 # Applied Combinatorics

## AppendixCList of Notation

Symbol Description Location
$$n!$$ $$n$$ factorial Paragraph
$$P(m,n)$$ number of permutations Paragraph
$$\binom{n}{k}$$ binomial coefficient Paragraph
$$C(n,k)$$ binomial coefficient (inline) Paragraph
$$\binom{n}{k_1,k_2,k_3,\dots,k_r}$$ multinomial coefficient Paragraph
$$\cgP$$ polynomial time problems Paragraph
$$\cgN\cgP$$ nondeterministic polynomial time problems Paragraph
$$\deg_\bfG(v)$$ degree of vertex $$v$$ in graph $$\bfG$$ Paragraph
$$\bfK_n$$ complete graph on $$n$$ vertices Paragraph
$$\bfI_n$$ independent graph on $$n$$ vertices Paragraph
$$\bfP_n$$ path with $$n$$ vertices Paragraph
$$\bfC_n$$ path with $$n$$ vertices Paragraph
$$\chi(\bfG)$$ chromatic number of a graph $$\bfG$$ Paragraph
$$\omega(\bfG)$$ clique number of $$\bfG$$ Paragraph
$$x\| y$$ $$x$$ and $$y$$ are incomparable Paragraph
$$\height(\bfP)$$ height of poset $$\bfP$$ Paragraph
$$\width(\bfP)$$ width of poset $$\bfP$$ Paragraph
$$D(x), D(S),D[x], D[S]$$ down set Paragraph
$$U(x), U(S), U[x], U[S]$$ up set Paragraph
$$\bfn$$ chain with $$n$$ points Paragraph
$$\bfP+\bfQ$$ disjoint sum of posets Paragraph
$$\phi(n)$$ Euler $$\phi$$ function Paragraph
$$\binom{p}{k}$$ generalized binomial coefficient Definition 8.9
$$Af(n)$$ advancement operator applied to $$f(n)$$ Paragraph
$$P(A|B)$$ probability of $$A$$ given $$B$$ Paragraph
$$C(X,k)$$ family of all $$k$$-element subsets of $$X$$ Paragraph
$$R(m,n)$$ Ramsey number Paragraph
$$\langle C\rangle$$ equivalence class of $$C$$ Paragraph
$$\stab_G(C)$$ stabilizer of $$C$$ under action of $$G$$ Paragraph
$$\overline{E}$$ complement of event $$E$$ Paragraph
$$x\in X$$ $$x$$ is a member of the set $$X$$ Paragraph
$$x\notin X$$ $$x$$ is not a member of the set $$X$$ Paragraph
$$X\cap Y$$ intersection of $$X$$ and $$Y$$ Paragraph
$$X\cup Y$$ union of $$X$$ and $$Y$$ Paragraph
$$\emptyset$$ empty set Paragraph
$$\posints$$ set of positive integers Paragraph
$$\ints$$ set of integers Paragraph
$$\rats$$ set of rational numbers Paragraph
$$\reals$$ set of real numbers Paragraph
$$\nonnegints$$ set of non-negative integers Paragraph
$$[n]$$ $$\{1,2,\dots,n\}$$ Paragraph
$$X\subseteq Y$$ $$X$$ is a subset of $$Y$$ Paragraph
$$X\subsetneq Y$$ $$X$$ is a proper subset of $$Y$$ Paragraph
$$X\times Y$$ cartesian product of $$X$$ and $$Y$$ Paragraph
$$f\colon X\rightarrow Y$$ $$f$$ is a function from $$X$$ to $$Y$$ Paragraph
$$f:X\injection Y$$ $$f$$ is an injection from $$X$$ to $$Y$$ Paragraph
$$f\colon X\surjection Y$$ $$f$$ is a surjection from $$X$$ to $$Y$$ Paragraph
$$f\colon X\bijection Y$$ $$f$$ is a bijection from $$X$$ to $$Y$$ Paragraph
$$|X|$$ cardinality of set $$X$$ Paragraph