List of Notations

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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 \(A\).

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