Blecher [‘Geometry for totally symmetric plane partitions (TSPPs) with self-conjugate main diagonal’, Util. Math. 88 (2012), 223–235] defined the combinatorial objects known as 1-shell totally symmetric plane partitions of weight $n$. He also proved that the generating function for $f(n), $ the number of 1-shell totally symmetric plane partitions of weight $n$, is given by

$$\begin{eqnarray*}\displaystyle \sum _{n\geq 0}f(n){q}^{n} = 1+ \sum _{n\geq 1}{q}^{3n- 2} \prod _{i= 0}^{n- 2} (1+ {q}^{6i+ 3} ).\end{eqnarray*}$$

$f(n)$

The long-standing open problem of finding an upper bound for the Wiener index of a graph in terms of its order and diameter is addressed. Sharp upper bounds are presented for the Wiener index, and the related degree distance and Gutman index, for trees of order $n$ and diameter at most $6$.

For a family of linear preferential attachment graphs, we provide rates of convergence for the total variation distance between the degree of a randomly chosen vertex and an appropriate power law distribution as the number of vertices tends to ∞. Our proof uses a new formulation of Stein's method for the negative binomial distribution, which stems from a distributional transformation that has the negative binomial distributions as the only fixed points.

We study the array of point-to-point distances in random regular graphs equipped with exponential edge lengths. We consider the regime where the degree is kept fixed while the number of vertices tends to ∞. The marginal distribution of an individual entry is now well understood, thanks to the work of Bhamidi, van der Hofstad and Hooghiemstra (2010). The purpose of this note is to show that the whole array, suitably recentered, converges in the weak sense to an explicit infinite random array. Our proof consists in analyzing the invasion of the network by several mutually exclusive flows emanating from different sources and propagating simultaneously along the edges.

We consider a large class of exponential random graph models and prove the existence of a region of parameter space corresponding to the emergent multipartite structure, separated by a phase transition from a region of disordered graphs. An essential feature is the formalism of graph limits as developed by Lovász et al. for dense random graphs.

In this paper we characterise the distributions of the number of predecessors and of the number of successors of a given set of vertices, A, in the random mapping model, TnD̂ (see Hansen and Jaworski (2008)), with exchangeable in-degree sequence (D̂1,D̂2,…,D̂n). We show that the exact formulae for these distributions and their expected values can be given in terms of the distributions of simple functions of the in-degree variables D̂1,D̂2,…,D̂n. As an application of these results, we consider two special examples of TnD̂ which correspond to random mappings with preferential and anti-preferential attachment, and determine the exact distributions for the number of predecessors and the number of successors in these cases. We also characterise, for these two special examples, the asymptotic behaviour of the expected numbers of predecessors and successors and interpret these results in terms of the threshold behaviour of epidemic processes on random mapping graphs. The families of discrete distributions obtained in this paper are also of independent interest.

This paper is based on works presented at the 2012 Applied Probability Trust Lecture in Sheffield; its main purpose is to survey the recent asymptotic results of Bertoin (2012a) and Bertoin and Uribe Bravo (2012b) about Bernoulli bond percolation on certain large random trees with logarithmic height. We also provide a general criterion for the existence of giant percolation clusters in large trees, which answers a question raised by David Croydon.

An $r$-ary necklace (bracelet) of length $n$ is an equivalence class of $r$-colourings of vertices of a regular $n$-gon, taking all rotations (rotations and reflections) as equivalent. A necklace (bracelet) is symmetric if a corresponding colouring is invariant under some reflection. We show that the number of symmetric $r$-ary necklaces (bracelets) of length $n$ is $\frac{1}{2} (r+ 1){r}^{n/ 2} $ if $n$ is even, and ${r}^{(n+ 1)/ 2} $ if $n$ is odd.

In 2007, Andrews and Paule introduced a new class of combinatorial objects called broken $k$-diamond partitions. Recently, Shishuo Fu generalised the notion of broken $k$-diamond partitions to combinatorial objects which he termed $k$ dots bracelet partitions. Fu denoted the number of $k$ dots bracelet partitions of $n$ by ${\mathfrak{B}}_{k} (n)$ and proved several congruences modulo primes and modulo powers of 2. More recently, Radu and Sellers extended the set of congruences proven by Fu by proving three congruences modulo squares of primes for ${\mathfrak{B}}_{5} (n)$, ${\mathfrak{B}}_{7} (n)$ and ${\mathfrak{B}}_{11} (n)$. In this note, we prove some congruences modulo powers of 2 for ${\mathfrak{B}}_{5} (n)$. For example, we find that for all integers $n\geq 0$, ${\mathfrak{B}}_{5} (16n+ 7)\equiv 0\hspace{0.167em} ({\rm mod} \hspace{0.334em} {2}^{5} )$.

While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, it turns out that the intersection of only the cyclic shifts of one Bruhat decomposition has many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown–Goodearl–Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties. Applying results from [A. Knutson, T. Lam and D. Speyer, Projections of Richardson varieties, J. Reine Angew. Math., to appear, arXiv:1008.3939 [math.AG]], we show that positroid varieties are normal, Cohen–Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plücker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This latter fact lets us connect Postnikov’s and Buch–Kresch–Tamvakis’ approaches to quantum Schubert calculus.

