# List of Notations Published on (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. $$(- \alpha)_+$$.