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Recall that a **partition** :math:`\mu` of :math:`n`, one writes :math:`\mu\vdash n` or :math:`n = |\mu|`, is an sequence of intergers:math:`(\mu_0,\mu_1,\dots,\mu_{k-1})` (the :math:`m_i`'s are the **parts** of :math:`\mu`) with :math:`\mu_0\geq\mu_1\geq\dots\geq\mu_{k-1} \geq0` and :math:`n = \mu_0 + \mu_1 + \dots + \mu_{k-1}`. The number :math:`\ell(\mu):= k` of parts of :math:`\mu` is said to be its **length**. A partition :math:`\mu` may also be described as a **Ferrers diagram**, which is the :math:`n`-subset of :math:`\mathbb{N}\times\mathbb{N}` :
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Recall that a **partition** :math:`\mu` of :math:`n`, one writes :math:`\mu\vdash n` or :math:`n = |\mu|`, is an sequence of integers:math:`(\mu_0,\mu_1,\dots,\mu_{k-1})` (the :math:`m_i`'s are the **parts** of :math:`\mu`) with :math:`\mu_0\geq\mu_1\geq\dots\geq\mu_{k-1} \geq0` and :math:`n = \mu_0 + \mu_1 + \dots + \mu_{k-1}`. The number :math:`\ell(\mu):= k` of parts of :math:`\mu` is said to be its **length**. A partition :math:`\mu` may also be described as a **Ferrers diagram**, which is the :math:`n`-subset of :math:`\mathbb{N}\times\mathbb{N}` :
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.. Math::
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\left\{(a,b)|0\leq a \leq\mu_i \text{ and } b < \ell(\mu)\right\}.
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Young Tableaux
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An A-valued **Young tableaux** of **shape** :math:`\mu` is a "filling" of the cells of a Ferrers diagram of :math:`\mu` with elements of an ordered set A. Hence, it is a function :math:`\tau:\mu\rightarrow A`. A tableau is said to be **standard** if it is bijective (hence A has cardinality equal to the number of cells of :math:`\mu`), and its entries on each row (and each column) are stricly increasing from left to right (from bottom to top in french convention). A tableau (not necessarily bijective) is said to be **semistandard** if its entries are weakly increasing from left to right on each row, and strictly increasing on each column. These object can be constructed in the following way.
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An A-valued **Young tableaux** of **shape** :math:`\mu` is a "filling" of the cells of a Ferrers diagram of :math:`\mu` with elements of an ordered set A. Hence, it is a function :math:`\tau:\mu\rightarrow A`. A tableau is said to be **standard** if it is bijective (hence A has cardinality equal to the number of cells of :math:`\mu`), and its entries on each row (and each column) are strictly increasing from left to right (from bottom to top in french convention). A tableau (not necessarily bijective) is said to be **semistandard** if its entries are weakly increasing from left to right on each row, and strictly increasing on each column. These object can be constructed in the following way.
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