Out of a set $S = \{1,2,3,4,5\}$, I am picking elements $1$ and $5$. I then want to create all k=2-permutations from the remaining elements. For instance:
$$ 1 \rightarrow \mathbf{2} \rightarrow \mathbf{3} \rightarrow 5 \\ 1 \rightarrow \mathbf{3} \rightarrow \mathbf{2} \rightarrow 5 \\ 1 \rightarrow \mathbf{4} \rightarrow \mathbf{3} \rightarrow 5 \\ \dots $$
In my notation, a permutation $1 \rightarrow \mathbf{2} \rightarrow \mathbf{3} \rightarrow 5$ correspons to the matrix elements $a_{21}a_{32}a_{53}$.
How can I describe the sum of the matrix elements corresponding to the permutations formulaically (along the lines of $\sum_{i=1}^{n-k}a_{ij}$)?
$$ a_{21}a_{32}a_{53} + a_{31}a_{23}a_{52} + a_{41}a_{34}a_{53} + \dots \ (\text{all permutations)} = \sum_ ? $$
For this specific permutation, I found the following representation sufficient:
$$ \sum_{n,m=1}^{5}a_{n1}a_{mn}a_{m5} \times (1-\delta_{mn})(1-\delta_{m1})(1-\delta_{m5})(1-\delta_{n1})(1-\delta_{n5}) $$
where $\delta$ is the Kronecker delta.