Ranks of partitions play an important role in the theory of partitions. They provide combinatorial interpretations for Ramanujan’s famous congruences for partition functions. We establish a family of congruences modulo powers of $5$ for ranks of partitions.

It is well known that the edge ideal $I(G)$ of a simple graph G has linear quotients if and only if $G^c$ is chordal. We investigate when the property of having linear quotients is inherited by homological shift ideals of an edge ideal. We will see that adding a cluster to the graph $G^c$ when $I(G)$ has homological linear quotients results in a graph with the same property. In particular, $I(G)$ has homological linear quotients when $G^c$ is a block graph. We also show that adding pinnacles to trees preserves the property of having homological linear quotients for the edge ideal of their complements. Furthermore, $I(G)$ has homological linear quotients for every graph G such that $G^c$ is a $\lambda $-minimal chordal graph.

We characterize totally symmetric self-complementary plane partitions (TSSCPP) as bounded compatible sequences satisfying a Yamanouchi-like condition. As such, they are in bijection with certain pipe dreams. Using this characterization and the recent bijection of Gao–Huang between reduced pipe dreams and reduced bumpless pipe dreams, we give a bijection between alternating sign matrices and TSSCPP in the reduced, 1432-avoiding case. We also give a different bijection in the 1432- and 2143-avoiding case that preserves natural poset structures on the associated pipe dreams and bumpless pipe dreams.

The Halpern–Läuchli theorem, a combinatorial result about trees, admits an elegant proof due to Harrington using ideas from forcing. In an attempt to distill the combinatorial essence of this proof, we isolate various partition principles about products of perfect Polish spaces. These principles yield straightforward proofs of the Halpern–Läuchli theorem, and the same forcing from Harrington’s proof can force their consistency. We also show that these principles are not ZFC theorems by showing that they put lower bounds on the size of the continuum.

We study heterogeneously interacting diffusive particle systems with mean-field-type interaction characterized by an underlying graphon and their finite particle approximations. Under suitable conditions, we obtain exponential concentration estimates over a finite time horizon for both 1- and 2-Wasserstein distances between the empirical measures of the finite particle systems and the averaged law of the graphon system.

We consider linear preferential attachment trees with additive fitness, where fitness is the random initial vertex attractiveness. We show that when the fitnesses are independent and identically distributed and have positive bounded support, the local weak limit can be constructed using a sequence of mixed Poisson point processes. We also provide a rate of convergence for the total variation distance between the r-neighbourhoods of a uniformly chosen vertex in the preferential attachment tree and the root vertex of the local weak limit. The proof uses a Pólya urn representation of the model, for which we give new estimates for the beta and product beta variables in its construction. As applications, we obtain limiting results and convergence rates for the degrees of the uniformly chosen vertex and its ancestors, where the latter are the vertices that are on the path between the uniformly chosen vertex and the initial vertex.

We study the asymptotic growth rate of the labels of high-degree vertices in weighted recursive graphs (WRGs) when the weights are independent, identically distributed, almost surely bounded random variables, and as a result confirm a conjecture by Lodewijks and Ortgiese (‘The maximal degree in random recursive graphs with random weights’, preprint, 2020). WRGs are a generalisation of the random recursive tree and directed acyclic graph models, in which vertices are assigned vertex-weights and where new vertices attach to $m\in\mathbb{N}$ predecessors, each selected independently with a probability proportional to the vertex-weight of the predecessor. Prior work established the asymptotic growth rate of the maximum degree of the WRG model, and here we show that there exists a critical exponent $\mu_m$ such that the typical label size of the maximum-degree vertex equals $n^{\mu_m(1+o(1))}$ almost surely as n, the size of the graph, tends to infinity. These results extend results on the asymptotic behaviour of the location of the maximum degree, formerly only known for the random recursive tree model, to the more general weighted multigraph case of the WRG model. Moreover, for the weighted recursive tree model, that is, the WRG model with $m=1$, we prove the joint convergence of the rescaled degree and label of high-degree vertices under additional assumptions on the vertex-weight distribution, and also extend results on the growth rate of the maximum degree obtained by Eslava, Lodewijks, and Ortgiese (Stoch. Process. Appl. 158, 2023).

We prove a weak version of the cross-product conjecture: $\textrm {F}(k+1,\ell ) \hskip .06cm \textrm {F}(k,\ell +1) \ge (\frac 12+\varepsilon ) \hskip .06cm \textrm {F}(k,\ell ) \hskip .06cm \textrm {F}(k+1,\ell +1)$, where $\textrm {F}(k,\ell )$ is the number of linear extensions for which the values at fixed elements $x,y,z$ are k and $\ell $ apart, respectively, and where $\varepsilon>0$ depends on the poset. We also prove the converse inequality and disprove the generalized cross-product conjecture. The proofs use geometric inequalities for mixed volumes and combinatorics of words.

