We work out a graph limit theory for dense interval graphs. The theory developed departs from the usual description of a graph limit as a symmetric function (, ) on the unit square, with and uniform on the interval (0, 1). Instead, we fix a and change the underlying distribution of the coordinates and . We find choices such that our limits are continuous. Connections to random interval graphs are given, including some examples. We also show a continuity result for the chromatic number and clique number of interval graphs. Some results on uniqueness of the limit description are given for general graph limits.

We provide some first experimental data about generating functions of restricted lattice walks with small steps in  .

In this paper, we expand on the notion of , first introduced explicitly by Aguiar and Mahajan in 2010 but already present in the literature in some other points of view. We do this by adapting the algebraic framework of species to the study of substructures in combinatorics. Afterwards, we consider functions that count the number of patterns of objects and endow the linear span of these functions with a product and a coproduct. In this way, any well-behaved family of combinatorial objects that admits a notion of substructure generates a Hopf algebra, and this association is functorial. For example, the Hopf algebra on permutations studied by Vargas in 2014 and the Hopf algebra on symmetric functions are particular cases of this construction. A specific family of pattern Hopf algebras of interest are the ones arising from . This includes the presheaves on graphs, posets and generalized permutahedra. Here, we show that all the pattern Hopf algebras corresponding to commutative presheaves are free. We also study a remarkable non-commutative presheaf structure on marked permutations, permutations with a marked element. These objects have a natural product called inflation, which is an operation motivated by factorization theorems for permutations. In this paper, we find new factorization theorems for marked permutations. We use these theorems to show that the pattern Hopf algebra for marked permutations is also free, using Lyndon words techniques.

In this paper, we explore intricate connections between Ramanujan's theta functions and a class of partition functions defined by the nature of the parity of their parts. This consequently leads us to the parity analysis of the crank of a partition and its correlation with the number of partitions with odd number of parts, self-conjugate partitions, and also with Durfee squares and Frobenius symbols.

An - is a type of rooted, arc-weighted, directed acyclic graph with leaf set , that is used in biology to represent the evolutionary history of a set  of species. In this paper, we introduce and investigate the space of equidistant -cactuses. This space contains, as a subset, the space of ultrametric trees on that was introduced by Gavryushkin and Drummond. We show that equidistant-cactus space is a CAT(0)-metric space which implies, for example, that there are unique geodesic paths between points. As a key step to proving this, we present a combinatorial result concerning rooted -cactuses. In particular, we show that such graphs can be encoded in terms of a pairwise compatibility condition arising from a poset of collections of pairs of subsets of  that satisfy certain set-theoretic properties. As a corollary, we also obtain an encoding of ranked, rooted -trees in terms of partitions of , which provides an alternative proof that the space of ultrametric trees on is CAT(0). We expect that our results will provide the basis for novel ways to perform statistical analyses on collections of equidistant -cactuses, as well as new directions for defining and understanding spaces of more general, arc-weighted phylogenetic networks.

A graph is cubical if it is a subgraph of a hypercube. For a cubical graph and a hypercube , is the largest number of edges in an -free subgraph of . If is at least a positive proportion of the number of edges in , then is said to have positive Turán density in the hypercube; otherwise it has zero Turán density. Determining and even identifying whether has positive or zero Turán density remains a widely open question for general . In this paper we focus on layered graphs, i.e., graphs that are contained in an edge layer of some hypercube. Graphs that are not layered have positive Turán density because one can form an -free subgraph of consisting of edges of every other layer. For example, a 4-cycle is not layered and has positive Turán density. However, in general, it is not obvious what properties layered graphs have. We give a characterization of layered graphs in terms of edge-colorings. We show that most non-trivial subdivisions have zero Turán density, extending known results on zero Turán density of even cycles of length at least 12 and of length 8. However, we prove that there are cubical graphs of girth 8 that are not layered and thus having positive Turán density. The cycle of length 10 remains the only cycle for which it is not known whether its Turán density is positive or not. We prove that , for a constant , showing that the extremal number for a 10-cycle behaves differently from any other cycle of zero Turán density.