# Interlacing two ordered sequences with constraints

How many ways can we create a sequence of length $2n$ from $\{x_i\}_{i=1}^n$ and $\{y_j\}_{j=1}^n$ so that $x_1<x_2<\cdots <x_n$, $\;y_1<y_2<\cdots <y_n$ and $x_i<y_i$ for all $1\leq i\leq n$ ?

This question is related to the following enter link description here, but with an extra condition on the ordering between the two sequences.

Using the wordings in the original question: Suppose we have two finite, ordered sequences $x=(x_1,\cdots,x_n)$ and $y=(y_1,\cdots,y_n)$. How many ways can we create a new sequence of length $2n$ from $x$ and $y$ so that the order of elements is preserved AND we have orderings $(x_i,y_i)$ for all $i$?

• Are you preserving the orders of the sequences as in the original question? Which conditions are you trying to insist upon and which properties are given? Jun 21 '18 at 14:52
• Thanks for the questions. I've added a paragraph to try to clarify. Jun 21 '18 at 15:43

$(\ )(\ )(\ ),\ (\ )((\ )),\ ((\ ))(\ ),\ ((\ )(\ )),\ (((\ )))$ or
$a_1b_1a_2b_2a_3b_3,\ a_1b_1a_2a_3b_2b_3,\ a_1a_2b_1b_2a_3b_3, \ a_1a_2b_1a_3b_2b_3,\ a_1a_2a_3b_1b_2b_3$