Let $\boldsymbol {k} := (k_1,\ldots ,k_s)$ be a sequence of natural numbers. For a graph G, let $F(G;\boldsymbol {k})$ denote the number of colourings of the edges of G with colours $1,\dots ,s$ such that, for every $c \in \{1,\dots ,s\}$, the edges of colour c contain no clique of order $k_c$. Write $F(n;\boldsymbol {k})$ to denote the maximum of $F(G;\boldsymbol {k})$ over all graphs G on n vertices. There are currently very few known exact (or asymptotic) results for this problem, posed by Erdős and Rothschild in 1974. We prove some new exact results for $n \to \infty $:

1. (i) A sufficient condition on $\boldsymbol {k}$ which guarantees that every extremal graph is a complete multipartite graph, which systematically recovers all existing exact results.

2. (ii) Addressing the original question of Erdős and Rothschild, in the case $\boldsymbol {k}=(3,\ldots ,3)$ of length $7$, the unique extremal graph is the complete balanced $8$-partite graph, with colourings coming from Hadamard matrices of order $8$.

3. (iii) In the case $\boldsymbol {k}=(k+1,k)$, for which the sufficient condition in (i) does not hold, for $3 \leq k \leq 10$, the unique extremal graph is complete k-partite with one part of size less than k and the other parts as equal in size as possible.

The goal of this paper is to go further in the analysis of the behavior of the number of descents in a random permutation. Via two different approaches relying on a suitable martingale decomposition or on the Irwin–Hall distribution, we prove that the number of descents satisfies a sharp large-deviation principle. A very precise concentration inequality involving the rate function in the large-deviation principle is also provided.

It is known that in an infinite very weakly cancellative semigroup with size $\kappa $, any central set can be partitioned into $\kappa $ central sets. Furthermore, if $\kappa $ contains $\lambda $ almost disjoint sets, then any central set contains $\lambda $ almost disjoint central sets. Similar results hold for thick sets, very thick sets and piecewise syndetic sets. In this article, we investigate three other notions of largeness: quasi-central sets, C-sets, and J-sets. We obtain that the statement applies for quasi-central sets. If the semigroup is commutative, then the statement holds for C-sets. Moreover, if $\kappa ^\omega = \kappa $, then the statement holds for J-sets.

A subset of positive integers F is a Schreier set if it is nonempty and $|F|\leqslant \min F$ (here $|F|$ is the cardinality of F). For each positive integer k, we define $k\mathcal {S}$ as the collection of all the unions of at most k Schreier sets. Also, for each positive integer n, let $(k\mathcal {S})^n$ be the collection of all sets in $k\mathcal {S}$ with maximum element equal to n. It is well known that the sequence $(|(1\mathcal {S})^n|)_{n=1}^\infty $ is the Fibonacci sequence. In particular, the sequence satisfies a linear recurrence. We show that the sequence $(|(k\mathcal {S})^n|)_{n=1}^\infty $ satisfies a linear recurrence for every positive k.

We show that certain sums of partition numbers are divisible by multiples of 2 and 3. For example, if $p(n)$ denotes the number of unrestricted partitions of a positive integer n (and $p(0)=1$, $p(n)=0$ for $n<0$), then for all nonnegative integers m,

$$ \begin{align*}\sum_{k=0}^\infty p(24m+23-\omega(-2k))+\sum_{k=1}^\infty p(24m+23-\omega(2k))\equiv 0~ (\text{mod}~144),\end{align*} $$

where $\omega (k)=k(3k+1)/2$.

We consider bond percolation on high-dimensional product graphs $G=\square _{i=1}^tG^{(i)}$, where $\square$ denotes the Cartesian product. We call the $G^{(i)}$ the base graphs and the product graph $G$ the host graph. Very recently, Lichev (J. Graph Theory, 99(4):651–670, 2022) showed that, under a mild requirement on the isoperimetric properties of the base graphs, the component structure of the percolated graph $G_p$ undergoes a phase transition when $p$ is around $\frac{1}{d}$, where $d$ is the average degree of the host graph.

In the supercritical regime, we strengthen Lichev’s result by showing that the giant component is in fact unique, with all other components of order $o(|G|)$, and determining the sharp asymptotic order of the giant. Furthermore, we answer two questions posed by Lichev (J. Graph Theory, 99(4):651–670, 2022): firstly, we provide a construction showing that the requirement of bounded degree is necessary for the likely emergence of a linear order component; secondly, we show that the isoperimetric requirement on the base graphs can be, in fact, super-exponentially small in the dimension. Finally, in the subcritical regime, we give an example showing that in the case of irregular high-dimensional product graphs, there can be a polynomially large component with high probability, very much unlike the quantitative behaviour seen in the Erdős-Rényi random graph and in the percolated hypercube, and in fact in any regular high-dimensional product graphs, as shown by the authors in a companion paper (Percolation on high-dimensional product graphs. arXiv:2209.03722, 2022).

