Let's say we have AEIOUXXXXX, 5 distinct vowels and 5 identical Xs. How can I represent a permutation using the two line notation?
I thought of doing this, but I don't think it works. $$\binom{AEIOUXXXXX}{XOXIOAXEXX}$$
2026-04-29 14:09:58.1777471798
How to write a permutation with both identical and distinct objects
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You would keep the top line as usual in two line notation, and the bottom line has the letters. So for your example $$\left(\begin{array}{llllllllll}1&2&3&4&5&6&7&8&9&10\\X&O&X&I&O&A&X&E&X&X\end{array}\right)$$ The indexing set still consists of $10$ distinguishable indices, it is the $X$'s in the permutation that are indistinguishable. The indices where they occur are still distinguishable.
The elements on the top and bottom in two line notation need not be the same.
There is such a thing as indistinguishable elements on the top in two line notation, but this is no longer a permutation, but rather a generalized permutation. See the Robinson-Schensted-Knuth correspondence.