Combinatorially interpreting generalized Stirling numbers

Asymptotic Normality of Some Graph Sequences

“…, f (n)}. This shows that ℓ and m are both bounded above by the clique number of H n , which is in turn bounded above by χ(H n ), which (by (2), which shows that G n and H n are co-chromatic) is equal to χ(G n ). The hypothesis g(n) → 0 now gives v n → ∞.…”

“…A Dyck path in R 2 is a staircase path (a path that proceeds by taking unit steps, either in the positive x direction or the positive y direction) that starts at (0, 0), ends on the line x = y, any never goes below this line. There is a natural correspondence between Dyck paths and Dyck words, given by mapping steps in the positive y direction to x, and steps in the positive x direction to D. For example, the word xxDxxDxDDD (which we will use as a running example for our interpretations) corresponds to the path that goes from (0, 0) to (0, 1) to (0, 2) to (1, 2) to (1,3) to (1,4) to (2,4) to (2,5) to (3,5) to (4, 5) to (5, 5).…”

“…The graph we end up with has a vertex, v 1 say, adjacent to each of four vertices, v 2 , v 3 , v 4 , v 5 say, with two edges among these four vertices, v 3 v 4 , v 4 v 5 say, inducing a path on three vertices; call this graph G ex . From [2] we have the following. As well as going from Dyck words to quasi-threshold graphs, we will need to go in the other direction.…”

“…, n} by putting an edge from i to j (i < j) if and only if (i, j) ∈ W w . For example, if w = xxDxxDxDDD then W w = {(1, 2), (2, 3), (2,4), (3,4), (3,5), (4,5)} and H w is the graph on vertex set {1, 2, 3, 4, 5} with edge set {12, 23, 24, 34, 35, 45}; call this graph G ex ′ (notice that G ex ′ is not isomorphic to G ex , since the former does not have a dominating vertex but the latter does). From [2] we have the following.…”

“…Using the symmetry of the standard normal, and the fact that the random variable (X n − µ n )/σ n in the definition of asymptotic normality is invariant under a shift in the sequence (a n,k ) k∈Z , (2) shows that asymptotic normality of Gn k k∈Z is implied by asymptotic normality of (m k (Γ n )) k∈Z . We cannot (yet) apply Theorem 2.6, because Γ n may have isolated vertices.…”