An \(X\)-string of length \(n\) is written \(x_1x_2x_3 \cdots x_n\) for \(x_i \in X\), where \(X\) is an alphabet its elements are characters. \[s: [n] \mapsto X, \ \ \ [n] = \{ 1, 2,3 \cdots, n \}\]
An enumeration such that:
Order of items matters.
Counts include duplication or removals of items.
If \(X\) is a finite set, the number of all \(X\)-strings of length \(n\) will be \(|X|^n\).
If \(X\) is an \(n\) element set, the number of \(X\)-strings of length \(m\) is, \[P(n,m) = \frac{n!}{(n-m)!}\]
Order of items matters.
Counts do not include duplication or removals of items.
Collection of counts could be stored in arrays.
The number of combinations of \(n\) objects taken \(k\) at a time. Given \(n, k \in \mathbb Z, 0 \leq k \leq n\) then the binomial coefficient is, \[C(n, k) = \binom{n}{k} = \frac{P(n,k)}{k!} = \frac{n!}{k!(n-k)!} = \binom{n}{n-k}\]
Order of items doesn’t matter.
Counts do not include duplication or removals of items.
Collection of counts could be stored in sets.
\[\binom{n+1}{k} = \binom{n}{k} + \binom{n}{k+1}\]
\[1 +2 +3 + \cdots + n = \frac{n(n+1)}{2}\]
\[1 +3 + 5 + \cdots + 2n -1 = n^2\]
A triangular array made up of the binomial coefficients in which the entry in the \(n\)th row and \(k\)th column is \({\tbinom {n}{k}}\).
\[\binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}\]
\[\sum_{i=0}^n \binom{n}{i} = 2^n\]
Let \(x,y \in \mathbb R, x+y \neq 0\). Then \(\forall n \in \mathbb Z^{+}\), \[(x + y)^n = \sum_{i=0}^n \binom{n}{i} x^{n-i} y^i\]
The number of combinations in which \(k\) elements are chosen from subsets of \(X\) with \(|X| = n\) will be, \[\binom{n}{k_1, k_2, \cdots, k_n} = \frac{n!}{k_1! k_2! \cdots k_n!}\]
Stars and bars is a graphical aid for deriving certain combinatorial theorems. It can be used to solve many simple counting problems, such as how many ways there are to put n indistinguishable balls into k distinguishable bins.
For any pair of positive integers n and k, the number of k-tuples of positive integers whose sum is n is equal to the number of \((k - 1)\)-element subsets of a set with \(n - 1\) elements, i.e. \(\binom{n - 1}{k-1}\).
For any pair of positive integers n and k, the number of k-tuples of non-negative integers whose sum is n is equal to the number of multisets of cardinality \(k - 1\) taken from a set of size \(n + 1\), i.e. \(\binom{n+1}{k-1}\).
Let \(x_1,\cdots x_n \in \mathbb R^+\). Then \(\forall n \in \mathbb N^+\), \[(x_1 + \cdots x_n)^n = \sum_{k_1 + ... + k_r = n} = \binom{n}{k_1, \cdots k_r} x_1^{k_1} x_2 ^{k_2} \cdots x_r^{k_r}\]
A sequence of ordered pairs \((m_1, n_1), (m_2, n_2), \cdots, (m_t, n_t)\) such that either:
\(m_{i+1} = m_{i}+1\) and \(n_{i+1} = n_{i}\)
\(m_{i+1} = m_i\) and \(n_{i+1} = n_i +1\).
The construction of lattice paths forms a bijection with \(X\)-strings where \(X = \{ H, V\}\) with \(H,V\) encoding horizontal or vertical moves. The number of lattice paths from \((m_1, n_1)\) to \((m_2,n_2)\) is, \[\binom{m_2 - m_1 + m_2 - m_1}{m_2-m_1}.\]
The number of lattice paths from \((0,0)\) to \((n,n)\) that do not go above the diagonal line \(y = x\), also known as Dyck paths, are equal to the \(n\)th Catalan number, \[C(n) = \frac{1}{n+1} \binom{2n}{n}\]
The rule of sum or addition principle is a basic counting principle. Stated simply, it is the idea that if we have A ways of doing something and B ways of doing another thing and we can not do both at the same time, then there are A + B ways to choose one of the actions.
