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+---
+date: 2019-03-15
+title: List of Notations
+template: default
+---
+
+Here I list meanings of notations that may have not been explained
+elsewhere.
+
+-   $\text{ty}$: type. Given a word $w \in [n]^\ell$,
+    $\ty w = (m_1, m_2, ..., m_n)$ where $m_i$ is the number of $i$\'s
+    in $w$. For example $\ty (1, 2, 2, 1, 4, 2) = (2, 3, 0, 1)$. The
+    definition of $\ty T$ for a tableau $T$ is similar.
+-   $[n]$: for $n \in \mathbb N_{>0}$, $[n]$ stands for the set
+    $\{1, 2, ..., n\}$.
+-   $i : j$: for $i, j \in \intg$, $i : j$ stands for the set
+    $\{i, i + 1, ..., j\}$, or the sequence $(i, i + 1, ..., j)$,
+    depending on the context.
+-   $k = i : j$: means $k$ iterates over $i$, $i + 1$,\..., $j$. For
+    example $\sum_{k = 1 : n} a_k := \sum_{k = 1}^n a_k$.
+-   $x_{i : j}$: stands for the set $\{x_k: k = i : j\}$ or the sequence
+    $(x_i, x_{i + 1}, ..., x_j)$, depending on the context. So are
+    notations like $f(i : j)$, $y^{i : j}$ etc.
+-   $\mathbb N$: the set of natural numbers / nonnegative integer
+    numbers $\{0, 1, 2,...\}$, whereas
+-   $\mathbb N_{>0}$ or $\mathbb N^+$: Are the set of positive integer
+    numbers.
+-   $x^w$: when both $x$ and $w$ are tuples of objects, this means
+    $\prod_i x_{w_i}$. For example say $w = (1, 2, 2, 1, 4, 2)$, and
+    $x = x_{1 : 7}$, then $x^w = x_1^2 x_2^3 x_4$.
+-   $LHS$, LHS, $RHS$, RHS: left hand side and right hand side of a
+    formula
+-   $e_i$: the $i$th standard basis in a vector space:
+    $e_i = (0, 0, ..., 0, 1, 0, 0, ...)$ where the sequence is finite or
+    infinite depending on the dimension of the vector space and the $1$
+    is the $i$th entry and all other entries are $0$.
+-   $1_{A}(x)$ where $A$ is a set: an indicator function, which
+    evaluates to $1$ if $x \in A$, and $0$ otherwise.
+-   $1_{p}$: an indicator function, which evaluates to $1$ if the
+    predicate $p$ is true and $0$ otherwise. Example: $1_{x \in A}$,
+    same as $1_A(x)$.
+-   $\xi \sim p$: the random variable $xi$ is distributed according to
+    the probability density function / probability mass function /
+    probability measure $p$.
+-   $\xi \overset{d}{=} \eta$: the random variables $\xi$ and $\eta$
+    have the same distribution.
+-   $\mathbb E f(\xi)$: expectation of $f(\xi)$.
+-   $\mathbb P(A)$: probability of event $A$.