Sergey Kitaev

Patterns in permutations and words

There has been considerable interest recently in the subject of patterns in permutations and words, a new branch of combinatorics with its roots in the works of Rotem, Rogers, and Knuth in the 1970s. Consideration of the patterns in question has been extremely interesting from the combinatorial point of view, and it has proved to be a useful language in a variety of seemingly unrelated problems, including the theory of Kazhdan—Lusztig polynomials, singularities of Schubert varieties, interval orders, Chebyshev polynomials, models in statistical mechanics, and various sorting algorithms, including sorting stacks and sortable permutations. The author collects the main results in the field in this up-to-date, comprehensive reference volume. He highlights significant achievements in the area, and points to research directions and open problems. The book will be of interest to researchers and graduate students in theoretical computer science and mathematics, in particular those working in algebraic combinatorics and combinatorics on words. It will also be of interest to specialists in other branches of mathematics, theoretical physics, and computational biology.

Multi-avoidance of generalised patterns

Recently, Babson and Steingrimsson introduced generalised permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. We investigate simultaneous avoidance of two or more 3-patterns without internal dashes, that is, where the pattern corresponds to a contiguous subword in a permutation.

Introduction to partially ordered patterns

We review selected known results on partially ordered patterns (POPs) that include co-unimodal, multi- and shuffle patterns, peaks and valleys ((modified) maxima and minima) in permutations, the Horse permutations and others. We provide several new results on a class of POPs built on an arbitrary flat poset, obtaining, as corollaries, the bivariate generating function for the distribution of peaks (valleys) in permutations, links to Catalan, Narayana, and Pell numbers, as well as generalizations of a few results in the literature including the descent distribution. Moreover, we discuss a q-analogue for a result on non-overlapping segmented POPs. Finally, we suggest several open problems for further research.

Word Problem of the Perkins Semigroup via Directed Acyclic Graphs

For a word w in an alphabet Γ, the alternation word digraph Alt(w), a certain directed acyclic graph associated with w, is presented as a means to analyze the free spectrum of the Perkins monoid

Crucial words and the complexity of some extremal problems for sets of prohibited words

\mathbf{B_2^1}

Graphs capturing alternations in words

. Let

Words and Graphs

(f_n^{\mathbf{B_2^1}})

Avoidance of boxed mesh patterns on permutations

denote the free spectrum of

Semi-transitive orientations and word-representable graphs

\mathbf{B_2^1}

Harmonic numbers, Catalan’s triangle and mesh patterns

, let an be the number of distinct alternation word digraphs on words whose alphabet is contained in {x1,..., xn}, and let pn denote the number of distinct labeled posets on {1,..., n}. The word problem for the Perkins semigroup