Annals of Combinatorics最新文献

We enumerate factorisations of the complete bipartite graph into spanning semiregular graphs in several cases, including when the degrees of all the factors except one or two are small. The resulting asymptotic behavior is seen to generalize the number of semiregular graphs in an elegant way. This leads us to conjecture a general formula when the number of factors is vanishing compared to the number of vertices. As a corollary, we find the average number of ways to partition the edges of a random semiregular bipartite graph into spanning semiregular subgraphs in several cases. Our proof of one case uses a switching argument to find the probability that a set of sufficiently sparse semiregular bipartite graphs are edge-disjoint when randomly labeled.

An animal is a planar shape formed by attaching congruent regular polygons along their edges. Usually, these polygons are a finite subset of tiles of a regular planar tessellation. These tessellations can be parameterized using the Schläfli symbol ({p,q}), where p denotes the number of sides of the regular polygon forming the tessellation and q is the number of edges or tiles meeting at each vertex. If ((p-2)(q-2)> 4), (=4), or (<4), then the tessellation corresponds to the geometry of the hyperbolic plane, the Euclidean plane, or the sphere, respectively. In 1976, Harary and Harborth studied animals defined on regular tessellations of the Euclidean plane, finding extremal values for their vertices, edges, and tiles, when any one of these parameters is fixed. They named animals attaining these extremal values as extremal animals. Here, we study hyperbolic extremal animals. For each ({p,q}) corresponding to a hyperbolic tessellation, we exhibit a sequence of spiral animals and prove that they attain the minimum numbers of edges and vertices within the class of animals with n tiles. We also give the first results on enumeration of extremal hyperbolic animals by finding special sequences of extremal animals that are unique extremal animals, in the sense that any animal with the same number of tiles which is distinct up to isometries cannot be extremal.

The immaculate basis of the Hopf algebra (textsf {NSym}) of noncommutative symmetric functions is a Schur-like basis of (textsf {NSym}) that has been applied in many areas in the field of algebraic combinatorics. The problem of determining a cancellation-free formula for the antipode of (textsf {NSym}) evaluated at an arbitrary immaculate function ( {mathfrak {S}}_{alpha } ) remains open, letting (alpha ) denote an integer composition. However, for the cases whereby we let (alpha ) be a hook or consist of at most two rows, Benedetti and Sagan (J Combin Theory Ser A 148:275–315, 2017) have determined cancellation-free formulas for expanding (S({mathfrak {S}}_{alpha })) in the ({mathfrak {S}})-basis. According to a Jacobi–Trudi-like formula for expanding immaculate functions in the ribbon basis that we had previously proved bijectively (Discrete Math 340(7):1716–1726, 2017), by applying the antipode S of (textsf {NSym}) to both sides of this formula, we obtain a cancellation-free formula for expressing (S({mathfrak {S}}_{(m^{n})})) in the R-basis, for an arbitrary rectangle ((m^{n})). We explore the idea of using this R-expansion, together with sign-reversing involutions, to determine combinatorial interpretations of the ({mathfrak {S}})-coefficients of antipodes of rectangular immaculate functions. We then determine cancellation-free formulas for antipodes of immaculate functions much more generally, using a Jacobi–Trudi-like formula recently introduced by Allen and Mason that generalizes Campbell’s formulas for expanding ({mathfrak {S}})-elements into the R-basis, and we further explore how new families of composition tableaux may be used to obtain combinatorial interpretations for expanding (S({mathfrak {S}}_{alpha })) into the ({mathfrak {S}})-basis.

Let (W, S) be a Coxeter system. We introduce the boolean complex of involutions of W which is an analogue of the boolean complex of W studied by Ragnarsson and Tenner. By applying discrete Morse theory, we determine the homotopy type of the boolean complex of involutions for a large class of (W, S), including all finite Coxeter groups, finding that the homotopy type is that of a wedge of spheres of dimension (vert Svert -1). In addition, we find simple recurrence formulas for the number of spheres in the wedge.

Goulden–Rattan polynomials give the exact value of the subdominant part of the normalized characters of the symmetric groups in terms of certain quantities ((C_i)) which describe the macroscopic shape of the Young diagram. The Goulden–Rattan positivity conjecture states that the coefficients of these polynomials are positive rational numbers with small denominators. We prove a special case of this conjecture for the coefficient of the quadratic term (C_2^2) by applying certain bijections involving maps (i.e., graphs drawn on surfaces).

Equivariant Euler characteristics are important numerical homotopy invariants for objects with group actions. They have deep connections with many other areas like modular representation theory and chromatic homotopy theory. They are also computable, especially for combinatorial objects like partition posets, buildings associated with finite groups of Lie types, etc. In this article, we make new contributions to concrete computations by determining the equivariant Euler characteristics for all subgroup complexes of symmetric groups (varSigma _n) when n is prime, twice a prime, or a power of two and several variants. There are two basic approaches to calculating equivariant Euler characteristics. One is based on a recursion formula and generating functions, and another on analyzing the fixed points of abelian subgroups. In this article, we adopt the second approach since the fixed points of abelian subgroups are simple in this case.

Phylogenetic (i.e., leaf-labeled) trees play a fundamental role in evolutionary research. A typical problem is to reconstruct such trees from data like DNA alignments (whose columns are often referred to as characters), and a simple optimization criterion for such reconstructions is maximum parsimony. It is generally assumed that this criterion works well for data in which state changes are rare. In the present manuscript, we prove that each binary phylogenetic tree T with (nge 20 k) leaves is uniquely defined by the set (A_k(T)), which consists of all characters with parsimony score k on T. This can be considered as a promising first step toward showing that maximum parsimony as a tree reconstruction criterion is justified when the number of changes in the data is relatively small.

In 1999, Merino and Welsh conjectured that evaluations of the Tutte polynomial of a graph satisfy an inequality. In this short article, we show that the conjecture generalized to matroids holds for the large class of all split matroids by exploiting the structure of their lattice of cyclic flats. This class of matroids strictly contains all paving and copaving matroids.