I’m working through Fulton-Harris and I’m kind of “stuck” at the following question. I’m looking for representations of $S_d$, the symmetric group on $d$ letters via Young Tableaux. The question is: “Show that for general $d$, the standard representation $V$ of $S_d$ corresponds to the partition $d = (d-1)+1$.” When I look at the hint […]
From the wiki page Catalan number, we know the number of Young tableaux whose diagram is a 2-by-n rectangle given $2n$ distinct numbers is $C_n$. In general, given $m\times n$ distinct numbers, how many Young tableaux whose diagram is a $m\times n$ rectangle are there? Also, what if these numbers can be repeated? Many thanks.