I have recently had the chance to attend a nice talk in Combinatorics, and once the speaker alluded to the famous 312-avoiding pattern problem, I was reminded of the following question I have had since I heard something about $312$-avoiding patterns for the first time:
Regarding the fact that D. E. Knuth, in "Art of computer Programming", proves for any arbitrary $\sigma \in \mathcal S_3$, the number of $\sigma$-avoiding permutations in $\mathcal S_n$ is given by the $n$-th Catalan number, and knowing that when different combinatorics are counted by the same Catalans, there exist bijections between their objects (which may not be trivial of course), I was wondering if there is any technical (convincing) reason that people still stick to $312$-avoiding patterns, rather than $\sigma$-avoiding patterns, where $\sigma \in \mathcal S_3$ is fixed.
This falls under the general notion of "pattern avoiding permutations," and the combinatorics of these can loosely be said to be inherited from reduced-decomposition structure for such permutations. For adjacent transpositions $s_i=(i,i+1)$, the generators $s_i$, $i=1,2,\cdots,n-1$ form a Coxeter group and generate $S_n$. Thus any permutation can be written as a minimal sequence of $s_i$'s. Example $312=s_2s_1$. It turns out that 312-avoiding permutations have very particular reduced-decomposition structure. Take a look at pages 7 and up of this reference on $c$-sortable permutations.