The inclusion–exclusion principle is a counting technique that generalizes the method of obtaining the number of elements in the union of two finite sets without double counting elements, \[|A\cup B|=|A|+|B|-|A\cap B|.\] The principle is particularly useful in the case of three sets, i.e. for the sets \(A\), \(B\) and \(C\), the count of the union is given by, \[|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|.\]
For a positive integer \(n\geq2\), Euler’s Totient function is defined by \[\phi(n)=|\{m\in \mathbb N: m\le n, \gcd(m,n)=1\}|.\]
\(\forall A \subset \mathbb N^+, A \neq \emptyset\), \(A\) must have a minimum element. This statement is equivalent to mathematical induction.
Let \(S_n\) be an open statement involving a positive integer \(n\). If \(S_1\) is true, and if for each positive integer \(k\), assuming that the statement \(S_k\) is true implies that the statement \(S_k + 1\) is true, then \(S_n\) is true for every positive integer \(n\).
Given \(m\) items and \(n\) containers, there exists a containers with at least \(\lceil \frac{m}{n} \rceil\) items.
Expressed in terms of functions and their domains and ranges: if \(f: X \mapsto Y\) and \(|X| > |Y|\) then there exists a \(y \in Y\) and \(x, x' \in X, x \neq x'\) so that \(f(x) = f(x') = y\)
If \(m,n \in \mathbb Z^+\), then any sequence of \(mn+1\) distinct real numbers either has an increasing sub-sequence of \(m+1\) items or a decreasing sub-sequence of \(n+1\) terms.
\[\phi_n = \phi_{n-1} +\phi_{n-2}\text{; } \phi_1 = 1 \text{, } \phi_2 = 2\]
The Euclidean algorithm is an efficient method for computing the greatest common divisor (GCD) of two integers.
Given \(m = q_i n_i + r_i\) where \(m_i,n_i \in \mathbb Z^+\) with \(0 \leq r_i < n_i\) we see that \(r_i\) represents the remainder and \(q_i\) represents the quotient. We use the fact that \(\gcd(m, n_i) = \gcd(n_i, r_i)\) to iteratively calculate the \(\gcd\) on the decomposition of the \(n_i\) term and the remainder \(r_i\).
Let \(G = (V, E)\) be a graph with \(V\) verticies and \(E\) edges.
\(H = (W, F)\) is a subgraph of \(G\) if \(W \subseteq V\) and \(F \subseteq E\). That is, a subgraph does not contain any additional vertices or edges but may contain fewer.
\(H = (W, F)\) is an induced subgraph if \(W \subseteq V\) and \(F = \{ xy \in E : xy \in W \}\). That is, an induced subgraph is formed from a subset of the vertices of the graph with all of the edges connecting pairs of vertices in the subset being kept from the original graph.
\(H = (W, F)\) is spanning when \(W = V\).
\(G\) is isomorphic to \(H\), denoted by \(G \cong H\), if there exists an edge-preserving bijective mapping, \(f: V \mapsto W\).
The subdivision of an edge \(e\) with endpoints \(\{u,v\}\) yields a graph containing one new vertex \(w\) with an edge set replacing \(e\) by two new edges, \(\{u,w\}\) and \(\{w,v\}\).
Two graphs \(G\) and \(H\) are homeomorphic if there is a graph isomorphism from some subdivision of \(G\) to some subdivision of \(H\).
\(G\) contains \(H\) if there exists a subgraph of \(G\) that is isomorphic to \(H\).
A complete graph is a graph in which each pair of graph vertices is connected by an edge, i.e. \(x, y \in E\) for all distinct \(x,y \in V\).
The complete graph with n graph vertices is denoted as \(K_n\). \(K_n\) has \(\binom{n}{2}\) edges.
An inedpendent graph, denoted \(I_n\), is a graph with \(xy \not\in E\) for all distinct pairs of vertices \(xy \in V\)
\(P_n = \{ x_1, x_2, \cdots x_n \}\) is a path of length \(n-1\).
\(C_n = \{ x_1, x_2, \cdots x_n, x_1 \}\) is a cycle with length \(n\).
A graph \(G\) is connected if every vertex can be reached by any other vertex in the graph, i.e a path exists connecting \(v_1\) to \(v_2\) for all \(v_1,v_2 \in V\).
A maximal connected subgraph of a disconnected graph.
A graph that does not contain any cycles.
The number of components is a forrest is \(|V| - |E|\)
A connected acyclic graph.
If \(G\) is a connected tree and \(H\) is a spanning subgraph, then we can call \(H\) a spanning tree.
