Given a sequence $1,1,2,2,3,3, …,k,k$, I am interested in counting the number of non-nesting permutations of the above sequence. Two intervals (determined by symbols $K$ and $L$) are nesting if one is completely contained inside the other. A multi-permutation is non-nesting if for any two symbols $K$ and $L$, the corresponding intervals ( $[K,K], [L,L]$) are non-nesting.
For instance, $122313$ is nesting permutation since the interval defined by the two copies of $2$ is contained inside the interval defined by $1$.
What is the count of non-nesting multi-permutations? What is known about these special multi-permutations?
P.S. I know that the total number of permutations of $1,1,2,2, ... ,k,k$ is given by $(2k)!/2^k$.
Update: I found this is equivalent to counting nonnesting matchings on [2n] which is equal to Catalan number$ C_n$ . See reference: Catalan numbers by Stanley.