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