Given a representation in $SU(N)$ with Young tableau $Y$, how many ways are there that I can write $$Y \in y_1\otimes y_2 \otimes \cdots \otimes y_n$$ where $n\leq |Y|$, the number of boxes there are in $Y$ and $y_r$ are smaller Young tableaux, ie of sizes $\leq |Y|$? Here $n$ is not fixed.
Is there an explicit formula for that, or is it connected to something else? Any help is appreciated.
Let the function in question be $c(Y) $. Then $c(\square) = 1$, $c\begin{pmatrix} \square \\\square \end{pmatrix} = 1 = c(\square\square) $, etc.
In fact I think I am interested in the weighted version, with the coefficient $N_{y_1,y_2,...,y_n} ^Y$ attached to the product $y_1\otimes y_2\otimes\cdots \otimes y_n$. Not sure how this is going to change the picture.
I mean I would like to find $$\sum _n \sum_{Y\in y_1\otimes \cdots \otimes y_n} N_{y_1,...,y_n} ^Y $$