We obtain an explicit formula for the variance of number $k$-peaks in a uniformly random permutation. This is then used to asymptotic length longest $k$-alternating subsequence permutations. Also central limit proved latter statistic.

An asymptotic lower bound for the maxrun function ρ(n) = max {number of runs in string x | all strings x of length n} is presented. More precisely, it is shown that for any ε > 0, (α−ε)n is an asymptotic lower bound, where α = 3 1+ √ 5 ≈ 0.927. A recent construction of an increasing sequence of binary strings “rich in runs” is modified and extended to prove the result.

An asymptotic lower bound for the maxrun function ρ(n) = max {number of runs in string x | all strings x of length n} is presented. More precisely, it is shown that for any ε > 0, (α−ε)n is an asymptotic lower bound, where α = 3 1+ √ 5 ≈ 0.927. A recent construction of an increasing sequence of binary strings “rich in runs” is modified and extended to prove the result.

In this note we show that the length spectrum for metric graphs exhibits a very high degree of degeneracy. More precisely, we obtain an asymptotic for the number of pairs of closed geodesics (or closed cycles) with the same metric length.

A set of words over a finite alphabet is called an unavoidable set if every word of sufficiently long length must contain some word from this set as a subword. Motivated by a theorem from automata theory, we introduce the notion of an almost unavoidable set and prove certain asymptotic estimates for the size of almost unavoidable sets of uniform length.

For any fixed integer d≥ 1, let ud(N) denote the number of permutations in the symmetric group S(N)which have no decreasing subsequence of length d+1.Regev determined the asymptotic form of ud(N) in the N→ ∞ limit; his argument relies on Selberg’s integral formula. In this article we give an elementary derivation of Regev’s formula via an asymptotic bijection with standard Young tableaux of rec...