Number of semistandard tableaux of all possible shapes fitting within some rectangle

Question

Suppose $n$ and $k$ are two integers. Then I am interested in having a closed form for the sum $$\sum_{\lambda \subset k \times n} S_\lambda (\mathbb{C}^n),$$ where $S_\lambda$ denotes the Schur functor and $\lambda$ is some partition, and the function should only depend on $n$ and $k$. Concretely, this comes down to enumerating all possible semistandard tableaux (with maximal entry $n$) of shape $\lambda$ where $\lambda$ is allowed to range through all possible diagrams fitting within the $k \times n$ diagram. For small values of $n$ and $k$ the sequences one gets can be found in OEIS but outside of the $k = 1,2$ it seems like closed forms are hard to come by (I expect such closed forms will involve superfactorials/analogues thereof). This seems like a rather natural combinatorial problem that I've been unable to find any mention of in the literature (potentially I'm using the wrong terminology).

I am aware of the fact that $\operatorname{Sym} (\mathbb{C}^n \oplus \bigwedge^2 \mathbb{C}^n) = \bigoplus_{\lambda} S_\lambda (\mathbb{C}^n)$, but of course the problem with this direct sum is that there is no restriction on the rows of the partitions that may appear.

Answer 1

I'm converting my comments to an answer.

Macdonald in his textbook on symmetric functions proved this identity of Schur functions: $$ \sum_{\lambda : \lambda_1 \leq k} s_{\lambda}(x_1,\ldots,x_n) = \frac{\det_{1\leq i,j \leq n}(x_i^{k+2n-j}-x_i^{j-1})}{\prod_{i=1}^{n}(x_i-1)\prod_{1\leq i < j \leq n}(x_i-x_j)(x_ix_j-1)}. \tag{*}$$ In fact he showed this sum is essentially an odd orthogonal character, and the determinant then follows from the Weyl character formula; for more details, see the discussion in the introduction of the paper by Okada.

Since this identity (*) is a generalization of Littlewood's identity $$\sum_{\lambda}s_{\lambda}(x_1,\ldots,x_n) = \prod_{i=1}^{n}\frac{1}{1-x_i} \prod_{1\leq i < j \leq n} \frac{1}{1-x_ix_j}$$ it has been called a "bounded Littlewood identity"; see the paper by Rains and Warnaar.

Taking a principal specialization of (*) gives $$\sum_{\lambda:\lambda_1\leq k} s_{\lambda}(q,q^3,\ldots,q^{2n-1}) = \prod_{i=1}^{n}\frac{1-q^{k+2i-1}}{1-q^{2i-1}} \prod_{1\leq i < j \leq n} \frac{1-q^{2(k+i+j-1)}}{1-q^{2(i+j-1)}}.$$ Specializing even further $q=1$ gives that the number of SSYTs of a shape $\lambda$ with $\lambda_1\leq k$ and with entries in $\{1,\ldots,n\}$, i.e., the number you are interested in, is: $$\prod_{1\leq i \leq j \leq n}\frac{k+i+j-1}{i+j-1}.$$ Note this is the same as the number of $n \times n$ symmetric plane partitions with maximum entry at most $k$. This product formula for bounded symmetric plane partitions was conjectured by MacMahon and proved independently by Macdonald and Andrews. Indeed, the arguments above prove MacMahon's conjecture, because it is easy to find a bijection from symmetric plane partitions to that set of SSYTs: the portion of a plane partition above the diagonal can be viewed as a Gelfand-Tsetlin pattern encoding the SSYT.

Macdonald, Ian Grant, Symmetric functions and Hall polynomials., Oxford: Clarendon Press. x, 475 p. (1995). ZBL0824.05059.

Okada, Soichi, Intermediate symplectic characters and shifted plane partitions of shifted double staircase shape, Combinatorial Theory 1, Paper No. 10, 42 p. (2021). ZBL1498.05022.

Rains, Eric M.; Warnaar, S. Ole, Bounded Littlewood identities, Memoirs of the American Mathematical Society 1317. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-4690-1/pbk; 978-1-4704-6522-3/ebook). vii, 115 p. (2021). ZBL1467.05001.