Background
----------


Many properties of permutations can be stated in terms of *classical patterns*.
For example:


* a permutation is *stack-sortable* if and only if it avoids 231 (Knuth 1975)
* a permutation corresponds to a *smooth* Schubert variety if and only if it
avoids 1324 and 2143 (Lakshmibai and Sandhya 1990)


For other properties we need a stronger notion of a pattern, e.g., the *mesh patterns*
introduced by Brändén and Claesson (2011). For example:


* a permutation corresponds to a factorial Schubert variety if and only if it
avoids 1324 and (2143,{(2,2)}) (These are the so-called *forest-like permutations*, Bousquet-Mélou and Butler 2007)
* a permutation is sortable in two passes through a stack if and only if
it avoids 2341 and (3241,{(1,4)}) (These are the so-called *West-2-stack-sortable permutations*, West 1990)


There are also properties which have not been translated into patterns (to my
knowledge):


* *meander* permutations (<http://theory.cs.uvic.ca/inf/perm/StampFolding.html>)
* the *involutions* in the symmetric group
* ...


The Question
------------



> 
> What permutation properties do you know that have not been described by the avoidance of patterns
> 
> 
> 


Motivation
----------


I recently wrote an algorithm that given a finite set of
permutations outputs the mesh patterns that the permutations avoid. This algorithm
is called BiSC (derived from the last names of three people that inspired me to
write the algorithm) and can conjecture the descriptions given in the
first two lists above. It is available at <http://staff.ru.is/henningu/programs/bisc/bisc.html> and described in the paper <http://arxiv.org/abs/1211.7110>.


This is a community wiki question since it there is obviously not a single best
answer



Properties of permutations with unknown pattern avoidance descriptions
Here's one idea. For every permutation $\pi$ of length $n$, there are $n^2+1$ permutation of length $n+1$ containing $\pi$. However, once you look at permutations of length $n+2$, this quantity depends on $\pi$. In their paper "[Posets of matrices and permutations with forbidden subsequences](https://doi.org/10.1007/s000260300005)", Ray and West gave a proof that for $\pi$ of length $n$ the number of permutations of length $n+2$ containing $\pi$ is
$$
(n^4+2n^3+n^2+4n+4-2j)/2,
$$
where $0\le j\le n-1$ depends on $\pi$. Perhaps you could give a description of this statistic in terms of patterns of $\pi$?


