The starting point is that it is known that the Hankel determinants for the Catalan sequence give the number of nested sequences of Dyck paths. I would like to promote this to symmetric functions.
This is motivated by some representation theory.


The naive idea is to start with the sequence of symmetric functions $s\_{n,n}$ and take the Hankel determinants using the inner product (that is product in the group ring of $S(2n)$) instead of the usual outer product. However this doesn't make sense.


Take the $2 \times 2$ case. Then the naive determinant is
$$ \left|\begin{array}{cc} s\_{n-1,n-1} & s\_{n,n} \\\ s\_{n,n} & s\_{n+1,n+1}\end{array}\right|$$
The inner product of the two diagonal terms is defined but the inner product of the two off-diagonal terms is not.


The idea that I want to test is that this is $\sum\_\lambda s\_\lambda$ where the sum is over the conjugates of the partitions $4^a2^{n-2a}$.


Any suggestions on how to fix this? If this does get fixed then I would like to know how to calculate the result. The difficulty is that I have not seen an implementation of the inner product in the computer algebra systems I use, Magma and Sage (which I think both use the same source for symmetric functions).



Hankel determinants of symmetric functions
It is unlikely to obtain for such a determinant the sum of all Schur functions indexed by partitions of $2n$ with four parts all even or all odd. Indeed, this sum is already equal to the inner product $s\_{n,n}\ast s\_{n,n}$ (see [arXiv:0809.3469](https://arxiv.org/abs/0809.3469)).


About your second question: you can compute the inner product of symmetric functions in SAGE using the "kronecker\_product" command. For instance, compute the inner product of Schur functions $s\_{6,2}\ast s\_{5,3}$ as follows:


s=SymmetricFunctionAlgebra(QQ,basis='schur')


s([6,2]).kronecker\_product(s([5,3]))


You may also use Maple with John Stembridge's package SF and the command "itensor".


