Toufik Mansour

Combinatorics of Compositions and Words

Introduction Historical Overview-Compositions Historical Overview-Words A More Detailed Look Basic Tools of the Trade Sequences Solving Recurrence Relations Generating Functions Compositions Definitions and Basic Results (One Variable) Restricted Compositions Compositions with Restricted Parts Connection between Compositions and Tilings Colored Compositions and Other Variations Research Directions and Open Problems Statistics on Compositions History and Connections Subword Patterns of Length 2: Rises, Levels, and Drops Longer Subword Patterns Research Directions and Open Problems Avoidance of Non-Subword Patterns in Compositions History and Connections Avoidance of Subsequence Patterns Generalized Patterns and Compositions Partially Ordered Patterns in Compositions Research Directions and Open Problems Words History and Connections Definitions and Basic Results Subword Patterns Subsequence Patterns-Classification Subsequence Patterns-Generating Functions Generalized Patterns of Type (2,1) Avoidance of Partially Ordered Patterns Research Directions and Open Problems Automata and Generating Trees History and Connections Tools from Graph Theory Automata Generating Trees The ECO Method Research Directions and Open Problems Asymptotics for Compositions History Tools from Probability Theory Tools from Complex Analysis Asymptotics for Compositions Asymptotics for Carlitz Compositions A Word on the Asymptotics for Words Research Directions and Open Problems Appendix A: Useful Identities and Generating Functions Appendix B: Linear Algebra and Algebra Review Appendix C: Chebychev Polynomials of the Second Kind Appendix D: Probability Theory Appendix E: Complex Analysis Review Appendix F: Using Mathematica and Maple Appendix G: C++ and Maple Programs Appendix H: Notation References Exercises appear at the end of each chapter.

A characterization of horizontal visibility graphs and combinatorics on words

A Horizontal Visibility Graph (HVG) is defined in association with an ordered set of non-negative reals. HVGs realize a methodology in the analysis of time series, their degree distribution being a good discriminator between randomness and chaos Luque et al. [B. Luque, L. Lacasa, F. Ballesteros, J. Luque, Horizontal visibility graphs: exact results for random time series, Phys. Rev. E 80 (2009), 046103]. We prove that a graph is an HVG if and only if it is outerplanar and has a Hamilton path. Therefore, an HVG is a noncrossing graph, as defined in algebraic combinatorics Flajolet and Noy [P. Flajolet, M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math., 204 (1999) 203–229]. Our characterization of HVGs implies a linear time recognition algorithm. Treating ordered sets as words, we characterize subfamilies of HVGs highlighting various connections with combinatorial statistics and introducing the notion of a visible pair. With this technique, we determine asymptotically the average number of edges of HVGs.

Counting occurrences of a pattern of type (1,2) or (2,1) in permutations

Babson and Steingrimsson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. Claesson presented a complete solution for the number of permutations avoiding any single pattern of type (1,2) or (2,1). For eight of these twelve patterns the answer is given by the Bell numbers. For the remaining four the answer is given by the Catalan numbers. With respect to being equidistributed there are three different classes of patterns of type (1,2) or (2,1). We present a recursion for the number of permutations containing exactly one occurrence of a pattern of the first or the second of the aforementioned classes, and we also find an ordinary generating function for these numbers. We prove these results both combinatorially and analytically. Finally, we give the distribution of any pattern of the third class in the form of a continued fraction, and we also give explicit formulas for the number of permutations containing exactly r occurrences of a pattern of the third class when r@?{1,2,3}.

Identities involving Narayana polynomials and Catalan numbers

We first establish the result that the Narayana polynomials can be represented as the integrals of the Legendre polynomials. Then we represent the Catalan numbers in terms of the Narayana polynomials by three different identities. We give three different proofs for these identities, namely, two algebraic proofs and one combinatorial proof. Some applications are also given which lead to many known and new identities.

Staircase tilings and k-Catalan structures

Many interesting combinatorial objects are enumerated by the k-Catalan numbers, one possible generalization of the Catalan numbers. We will present a new combinatorial object that is enumerated by the k-Catalan numbers, staircase tilings. We give a bijection between staircase tilings and k-good paths, and between k-good paths and k-ary trees. In addition, we enumerate k-ary paths according to DD, UDU, and UU, and connect these statistics for k-ary paths to statistics for the staircase tilings. Using the given bijections, we enumerate statistics on the staircase tilings, and obtain connections with Catalan numbers for special values of k. The second part of the paper lists a sampling of other combinatorial structures that are enumerated by the k-Catalan numbers. Many of the proofs generalize from those for the Catalan structures that are being generalized, but we provide one proof that is not a straightforward generalization. We propose a web site repository for these structures, similar to those maintained by Richard Stanley for the Catalan numbers [R.P. Stanley, Catalan addendum. Available at: http://www-math.mit.edu/~rstan/ec/] and by Robert Sulanke for the Delannoy numbers [R. Sulanke, Objects counted by the central Delannoy numbers, J. Integer Seq. 6 (1) (2003), Article 03, 1, 5, 19 pp. Available also at: math.boisestate.edu/~sulanke/infowhowasdelannoy.html]. On the website, we list additional combinatorial objects, together with hints on how to show that they are indeed enumerated by the k-Catalan numbers.

Umbral calculus associated with Frobenius-type Eulerian polynomials

In this paper, we study some properties of several polynomials arising from umbral calculus. In particular, we investigate the properties of orthogonality type of the Frobeniustype Eulerian polynomials which are derived from umbral calculus. By using our properties, we can derive many interesting identities of special polynomials associated with Frobeniustype Eulerian polynomials. An application to normal ordering is presented.

Finite automata and pattern avoidance in words

We say that a word w on a totally ordered alphabet avoids the word v if there are no subsequences in w order-equivalent to v. In this paper we suggest a new approach to the enumeration of words on at most k letters avoiding a given pattern. By studying an automaton which for fixed k generates the words avoiding a given pattern we derive several previously known results for these kind of problems, as well as many new. In particular, we give a simple proof of the formula (Electron. J. Combin. 5(1998) #R15) for exact asymptotics for the number of words on k letters of length n that avoids the pattern 12...(l + 1). Moreover, we give the first combinatorial proof of the exact formula (Enumeration of words with forbidden patterns, Ph.D. Thesis, University of Pennsylvania, 1998) for the number of words on k letters of length n avoiding a three letter permutation pattern.

Umbral calculus and Sheffer sequences of polynomials

In this paper, we investigate some properties of Sheffer sequences of polynomials arising from umbral calculus. From these properties, we derive new and interesting identities between Sheffer sequences of polynomials. An application to normal ordering is presented.

Modified approach to generalized Stirling numbers via differential operators

Abstract In this paper we give a modified approach to the generalized Stirling numbers of the second kind S r , s ( n , k ) and S r , s ( k ) . These numbers were firstly defined by Carlitz and recently studied extensively by Blasiak, Penson and Solomon. This approach depends on the previous results obtained by Carlitz, Toscano and Cakic. We show that Blasiak’s results can be investigated from Carlitz and Cakic’s results. Some interesting combinatorial identities are obtained.

Enumeration of (k,2)-noncrossing partitions

A set partition is said to be (k,d)-noncrossing if it avoids the pattern 12...k12...d. We find an explicit formula for the ordinary generating function of the number of (k,d)-noncrossing partitions of {1,2,...,n} when d=1,2.