Let $G$ be a finite group acting vertex-transitively on a graph. We show that bounding the order of a vertex stabiliser is equivalent to bounding the second singular value of a particular bipartite graph. This yields an alternative formulation of the Weiss conjecture.

The saturation theorem of Knutson and Tao concerns the nonvanishing of Littlewood–Richardson coefficients. In combination with work of Klyachko, it implies Horn’s conjecture about eigenvalues of sums of Hermitian matrices. This eigenvalue problem has a generalization to majorized sums of Hermitian matrices, due to S. Friedland. We further illustrate the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians. Our main result gives a Schubert calculus interpretation of Friedland’s problem, via equivariant cohomology of Grassmannians. In particular, we prove a saturation theorem for this setting. Our arguments employ the aforementioned work together with recent work of H. Thomas and A. Yong.

We give a simple graph-theoretic proof of a classical result due to Nash-Williams on covering graphs by forests. Moreover, we derive a slight generalization of this statement where some edges are preassigned to distinct forests.

Local models are schemes, defined in terms of linear-algebraic moduli problems, which are used to model the étale-local structure of integral models of certain $p$-adic PEL Shimura varieties defined by Rapoport and Zink. In the case of a unitary similitude group whose localization at ${ \mathbb{Q} }_{p} $ is ramified, quasi-split $G{U}_{n} $, Pappas has observed that the original local models are typically not flat, and he and Rapoport have introduced new conditions to the original moduli problem which they conjecture to yield a flat scheme. In a previous paper, we proved that their new local models are topologically flat when $n$ is odd. In the present paper, we prove topological flatness when $n$ is even. Along the way, we characterize the $\mu $-admissible set for certain cocharacters $\mu $ in types $B$ and $D$, and we show that for these cocharacters admissibility can be characterized in a vertexwise way, confirming a conjecture of Pappas and Rapoport.

Let Ω be a finite set and let G be a permutation group acting on it. A subset H of G is called t-intersecting if any two elements in H agree on at least t points. Let SDn and SBn be the classical Coxeter group of type Dn and type Bn, respectively. We show that the maximum-sized (2t)-intersecting families in SDn and SBn are precisely cosets of stabilizers of t points in [n] ≔ {1, 2, …, n}, provided n is sufficiently large depending on t.

In this short paper, we characterise graphs of order $pq$ with $p, q$ prime which are self-complementary and vertex-transitive.

Let $p$ be a real number greater than one and let $\Gamma $ be a graph of bounded degree. We investigate links between the $p$-harmonic boundary of $\Gamma $ and the ${D}_{p} $-massive subsets of $\Gamma $. In particular, if there are $n$ pairwise disjoint ${D}_{p} $-massive subsets of $\Gamma $, then the $p$-harmonic boundary of $\Gamma $ consists of at least $n$ elements. We show that the converse of this statement is also true.

In this paper, we determine when $\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} $, the complement of the zero divisor graph ${\Gamma }_{I} (L)$ with respect to a semiprime ideal $I$ of a lattice $L$, is connected and also determine its diameter, radius, centre and girth. Further, a form of Beck’s conjecture is proved for ${\Gamma }_{I} (L)$ when $\omega (\mathop{({\Gamma }_{I} (L))}\nolimits ^{c} )\lt \infty $.

We consider the bipartite matching model of customers and servers introduced by Caldentey, Kaplan and Weiss (2009). Customers and servers play symmetrical roles. There are finite sets C and S of customer and server classes, respectively. Time is discrete and at each time step one customer and one server arrive in the system according to a joint probability measure μ on C× S, independently of the past. Also, at each time step, pairs of matched customers and servers, if they exist, depart from the system. Authorized matchings are given by a fixed bipartite graph (C, S, E⊂ C × S). A matching policy is chosen, which decides how to match when there are several possibilities. Customers/servers that cannot be matched are stored in a buffer. The evolution of the model can be described by a discrete-time Markov chain. We study its stability under various admissible matching policies, including ML (match the longest), MS (match the shortest), FIFO (match the oldest), RANDOM (match uniformly), and PRIORITY. There exist natural necessary conditions for stability (independent of the matching policy) defining the maximal possible stability region. For some bipartite graphs, we prove that the stability region is indeed maximal for any admissible matching policy. For the ML policy, we prove that the stability region is maximal for any bipartite graph. For the MS and PRIORITY policies, we exhibit a bipartite graph with a non-maximal stability region.

This paper discusses a flaw in Murasugi–Przytycki’s Memoir ‘An index of a graph with applications to knot theory’ [Mem. Amer. Math. Soc. 106 (1993)]. We point out and partly fix a gap occurring in the proof of Murasugi–Przytycki’s braid index inequalities involving the graph index. We explain why their notion of index fails to precisely reflect the reduction of Seifert circles by their diagram move, and redefine the index to account for that discrepancy.