Avoidance of Partially Ordered Generalized Patterns of the form   $k$-$\sigma$-$k$
  Sergey Kitaev has shown that the exponential generating function for permutations avoiding the generalized pattern $\sigma$-$k$, where $\sigma$ is a pattern without dashes and $k$ is one greater than the biggest element in $\sigma$, is determined by the exponential generating function for permutations avoiding $\sigma$.   We show that this also holds for permutations avoiding all the generalized patterns $\sigma_1$-$k_1$, $...$, $\sigma_n$-$k_n$, where $\sigma_1$, $...$, $\sigma_n$ are patterns without dashes and $k_i$ is one greater than the biggest element in $\sigma_i$. Similarly the exponential generating function for permutations avoiding the partially ordered generalized patterns $k_1$-$\sigma_1$-$k_1$, $...$, $k_n$-$\sigma_n$-$k_n$ can be determined from the exponential generating function for permutations avoiding the generalized patterns $\sigma_1$, $...$, $\sigma_n$, where $\sigma_1$, $...$, $\sigma_n$ are patterns without dashes and $k_i$ is one greater than the largest element in $\sigma_i$.   Using this we construct a bijection between bicolored set partitions and permutations avoiding the partially ordered generalized pattern 3-12-3 (that is, permutations avoiding both the patterns 3-12-4 and 4-12-3). By using this method twice, we find a closed formula for the exponential generating function for permutations avoiding the partially ordered generalized pattern 3-121-3.   Finally, we give a complete classification of when single partially ordered generalized patterns have the same set of avoiders. 
Marteinn T. Hardarson