All Questions

In a complementary proof for a matrix determinant of $a_{i,j}=\binom{n-1+i}j$, raised by BillyJoe, I showed the more general evaluation $$\det\left(\binom{i+p}{j+k-1}\right)_{1\leq i,j\leq m} =\prod_{...

Given a partition $\lambda$, the Schur polynomials can be defined, among many other ways, as $$S_{\lambda}(\xi_1,\dots,\xi_a)=\frac{\det\left(\xi_i^{\lambda_j+a-j}\right)_{i,j=1}^a}{\det\left(\xi_i^{a-...

Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. A partition $\lambda$ is called a $t$-core if none of its hook lengths are multiples of $t$. QUESTION. Consider the ...

Given an (unrestricted) integer partition $\lambda$ of $n$, let $f_{\lambda}$ denote the number of standard Young tableaux (SYT) of shape the Young diagram $Y(\lambda)$ of $\lambda$. Then, $$\sum_{\...

This is the second installment of my earlier MO question. Let $s_{\lambda}$ denote the Schur polynomial associated to a partition $\lambda$. Denote the set of all partitions with distinct parts by $\...

In my earlier quest, we looked at $\chi_{\mu}^{\lambda}=$value of an irreducible character of the symmetric group $\frak{S}_n$, where $\mu$ and $\lambda$ are (unrestricted) partitions of $n$. Then, ...