The main theme of this paper is the enumeration of the order-isomorphic occurrence of a pattern in words and permutations. We mainly focus on asymptotic properties of the sequence $f_r^v(k,n),$ the number of $n$-array $k$-ary words that contain a given pattern $v$ exactly $r$ times. In addition, we study the asymptotic behavior of the random variable $X_n,$ the number of pattern occurrences in a random $n$-array word. The two topics are closely related through the identity $P(X_n=r) = $ $\frac{1}{k^n}f_r^v(k,n).$ In particular, we show that for any $r\geq 0,$ the Stanley--Wilf sequence $\bigl(f_r^v(k,n)\bigr)^{1/n}$ converges to a limit independent of $r,$ and we determine the value of the limit. We then obtain several limit theorems for the distribution of $X_n,$ including a central limit theorem, large deviation estimates, and the exact growth rate of the entropy of $X_n.$ Furthermore, we introduce a concept of weak avoidance and link it to a certain family of nonproduct measures on words that penalize pattern occurrences but do not forbid them entirely. We analyze this family of probability measures in a small parameter regime, where the distributions can be understood as a perturbation of a uniform measure. Finally, we extend some of our results for words, including the one regarding the equivalence of the limits of the Stanley--Wilf sequences, to pattern occurrences in permutations.

We formulate conditions on a set of log-concave sequences, under which any linear combination of those sequences is log-concave, and further, of conditions under which linear combinations of log-concave sequences that have been transformed by convolution are log-concave. These conditions involve relations on sequences called synchronicity and ratio-dominance, and a characterization of some bivariate sequences as lexicographic. We are motivated by the 25-year-old conjecture that the genus distribution of every graph is log-concave. Although calculating genus distributions is NP-hard, they have been calculated explicitly for many graphs of tractable size, and the three conditions have been observed to occur in the partitioned genus distributions of all such graphs. They are used here to prove the log-concavity of the genus distributions of graphs constructed by iterative amalgamation of double-rooted graph fragments whose genus distributions adhere to these conditions, even though it is known that the genus polynomials of some such graphs have imaginary roots. A blend of topological and combinatorial arguments demonstrates that log-concavity is preserved through the iterations.

In this paper we consider the absolute variation statistics of a composition σ = σ1 ··· σm of n which is a measure of the sum of absolute differences between each consecutive pair of parts in a composition. This and some related statistics which we d