Suppose we have a set of $k$ objects $\{1,2,3,\dots, k\}.$ We want to compute the number of ways to make a list of $2n$, with each entry chosen from the set of $k$ objects, such that each object chosen appears an even number of times. For example, $1,1,1,1,2,2$ is a valid list and is different from $1,2,2,1,1,1.$ In addition, $1,2,2,2,3,3$ is not a valid list. For the case without ordering, this is just a straightforward stars and bars problem. In particular, there is a bijection to the problem of finding $x_1 + \dots + x_k = n$ which is just $\binom{n+k-1}{k-1}$ ways. However, I am unable to find a solution which accounts for ordering. I'm unsure if there is a closed form, but if not, any asymptotic results would also be appreciated.
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for clarification is a list of say all ones (1,1,1,1,1,1) valid since it has an even number of ones? – Evan Semet Apr 03 '23 at 05:22
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yes, that's correct – sde7 Apr 03 '23 at 05:26
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(I may be wrong) We can get a messy expression for the number of ways. Can you provide some values for different $n$ and $k$ so I can verify my answer? – D S Apr 03 '23 at 06:01