Silvia Heubach

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.

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.

INVERSIONS IN COMPOSITIONS OF INTEGERS

A composition of the positive integer n is a representation of n as an ordered sum of positive integers n = a1 + a2 + … + am. It is well known that there are 2 n −1 compositions of n. An inversion in a composition is a pair of summands {ai,aj } for which i < j and a i > aj . The number of inversions of a composition is an indication of how far the composition is from a partition of n, which by convention uses a sequence of nondecreasing summands and has no inversions. We consider counting techniques for determining both the number of inversions in the set of compositions of n and the number of compositions of n with a given number of inversions. We provide explicit bijections to resolve several conjectures, and also consider asymptotic results.

A stochastic model for the movement of a white blood cell

We present a stochastic model for the movement of a white blood cell both in uniform concentration of chemoattractant and in the presence of a chemoattractant gradient. It is assumed that the rotational velocity is proportional to the weighted difference of the occupied receptors in the two halves of the cell and that each of the receptors stays free or occupied for an exponential length of time. We define processes corresponding to a cell with 2nP + 1 receptors (receptor sites). In the case of constant concentration, we show that the limiting process for the rotational velocity is an Omstein-Uhlenbeck process. Its drift coefficient depends on the parameters of the exponential waiting times and its diffusion coefficient depends in addition also on the weight function. In the inhomogeneous case, the velocity process has a diffusion limit with drift coefficient depending on the concentration gradient and diffusion coefficient depending on the concentration and the weight function.

A Misère-Play \(\star \)-Operator

We study the \(\star \)-operator (Larsson et al. in Theoret. Comp. Sci. 412:8–10, 729–735, 2011) of impartial vector subtraction games (Golomb in J. Combin. Theory 1:443–458, 1965). Here, we extend the operator to the misere-play convention and prove convergence and other properties; notably, more structure is obtained under misere-play as compared with the normal-play convention (Larsson in Theoret. Comput. Sci. 422:52–58, 2012).