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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 partitions the conjugates of $2n$ with four parts such that $\lambda_i-\lambda_j$ is even for $i,j$ (or maybe the partitions conjugate to these).$4^a2^{n-2a}$.

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).
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Henkel Hankel determinants of symmetric functions

The starting point is that it is known that the Henkel 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 Henkel 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 partitions of $2n$ with four parts such that $\lambda_i-\lambda_j$ is even for $i,j$ (or maybe the partitions conjugate to these).

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).
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Henkel determinants of symmetric functions

The starting point is that it is known that the Henkel 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 Henkel 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 partitions of $2n$ with four parts such that $\lambda_i-\lambda_j$ is even for $i,j$ (or maybe the partitions conjugate to these).

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).