| symbol | latex code | explanation | |
|---|---|---|---|
n! | n! | Factorial of n | |
nPk | {n \mathcal{P} k} | Number of permutations for k elements chosen from n | |
nCk | \binom{n}{k} | Number of combinations for k elements chosen from n | |
P(A) | P(A) | Probability of event A | |
|A| | |A| | Cardinality of set A (number of elements) | |
∑ | \sum | Summation operator | |
Π | \prod | Product operator | |
(x + y)ⁿ | (x + y)^n | Expansion of a binomial raised to the nth power | |
∑ₖ₌₀ⁿ (nCk) xⁿ⁻ᵏ yᵏ | \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k | Binomial theorem expansion | |
nCk = n! / (k!(n − k)!) | \binom{n}{k} = \frac{n!}{k!(n-k)!} | Formula for binomial coefficient | |
n! | n! | Number of permutations for n elements | |
P(n,k)=(n−k)!n! | P(n, k) = \frac{n!}{(n-k)!} | Number of permutations for k elements chosen from n | |
C(n,k)=k!(n−k)!n! | C(n, k) = \frac{n!}{k!(n-k)!} | Number of combinations for k elements chosen from n | |
nHk | \binom{n+k-1}{k} | Number of ways to distribute k identical items into n distinct groups (combinations with replacement) | |
P(n,r) | P(n,r) | ||
B(n) | B(n) | Bell number, the number of ways to partition a set of n elements | |
S(n,k) | S(n, k) | Stirling number of the second kind, the number of ways to partition a set of n elements into k non-empty subsets | |
|A ∪ B| = |A| + |B| − |A ∩ B| | |A \cup B| = |A| + |B| - |A \cap B| | Inclusion-Exclusion principle for two sets | |
|A₁ ∪ A₂ ∪ ⋯ ∪ Aₙ| | |A_1 \cup A_2 \cup \cdots \cup A_n| | Cardinality of the union of multiple sets | |
∑(−1)ⁿ⁺¹ |Aᵢ₁ ∩ ⋯ ∩ Aᵢₖ| | \sum (-1)^{n+1} |A_{i_1} \cap \cdots \cap A_{i_k}| | General formula for Inclusion-Exclusion principle | |
G(x) = ∑ₙ₌₀ aₙxⁿ | G(x) = \sum_{n=0}^\infty a_n x^n | Ordinary generating function | |
H(x) = ∏ₙ₌₁ (1 − xⁿ)⁻¹ | H(x) = \prod_{n=1}^\infty (1 - x^n)^{-1} | Exponential generating function | |
F(x, y) = ∑ₘₙ aₘₙ xᵐ yⁿ | F(x, y) = \sum_{m, n} a_{m, n} x^m y^n | Bivariate generating function | |
p(n) | p(n) | Number of partitions of integer n | |
q(n) | q(n) | Number of distinct partitions of n | |
λ ⊢ n | \lambda \vdash n | Partition λ of n | |
n = λ₁ + λ₂ + ... + λₖ | n = \lambda_1 + \lambda_2 + \cdots + \lambda_k | A specific partition of n | |
Cₙ | C_n | nth Catalan number | |
Cₙ = (2n)! / ((n+1)!n!) | C_n = \frac{(2n)!}{(n+1)!n!} | Formula for Catalan number | |
C₀, C₁, C₂, ... | C_0, C_1, C_2, \ldots | Sequence of Catalan numbers | |
aₙ = aₙ₋₁ + aₙ₋₂ | a_n = a_{n-1} + a_{n-2} | Example recurrence relation (Fibonacci numbers) | |
T(n) = 2T(n/2) + n | T(n) = 2T(n/2) + n | Divide-and-conquer recurrence relation | |
a₀ = c₀, aₙ = Σₖ aₖfₖₙ | a_0 = c_0, a_n = \sum_k a_k f_{kn} | General recurrence relation | |
G = (V, E) | G = (V, E) | Graph G with vertices V and edges E | |
|V| | |V| | Number of vertices in a graph | |
|E| | |E| | Number of edges in a graph | |
deg(v) | \text{deg}(v) | Degree of vertex v | |
χ(G) | \chi(G) | Chromatic number of a graph | |
T(G) | T(G) | Number of spanning trees in graph G |