Journal of Combinatorial Theory Series A最新文献

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.

The goal of the classic football-pool problem is to determine how many lottery tickets are to be bought in order to guarantee at least $n-r$ correct guesses out of a sequence of n games played. We study a generalized (second-order) version of this problem, in which any of these n games consists of two sub-games. The second-order version of the football-pool problem is formulated using the notion of generalized-covering radius, recently proposed as a fundamental property of linear codes. We consider an extension of this property to general (not necessarily linear) codes, and provide an asymptotic solution to our problem by finding the optimal rate function of second-order covering codes given a fixed normalized covering radius. We also prove that the fraction of second-order covering codes among codes of sufficiently large rate tends to 1 as the code length tends to ∞.

List-decoding and list-recovery are important generalizations of unique decoding and receive considerable attention over the years. We study the optimal trade-off among the list-decoding (resp. list-recovery) radius, the list size, and the code rate, when the list size is constant and the alphabet size is large (both compared with the code length). We prove a new Singleton-type bound for list-decoding, which, for a wide range of parameters, is asymptotically tight up to a $1+o\left(1\right)$ factor. We also prove a Singleton-type bound for list-recovery, which is the first such bound in the literature. We apply these results to obtain near optimal lower bounds on the list size for list-decodable and list-recoverable codes with rates approaching capacity.

Moreover, we show that under some indivisibility condition of the parameters and over a sufficiently large alphabet, the largest list-decodable nonlinear codes can have much more codewords than the largest list-decodable linear codes. Such a large gap is not known to exist in unique decoding. We prove this by a novel connection between list-decoding and the notion of sparse hypergraphs in extremal combinatorics.

Lastly, we show that list-decodability or recoverability implies in some sense good unique decodability.

We apply the methods of partition analysis to partitions with n copies of n. This allows us to obtain multivariable generating functions related to classical Rogers-Ramanujan type identities. Also, partitions with n copies of n are extended to partition diamonds yielding numerous new results including a natural connection to overpartitions and a variety of partition congruences.