I am interested to find the number of non-crossing pair partitions on the set $\{1,2,...,2k\}$ such that none of these pair partitions has a pair of the form $(2r-1,2r)$ and there are exactly $t$ pairs of the form $(2i-1,2j)$ where $2i-1<2j$.

Let's go through this slowly. I think we understand what non-crossing pair partitions are, but briefly, a partition whose any block is a pair and if $i<j<k<l$ then $(i,k)$ and $(j,l)$ cannot be blocks, because if we join the elements in the same block by lines, we see that $i,k$ will be joined by a line, and $j,l$ will be joined by a line, which intersect. It is known that the number of non-crossing pair partitions on a set of size $2k$ is equal to $C_k$, the $k$-th Catalan number defined by $C_k=\dfrac{1}{k+1}{2k\choose k}$.

Clearly, since I am talking about pair partitions, my set has to be of even cardinality, hence the $2k$.

I am demanding that my non-crossing pair partitions will not have a pair of the form $(2r-1,2r)$ i.e. $(1,2),(3,4),...,(2k-1,2k)$ cannot be blocks in my partition. I can count the number of such non-crossing pair partitions, because it is easy to count the number of non-crossing pair partitions having at least one of these pairs as a block. That's because, say, we fix $(1,2)$ to be a block, then we are really counting the number of non-crossing pair partitions on a set of size $2k-2$ So this part is fine and can be tackled.

I am also demanding that my non-crossing pair partitions will have exactly $t$ pairs of the form $(2i-1,2j)$ where $2i-1$ is to the left of $2j$, that is, a line joining $2i-1$ and $2j$ begins at $2i-1$. It is well known that the number of such non-crossing pair partitions is equal to a Narayana number.

But can we compute the number of non-crossing pair partitions such that BOTH the above happen? Any help would be appreciated. All search results were giving me how to count 132 avoiding permutations, which I believe is unrelated to this problem.