The number of permutations of the set $\{1,2,3,4\}$ in which no two adjacent positions are filled by consecutive integers (increasing order)

It means the pattern $12, 23, 34$ do not appear..

Let $S_i$ denote the set that $i(i+1)$ appear together. To find number of elements in $S_i,$ we have to permute $2$ numbers and the entity $i(i + 1),$ this can be done in $3!$ ways.

Also, $\left(12\right), \left[or \left(23\right), \left(34\right)\right]$ can occur together in $3!$ ways, and $\left(12\right), \left(34\right)$ can occur together in $2!$ ways.

Next, $12, 23$ and $34$ can occur together (in that order) in just one way. Thus, the required number of ways

  $=4!-3(3!)+2(3!)+2!-1!=19$ ways.