The degree sum formula states that, given a graph \(G=(V,E)\),
\[\sum _{v\in V}\deg(v)=2|E|.\]
The formula implies that in any undirected graph, the number of vertices with odd degree is even, which is known as the Handshaking lemma. The name comes from a popular problem: in any group of people the number of people who have shaken hands with an odd number of other people from the group must be even.
A Eulerian cycle is a cycle that contains every edge in a given graph exactly once. A Eulerian graph is a graph that contains a Eulerian cycle.
A graph admits a Eularian cycle if and only if all vertices have even degree
A Hamiltonian cycle is a cycle that traverses each vertex exacltly once. A Hamiltonian graph is a graph that contains a Hamiltonian cycle.
If a graph has \(n\) vertices and \(\forall v \in V, \deg(v) \geq \lceil \frac{n}{2} \rceil\), then it must be Hamiltonian.
(Ore’s Theorem) If for all non-adjacent vertices \(v,w \in V\), \(deg(v) + deg(w) \geq n\), then \(G\) must be Hamiltonian.
A graph is bipartite if there exists a partition of \(V\) so that the resulting induced subgraphs are independent.
A graph is complete bipartite, denoted \(K_{m,n}\), if it is bipartite with \(V = V_1 \cup V_2, |V_1| = m, |V_2|= n\) and has edges \(xy\) for all \(x \in V_1, y \in V_2\).
A graph can be colored by assigning a color or label to each of its vertices. A proper coloring exists when no adjacent vertices have the same color.
The chromatic number of a graph, denoted \(\chi (G)\), is the smallest number of \(k\) colors so that \(G\) is k-colorable.
Recall \(C_n\) is a cycle of \(n\) vertices. Then, \(\chi (C_n) = \begin{cases} 2 ,\ \ \text{if } n \text{ is even} \\ 3,\ \ \text{if } n \text{ is odd } \end{cases}\)
The clique number of a graph, denoted \(\omega(G)\), is the maximum number \(n\) so that \(G\) has a subgraph isomorphc to the complete graph \(K_m\).
For any graph \(G\), $ (G) (G)$.
If $ (G) = (G)$, then \(G\) is called perfect or reduced.
For any \(t \geq 3\), there exists a graph \(G\) so that \(\omega(G) = 2\) and \(\chi(G) = t\).
This means that graphs can have an arbitrarily large chromatic number. If the original graph is triangle-free, a Mycielskian graph may be constructed to increase the graph’s chromatic number.
A graph is planar if it has a drawing or embedding so that its edges only intersect one another at one of the graph’s vertices.
Given a finite, connected, planar graph where \(v\) is the number of vertices, \(e\) is the number of edges and \(f\) is the number of faces, \[v-e+f=2.\]
A planar graph on \(n\) verticies has at most \(3n-6\) edges, for \(n \geq 3\).
A graph is planar if and only if it does not contain a subgraph homeomorphic to \(K_5\) or \(K_{3,3}\).
Every planar graph has chromatic number of at most 4.
A Petersen graph has the following properties: \(|V| = 10, |E| = 15\), there are no cycles of length 3 or 4, and \(\deg(v) = 3, \forall v \in V\). A Petersen graph is never Hamiltonian.
A dense graph is a graph in which the number of edges is close to the maximal number of edges. For undirected simple graphs, the graph density is: \[D={\frac {|E|}{\binom {|V|}{2}}}={\frac {2|E|}{|V|(|V|-1)}}\]
Conversely, a sparse graph is a graph in which the number of edges is close to the minimal number of edges. A sparse graph can be a disconnected graph.
Given a sequence \(\sigma=\{a_n:n\ge0\}\) of real numbers, we define the ordinary generating function \(F(x)\) of the sequence \(\sigma\) as, \[F(x)=\sum_{n=0}^\infty a_n x^n.\]
This differs from a formal power series in that we are not concerned with the series convergence or divergence and are generally not interested in substituting a value of \(x\) and obtaining a specific value for \(F(x)\). Instead we may use \(F(x)\) to encode and generate the sequence \(\sigma\).
The geometric series: \[\frac {1}{1-x}=\sum _{n=0}^{\infty }x^{n}=1+x+x^{2}+x^{3}+\cdots\]
For all real \(p\) with \(p\neq 0\), \[(1+x)^p=\sum_{n=0}^\infty\binom{p}{n}x^n.\]
A partition \(P\) of an integer is a collection of (not necessarily distinct) positive integers such that, \[\sum_{i\in P} i = n.\] We can find the number of partitions of a given integer using a generator function \[P(x) = \left(\sum_{m=0}^\infty x^{m}\right)\left(\sum_{m=0}^\infty x^{2m}\right) \left(\sum_{m=0}^\infty x^{3m}\right)\cdots \left(\sum_{m=0}^\infty x^{km}\right)\cdots = \prod_{m=1}^\infty\frac{1}{1-x^m}.\]
For each \(n\geq 1\), the number of partitions of \(n\) into distinct parts is equal to the number of partitions of \(n\) into odd parts.
