Say you have the permutation $(54231)$ element of $S_5$
Now you drop say the "4" and then re-rank the remnant permutation on the other elements.
Then you are left with, $(4231)$ element of $S_4$
Anyone has seen this operation and has any insights about this? Is this known by some name? Why is this an useful idea?
Also note this related article: http://groupprops.subwiki.org/wiki/Derived_permutation
These are called permutation patterns. The smaller permutation you obtain is called the pattern. It is usually considered as a subsequence of the permutation, and we say that the permutation "contains" the pattern, and if it does not contain the pattern then it "avoids" it. Thus $54231$ contains the pattern $4231$ at positions $1,3,4,5$, but it avoids the pattern $3412$.
As simple as this is, this is very deep. Though it of course wasn't always thought of this way, this is a Coxeter group thing, which means it has combinatorial, geometric, and algebraic aspects to it.
For the combinatorial aspect, see the Wikipedia article on permutation patterns. This is not my specialty but that article seems to have a lot of info.
For the geometry, permutation patterns allow one to determine certain properties of Schubert varieties in homogeneous spaces of complex semisimple linear algebraic groups (see the Wikipedia article on Schubert calculus). Schubert varieties in a flag variety of type $A$ are indexed by permutations, and, for example, a Schubert variety is smooth if and only if its permutation avoids the patterns $4231$ and $3412$. I was able to prove that rational smoothness/smoothness of Schubert varieties in generalized flag varieties (possibly infinite-dimensional) is a pattern avoidance property, meaning that if a Schubert variety is (rationally) smooth, then a Schubert variety corresponding to any pattern the element contains is also (rationally) smooth. (This was proved for finite-dimensional flag varieties by Billey and Postnikov.) It was never published.
Algebraically, these kind of maps are homomorphisms of weak order and can be easily classified. I will now present the situation in ridiculous generality, but you have to promise not to laugh. The following was going to be the basis of my dissertation but I ended up doing something else instead.
A Coxeter system is a pair $(W,S)$ where $W$ is a group and $S$ is a generating set such that $W$ has a presentation involving only the relations $s^2=1$ for all $s\in S$ as well as zero or more relations of the form $(ss')^m=1$ for $s,s'\in S$ and some integer $m\geq 2$ depending on $s,s'$. Patterns arise from group homomorphisms that are easiest to describe in terms of abstract root systems.
$W$ has a length function $\ell$ sending an element to the smallest number of elements of $S$ required to express it as a product. Let $T$ be the set of elements conjugate to some element of $S$ (which are all of order 2) and let $R=T\times \{-1,1\}$. Elements of $R$ will be called roots, elements of $T\times \{1\}$ positive roots, and elements of $T\times \{-1\}$ negative roots.
We define a (nonassociative) product $\ast$ on $R$ by the rule $$(t,e)\ast (t',e')=\left\{\begin{array}{ll}(tt't,e')&\mbox{ if }\ell(tt')>\ell(t)\\ (tt't,-e')&\mbox{ otherwise}\end{array}\right.$$ This product extends to an action of $W$ on $R$ such that $$w\ast (t',e')=\left\{\begin{array}{ll}(wt'w^{-1},e')&\mbox{ if }\ell(wt')>\ell(w)\\ (wt'w^{-1},-e')&\mbox{ otherwise}\end{array}\right.$$
The set $$I(w)=\{(t,1):\ell(wt)<\ell(w)\}$$ is called the inversion set of $w$ and completely characterizes $w$. If $R,R'$ are root systems of the Coxeter systems $(W,S),(W',S')$ then a function $f:R\to R'$ such that $$f(r\ast r')=f(r)\ast f(r')$$ such that $f(r)$ is positive whenever $r$ is positive is called a homomorphism of root systems. The pattern map $f^{\ast}:W'\to W$ is the function such that $f^{\ast}(w')=w$, where $w$ is the unique element such that $$I(w)=f^{-1}(I(w'))$$
For $W=S_n$ we have that $S$ consists of the adjacent transpositions $(i,i+1)$ and $T$ consists of all transpositions. There are two kinds of homomorphism $R_m\to R_n$ (where $R_i$ is the abstract root system corresponding to $S_i$): ones where there is a sequence $1\leq a_1<a_2<\cdots<a_m\leq n$ such that $f((i,j),e)=((a_i,a_j),e)$, thus embedding the smaller permutation into a larger permutation, and ones where the sequence of $a_i$ is decreasing instead. These are equivalent up to automorphism. For the increasing homomorphisms, the pattern map is exactly the construction you describe, deleting multiple indices at a time, leaving the flattened permutation at $a_1,a_2,\ldots,a_m$ behind.
Let me know if you have any further questions.