Consider lattice paths in Z taking unit steps north (N) and east (E). Fix positive integers r, s and put an equivalence relation on points of Z by letting v, w be equivalent if v − w = `(r, s) for some ` ∈ Z. Call a lattice path valid if whenever it enters a point v with an E-step, then any further points of the path in the equivalence class of v are also entered with an E-step. Loehr and Warri...

In this paper we characterize the divisibility by 2 of the Stirling number of the second kind, S(n; k); where n is a su ciently high power of 2. Let 2(r) denote the highest power of 2 which divides r. We show that there exists a function L(k) such that for all n L(k); 2 k!S(2; k) = k 1 hold, independently from n: (Here the independence follows from the periodicity of the Stirling numbers modulo...

The monochromatic tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum integer k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex-disjoint monochromatic trees. In general, to determine this number is very difficult. For 2-edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the ex...

The monochromatic tree partition number of an r-edge-colored graph G, denoted by tr(G), is the minimum integer k such that whenever the edges of G are colored with r colors, the vertices of G can be covered by at most k vertex-disjoint monochromatic trees. In general, to determine this number is very difficult. For 2edge-colored complete multipartite graph, Kaneko, Kano, and Suzuki gave the exa...

Abstract In this paper, we study asymmetric Ramsey properties of the random graph $G_{n,p}$ . Let $r \in \mathbb{N}$ and $H_1, \ldots, H_r$ be graphs. We write $G_{n,p} \to (H_1, H_r)$ to denote property that whenever colour edges with colours from set $[r] \,{:\!=}\, \{1, r\}$ there exists $i [r]$ a copy $H_i$ in monochromatic $i$ There has been much interest determining asymptotic threshold f...

If σ is a permutation of n letters, say that j is an outstanding element (Fr. élément saillant) of σ if whenever i < j we have σ(i) < σ(j). A famous theorem of A. Rényi [R] (see also [C]) states that the number of permutations of n letters that have k outstanding elements is equal to the number that have k cycles, and therefore there are [ n k ] such permutations, the latter being the unsigned ...

Let k ≥ l ≥ 1 and n ≥ 1 be integers. Let G(k, n) be the complete k-partite graph with n vertices in each colour class. An l-decomposition of G(k, n) is a set X of copies of Kk in G(k, n) such that each copy of Kl in G(k, n) is a subgraph of exactly one copy of Kk in X. This paper asks: when does G(k, n) have an l-decomposition? The answer is well known for the l = 2 case. In particular, G(k, n)...

Let r ≥ 2 be a fixed integer. For infinitely many n, let k = (k1, . . . , kn) be a vector of nonnegative integers such that their sum M is divisible by r. We present an asymptotic enumeration formula for simple r-uniform hypergraphs with degree sequence k. (Here “simple” means that all edges are distinct and no edge contains a repeated vertex.) Our formula holds whenever the maximum degree kmax...