I would like to know if there is a formula for the number of Young tableaux of size $n$ with a given number of rows, each row having a distinct number of boxes. I have seen the Hook length formula which gives the number of Young tableaux of shape $\lambda$, but this doesn't seem to help me as I'm not interested in a Young filling.
For example, say I want to know how many Young tableaux of size 6 there are with 2 rows. Then the possible tableaux are those with shape (4,2) and (5,1), hence there are 2 Young Tableaux of size 6 with 2 rows.
Thank you very much in advance for any help.
Edit: I've also seen the recurrence relation $a(n+1)=a(n)+na(n-1)$, with $a(0)=a(1)=1$, where $a(n)$ gives the number of Young tableaux of size $n$.