwitz numbers relative to simple splittings, relative to arbitrary splittings and real mixed double Hurwitz numbers. Under certain conditions, we find lower bounds for these real analogues, and obtain logarithmic asymptotics for real monotone double Hurwitz numbers relative to arbitrary splittings and real mixed double Hurwitz numbers. In particular, under given conditions real mixed double Hurwitz numbers are logarithmically equivalent to complex double Hurwitz numbers. We construct a family of real tropical covers and use them to show that real monotone double Hurwitz numbers relative to simple splittings are logarithmically equivalent to monotone double Hurwitz numbers with specific conditions. This is consistent with the logarithmic equivalence of real double Hurwitz numbers and complex double Hurwitz numbers.

We give a criterion for when the smallest regular cover of a chiral polytope $P$ is itself a polytope, using only information about the facets and vertex-figures of $P$.

In this paper, we introduce new interpretations for the sum of the parts with the same parity in all the partitions of n.

In his 1984 AMS memoir, Andrews introduced the family of k-colored generalized Frobenius partition functions. For any positive integer k, let $c{\varphi }_k\left(n\right)$ denote the number of k-colored generalized Frobenius partitions of n. Among many other things, Andrews proved that for any $n\ge 0$, $c{\varphi }_2(5n+3)\equiv 0\left(\mathrm{mod}5\right)$. Since then, many scholars subsequently considered congruence properties of various k-colored generalized Frobenius partition functions, typically with a small number of colors.

In 2019, Chan, Wang and Yang systematically studied arithmetic properties of $\text{C}{\Phi }_k\left(q\right)$ with $2\le k\le 17$ by employing the theory of modular forms, where $\text{C}{\Phi }_k\left(q\right)$ denotes the generating function of $c{\varphi }_k\left(n\right)$. We notice that many coefficients in the expressions of $\text{C}{\Phi }_k\left(q\right)$ are not integers. In this paper, we first observe that $\text{C}{\Phi }_k\left(q\right)$ is related to the constant term of a family of bivariable functions, then establish a general symmetric and recurrence relation on the coefficients of these bivariable functions. Based on this relation, we next derive many bivariable identities. By extracting and computing the constant terms of these bivariable identities, we establish the expressions of $\text{C}{\Phi }_k\left(q\right)$ with integral coefficients. As an immediate consequence, we prove some infinite families of congruences satisfied by $c{\varphi }_k\left(n\right)$, where k is allowed to grow arbitrary large.

We determine asymptotic growth rates for lengths of monochromatic arithmetic progressions in certain automatic sequences. In particular, we look at (one-sided) fixed points of aperiodic, primitive, bijective substitutions and spin substitutions, which are generalisations of the Thue–Morse and Rudin–Shapiro substitutions, respectively. For such infinite words, we show that there exists a subsequence $\left\{d_n\right\}$ of differences along which the maximum length $A\left(d_n\right)$ of a monochromatic arithmetic progression (with fixed difference $d_n$) grows at least polynomially in $d_n$. Explicit upper and lower bounds for the growth exponent can be derived from a finite group associated to the substitution. As an application, we obtain bounds for a van der Waerden-type number for a class of colourings parametrised by the size of the alphabet and the length of the substitution.