By Geir T. Helleloid

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**Example text**

A tournament is a directed graph on the vertex set [n] with exactly one of the arcs (i, j) and (j, i) for each i = j. The weight w(a) of an arc (i, j) is xi and the sign sgn(a) of the arc is +1 if i < j and −1 if i > j. The weight of a tournament T is w(T ) = w(a) and the sign of a tournament T is sgn(T ) = sgn(a). The weight of a tournament is xa11 · · · xann , where ai is the out-degree of i. Then A= sgn(T )w(T ). T A tournament is transitive if whenever arcs (i, j) and (j, k) are present, so is (i, k).

31, let f (x) = g(x) = 1. Then h(n) = B(n) (the n-th Bell number). But Ef (x) = ex − 1 and Eg (x) = ex , so the EGF for the Bell numbers is ∞ Eh (x) = ee x −1 = B(n) n=0 xn . n! Roughly, the idea of the Compositional Formula is that if we have a set of objects (like set partitions) that are disjoint unions of connected components (like blocks of a partition), and if there are f (j) possibilities for a component of size j, and there are g(k) ways to stick together k components to form an object, then h(n) is the total number of objects.

Types of questions we can ask about permutation patterns include: 1. How many π ∈ Sn do not contain σ? 2. How many times does π ∈ Sn contain σ? 3. How many distinct patterns does π ∈ Sn contain? 43 M390C Algebraic Combinatorics Fall 2008 Instructor: Geir Helleloid We will only discuss the first question. Several chapters on this subject can be found in Bona [2]. Given a pattern σ, let Nn (σ) denote the number of π ∈ Sn that avoid (do not contain) σ. The first result is easy. 1. Let σ, σ ∈ Sk correspond to permutation matrices that are equivalent under dihedral symmetries.