# List of Notations

Here I list meanings of notations that may have not been explained elsewhere.

• $$\ty$$: Is for 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 $$n \in \pint_{>0}$$, $$[n]$$ 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.
• $$\pint$$: The set of natural numbers / nonnegative integer numbers $$\{0, 1, 2,...\}$$, whereas
• $$\pint_{>0}$$ or $$\pint^+$$: 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
• $$e_i$$: 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 $$0$$.