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+#+title: List of Notations
+
+#+date: 2019-03-15
+
+Here I list meanings of notations that may have not been explained
+elsewhere.
+
+- \(\text{ty}\): type. Given a word \(w \in [n]^\ell\),
+  \(\text{ty} w = (m_1, m_2, ..., m_n)\) where \(m_i\) is the number of
+  \(i\)'s in \(w\). For example
+  \(\text{ty} (1, 2, 2, 1, 4, 2) = (2, 3, 0, 1)\). The definition of
+  \(\text{ty} T\) for a tableau \(T\) is similar.
+- \([n]\): for \(n \in \mathbb N_{>0}\), \([n]\) stands for the set
+  \(\{1, 2, ..., n\}\).
+- \(i : j\): for \(i, j \in \mathbb Z\), \(i : j\) 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 \(A\).
+- \(a \wedge b\): \(\min\{a, b\}\).
+- \(a \vee b\): \(\max\{a, b\}\).
+- \((\alpha)_+\): the positive part of \(\alpha\),
+  i.e. \(\alpha \vee 0\).
+- \((\alpha)_-\): the negative part of \(\alpha\),
+  i.e. \((- \alpha)_+\).