# Number of Young Tableaux of size n with a given number of rows and a distinct number of boxes in each row

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 $n$.

• For future readers: There seems to be a confusion between Young tableaux and Young diagrams in this post, and the author seems to be interested in Young diagrams. – darij grinberg Aug 19 '15 at 18:41

This is equivalent to the number of partitions of $n$ into $k$ distinct positive terms.
This is the same as the number of partitions of $n-\frac{k(k-1)}{2}$ into $k$ positive terms, since you can subtract ${k-1}$ from the largest distinct part, ${k-2}$ from the second largest, and so on down to subtracting $0$ from the smallest distinct part.
So you want the co-efficient of $x^{n-k(k-1)/2}$ in the expansion of $$\frac{x^k}{\displaystyle\prod_{i=1}^{k} (1-x^i)}.$$
You can use recursion, as in in my Java applet at http://www.se16.info/js/partitions.htm : so if you ask for partitions of $6$ into exactly $2$ distinct parts you get a result of $2$; asking for partitions with distinct terms of $6000$ with exactly $20$ parts gives $1470489673468747936729666225918528498$.