For any branched double covering of compact Riemann surfaces, we consider the associated character varieties that are unitary in the global sense, which we call $\operatorname {\mathrm {GL}}_n\rtimes \!<\!\sigma {>}$-character varieties. We restrict the monodromies around the branch points to generic semi-simple conjugacy classes contained in $\operatorname {\mathrm {GL}}_n\sigma $ and compute the E-polynomials of these character varieties using the character table of $\operatorname {\mathrm {GL}}_n(q)\rtimes \!<\!\sigma \!>\!$. The result is expressed as the inner product of certain symmetric functions associated to the wreath product $(\mathbb {Z}/2\mathbb {Z})^N\rtimes \mathfrak {S}_N$. We are then led to a conjectural formula for the mixed Hodge polynomial, which involves (modified) Macdonald polynomials and wreath Macdonald polynomials.

A quiver representation assigns a vector space to each vertex, and a linear map to each arrow of a quiver. When one considers the category $\mathrm {Vect}(\mathbb {F}_1)$ of vector spaces “over $\mathbb {F}_1$” (the field with one element), one obtains $\mathbb {F}_1$-representations of a quiver. In this paper, we study representations of a quiver over the field with one element in connection to coefficient quivers. To be precise, we prove that the category $\mathrm {Rep}(Q,\mathbb {F}_1)$ is equivalent to the (suitably defined) category of coefficient quivers over Q. This provides a conceptual way to see Euler characteristics of a class of quiver Grassmannians as the number of “$\mathbb {F}_1$-rational points” of quiver Grassmannians. We generalize techniques originally developed for string and band modules to compute the Euler characteristics of quiver Grassmannians associated with $\mathbb {F}_1$-representations. These techniques apply to a large class of $\mathbb {F}_1$-representations, which we call the $\mathbb {F}_1$-representations with finite nice length: we prove sufficient conditions for an $\mathbb {F}_1$-representation to have finite nice length, and classify such representations for certain families of quivers. Finally, we explore the Hall algebras associated with $\mathbb {F}_1$-representations of quivers. We answer the question of how a change in orientation affects the Hall algebra of nilpotent $\mathbb {F}_1$-representations of a quiver with bounded representation type. We also discuss Hall algebras associated with representations with finite nice length, and compute them for certain families of quivers.

The Hoffman ratio bound, Lovász theta function, and Schrijver theta function are classical upper bounds for the independence number of graphs, which are useful in graph theory, extremal combinatorics, and information theory. By using generalized inverses and eigenvalues of graph matrices, we give bounds for independence sets and the independence number of graphs. Our bounds unify the Lovász theta function, Schrijver theta function, and Hoffman-type bounds, and we obtain the necessary and sufficient conditions of graphs attaining these bounds. Our work leads to some simple structural and spectral conditions for determining a maximum independent set, the independence number, the Shannon capacity, and the Lovász theta function of a graph.

We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the underlying treewidth of a graph class $\mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some function $f$, for every graph $G \in \mathcal{G}$ there is a graph $H$ with $\textrm{tw}(H) \leqslant c$ such that $G$ is isomorphic to a subgraph of $H \boxtimes K_{f(\textrm{tw}(G))}$. We introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth $3$; the class of $K_{s,t}$-minor-free graphs has underlying treewidth $s$ (for $t \geqslant \max \{s,3\}$); and the class of $K_t$-minor-free graphs has underlying treewidth $t-2$. In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a subdivided star, and that the class of graphs with no induced $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a star.

We solve a fundamental question posed in Frohardt’s 1988 paper [6] on finite $2$-groups with Kantor familes, by showing that finite groups K with a Kantor family $(\mathcal {F},\mathcal {F}^*)$ having distinct members $A, B \in \mathcal {F}$ such that $A^* \cap B^*$ is a central subgroup of K and the quotient $K/(A^* \cap B^*)$ is abelian cannot exist if the center of K has exponent $4$ and the members of $\mathcal {F}$ are elementary abelian. Then we give a short geometrical proof of a recent result of Ott which says that finite skew translation quadrangles of even order $(t,t)$ (where t is not a square) are always translation generalized quadrangles. This is a consequence of a complete classification of finite cyclic skew translation quadrangles of order $(t,t)$ that we carry out in the present paper.

Fix a prime $p\geq 11$. We show that there exists a positive integer $m$ such that any subset of $\mathbb {F}_p^n\times \mathbb {F}_p^n$ containing no nontrivial configurations of the form $(x,y)$, $(x,y+z)$, $(x,y+2z)$, $(x+z,y)$ must have density $\ll 1/\log _{m}{n}$, where $\log _{m}$ denotes the $m$-fold iterated logarithm. This gives the first reasonable bound in the multidimensional Szemerédi theorem for a two-dimensional four-point configuration in any setting.