# Major indices of standard tableaux of shapes obtained from addable cells of a given Young diagram

I have a "very" indirect proof that the following fact is true for every Young diagram $$\lambda \vdash n$$ and every $$r \in \{0,\dotsc,n\}$$:

$$\begin{equation} d_\lambda = \sum_{a \in \mathrm{AC}(\lambda)} | \{T \in \mathrm{SYT}(\lambda \cup a) \, | \, \mathrm{maj}_{n+1}(T) = r \} |, \end{equation}$$ where $$d_\lambda$$ is the dimension of the symmetric group irrep $$\lambda$$ or, in other words, $$d_\lambda = |\mathrm{SYT}(\lambda)|$$; $$\mathrm{SYT}(\lambda)$$ is the set of standard Young tableaux of shape $$\lambda$$; $$\mathrm{AC}(\lambda)$$ is the set of addable cells of the Young diagram $$\lambda$$; $$\lambda \cup a$$ is the Young diagram obtained from $$\lambda$$ by adding the cell $$a$$; $$\mathrm{maj}_{n+1}(T)$$ is the major index of the standard Young tableaux $$T$$ modulo $$n+1$$.

I assume this fact must be proven somewhere in combinatorics literature, but I am unable to find the relevant paper/book. A reference would be highly appreciated.

Moreover, I am interested in understanding the following number for every $$\lambda \vdash n$$, every $$r \in \{0,\dotsc,n\}$$ and every $$d \leq \ell(\lambda)$$: $$\begin{equation} f_{\lambda,d}(r) = \sum_{a \in \mathrm{AC}_d(\lambda)} | \{T \in \mathrm{SYT}(\lambda \cup a) \, | \, \mathrm{maj}_{n+1}(T) = r \} |, \end{equation}$$ where $$\ell(\lambda)$$ is the number of rows of the Young diagram $$\lambda$$ and $$\mathrm{AC}_d(\lambda)$$ is the set of addable cells of the Young diagram $$\lambda$$ with row coordinate at most $$d$$. Clearly, $$f_{\lambda,\ell(\lambda)+1}(r) = d_\lambda$$ for every $$r$$, but what can we say about other values of $$d$$?

In particular, the number $$f_{\lambda,\ell(\lambda)}(r)$$ is especially interesting to me. Note, that understanding $$f_{\lambda,\ell(\lambda)}(r)$$ is equivalent to understanding $$\begin{equation} d_\lambda - f_{\lambda,\ell(\lambda)}(r) = | \{T \in \mathrm{SYT}(\lambda \cup (\ell(\lambda)+1,1)) \, | \, \mathrm{maj}_{n+1}(T) = r \} |, \end{equation}$$ where $$(\ell(\lambda)+1,1)$$ is an addable cell at row $$\ell(\lambda)+1$$ and column $$1$$.

• Joshua Swanson, jpswanson.org should be able to answer, or give a reference. Nov 19 at 10:04
• One observation, which you might already know, is that the number of SYT in your union over $a$ in $AC(\lambda)$ is $(n+1)d_{\lambda}$. This is because they index a basis for the induced representation from $S_n$ to $S_{n+1}$. So your claim is equivalent to knowing that $\mathrm{maj} \mod (n+1)$ is equidistributed. Nov 19 at 11:57
• Using standard results on major indices (Proposition 7.19.11 in Stanley's EC2) and the Pieri rule, your claim is equivalent to the following: For any partition $\lambda$ of $n$, the formal power series $\left(1-q^2\right)\left(1-q^3\right)\cdots\left(1-q^{n+1}\right) s_\lambda\left(1,q,q^2,q^3,\ldots\right)$ is a polynomial in $q$ that is divisible by $1 + q + q^2 + \cdots + q^n$. But this is easy, since the $\left(1-q^{n+1}\right)$ factor on the left hand side can be rewritten as $\left(1-q\right)\left(1 + q + q^2 + \cdots + q^n\right)$, and the ... Nov 19 at 17:42
• ... power series $\left(1-q^2\right)\left(1-q^3\right)\cdots\left(1-q^n\right) \left(1-q\right) s_\lambda\left(1,q,q^2,q^3,\ldots\right)$ is just the generating function for the major index on SYTs of shape $\lambda$ (again Proposition 7.19.11), thus a polynomial. Nov 19 at 17:44
• Quick remark on how I lost the $1-q$ factor at the beginning: Summing over all $a \in \operatorname{AC}\left(\lambda\right)$ is equivalent to summing over all partitions $\mu$ of the form "$\lambda$ plus one box". But on the level of Schur functions, this is equivalent to multiplying by $h_1 = x_1 + x_2 + x_3 + \cdots$ (by the Pieri rule). Upon principal specialization, this means multiplying by $h_1\left(1, q, q^2, q^3, \ldots\right) = 1 / \left(1-q\right)$. Thus, the $1-q$ factor in Proposition 7.19.11 gets cancelled. Nov 19 at 17:46