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Background Let $\mu = (\mu_1, \ldots, \mu_k)$ be a partition, meaning that $\mu_1 \geq \ldots \mu_k \geq 1$. The Young diagram associated to $\mu$ is given by the set $(r,c) \in \mathbb{N} \times \mat …

So I just learn about Jucys-Murphy elements. They are elements of $\mathbb{C}[\mathfrak{S}_n]$, the group algebra of the symmetric group, defined as: $$ X_i = \displaystyle \sum_{k=1}^{i-1} (k,i) $$ f …

I am working on an equation that would be solved if I show the following. Let $k \geq l$, and consider the set of $\{0,1\}$-matrices of size $k \times l$ with exactly $i$ 1’s. Consider the subset $\ma …

I am a PhD student. During my researches, I often have to deal with inequalities involving sums of binomial coefficients, where the sums are indexed by some set of integer compositions. For example, l …

Littlewood-Richardson coefficients $c_{\nu\mu}^{\lambda}$ (where $\nu$,$\mu$ and $\lambda$ are integer partitions such that $|\nu| + |\mu| = |\lambda|$) are well-known coefficients appearing in variou …

Let $s_{\lambda}$ be a Schur function. The set of all such functions are known to be a linear basis of the algebra of symmetric functions. The Littlewood-Richardson coefficients $c_{\nu\mu}^{\lambda}$ …

I have always see the following Murnaghan-Nakayama rule for a partition $\lambda$ and a permutation $\sigma \in \mathfrak{S}_n$ of cycle structure $(\sigma_1, ..., \sigma_n)$: $$ \chi_{\lambda}(\sigma …