The exponential generating function for the sequence \(\{a_n\colon n\geq 0\}\) is \(\displaystyle \sum_n \frac{a_n x^n}{n!}\).
The exponential function formula: \[e^{x} =\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots.\]
A recurrence tells us how to compute the \(n\)th term of a sequence given some number of previous values. A recursion is linear when it has the following form, \[c_0a_{n+k}+ c_1a_{n+k-1} + \dots+c_ka_{n} = g(n),\] where \(k \geq 1\) is an integer, \(c_0,c_1,\dots,c_k\) are constants with \(c_0, c_k \neq 0\), and \(g: \mathbb Z \rightarrow \mathbb R\) is a function.
A linear equation is homogeneous if the function \(g(n)\) is the zero function is non-homogeneous otherwise. The process of solving recurrence equations is similar to solving ordinary differential equations as both functions express future states in relation to differences to their previous states.
The advancement operator is a linear map that shifts or advances a function’s input forward by one, i.e. \(A: \mathbb R \mapsto \mathbb R\), where \((Af)(n) = f(n+1)\).
It is commutative and has the properties that \(A^k = \underbrace{A \circ \cdots \circ A }_\text{k times}\) and \((A^k f)(n) = f(n+k)\).
The set of all solutions to the homogeneous linear equation of degree \(k\) is a \(k\)-dimensional subspace of \(V\).
This means that we may solve a linear recurrence stated in terms of advancement operations by finding a basis for the vector space of solutions.
Let \(r \neq 0\) , and let \(f\) be a solution to the operator equation \((A - r)f=0\). If \(c=f(0)\), then \(f(n)=c r^n\) for every \(n \in \mathbb Z\).
This means that the roots of the advancement operator equation will give us solutions to its corresponding recurrence of the form \(f(n) = c r^n\).
Consider a non-homogeneous operator equation and let \(W\) be the subspace of \(V\) consisting of all solutions to its corresponding homogeneous equation.
If \(f_0\) is a solution to the homogeneous equation, then every solution \(f\) to has the form \(f = f_0 + f_1\) where \(f_1 \in W\) is a particular solution.
If the polynomial in the advancement operator has distinct roots we may use the following theorem,
Consider the following advancement operator equation \(p(A) f = (A-r_1)(A-r_2)\cdots(A-r_k)f=0\). with \(r_1 , r_2 , \cdots , r_k\) distinct non-zero constants. Then every solution has the form \(f(n) = c_1 r_1^n + c_2 r_2^n + c_3 r_3^n + \cdots + c_k r_k^n\).
If there are repeated roots we use the following theorem,
Let \(k \geq 1\) and consider the equation \(( A - r )^k f = 0\). Then the general solution has the form \(f(n) = c_1 r_n + c_2 n r_n + c_3 n^2r^n + c_4 n^3 r^n + \cdots + c_k n^{k - 1} r ^n\).
Alternatively, we may use generating function to solve recursive equations. This approach does not require determining an appropriate form for the particular solution but instead the method of generating functions often requires that the resulting generating function be expanded using partial fractions.
Generating functions can be used in solving certain nonlinear recurrence equation, in particular, we can use it to determine the number of rooted, unlabeled, binary, ordered trees. A tree is rooted if we have designated a special vertex called its root. An unlabeled tree is one in which we do not make distinctions based upon names given to the vertices. A binary tree is one in which each vertex has 0 or 2 children, and an ordered tree is one in which the children of a vertex have some ordering (first, second, third, etc.).
The generating function for the number \(c_n\) of rooted, unlabeled, binary, ordered trees with \(n\) leaves is \[C(x) = \frac{1-\sqrt{1-4x}}{2} = \sum_{n=1}^\infty \frac{1}{n}\binom{2n-2}{n-1}x^n.\]
Notice that \(c_n\) is a Catalan number, which also counts the number of lattice paths that do not cross the diagonal line \(y=x\).
Keller, M.T. and Trotter, W.T., Applied Combinatorics, Open Textbook Library, ISBN9781534878655. http://www.rellek.net/book/app-comb.html