Question: The quintic del Pezzo $3$-fold $V(5)$ of degree $5$ is defined as the intersection of $Grass(2,5)$ and a codimension $3$ linear subspace. I would like to show the following:

**For every point $p$ in $V(5)$ there is a line contained in $V(5)$ not passing through $p$.** 

I know that $\operatorname{Gr}(2,5)$ is defined by $5$ quadrics in $\mathbb{P}^9$, thus we can view $V(5)$ as defined by $5$ quadrics in $\mathbb{P}^6$. Now given a point $p=[1,0,\dots,0]$, write the five quadrics as $$Q\_i= q\_{i,2}(x\_1,\dots, x\_n)+ x\_0 q\_{i,1}(x\_1,\dots, x\_n),$$ where $q\_{i,k}$ is homogeneous of degree $k$. There is a line contained in $V(5)$ passing through $p$ if and only if the $\{q\_{i,2},q\_{i,1}: i=1,2,3,4,5\}$ have a common zero in $\mathbb{P}^5$. Indeed, for a common zero $[a\_0,a\_1,\dots,a\_n]$, the line joining $p$ and $[a\_0,a\_1,\dots,a\_n]$ is contained in $V(5)$. Since the quadrics are linearly independent, the dimension of the subset defined by these $10$ equations is $\leq0$ which means there are only finitely many lines in $V(5)$ passing through a given point. I'm thinking whether the number of lines on $V(5)$ is finite or infinite so that I can compare the number of lines to get a line not passing through $p$. How to determine the number of lines on $V(5)$? Or is there a better approach to this question?

Answer: 
For Q1, the sequence corresponding to [Stat19](http://www.findstat.org/StatisticsDatabase/St000019) is given in [A263484](http://oeis.org/A263484) and comes from a 2005 article by Richard Stanley (developing ideas from Comtet) on connectivity sets, connectedness, cut points, etc.

It may not have a standard name because it is the reflection of the decomposition/block number [Stat56](http://www.findstat.org/StatisticsDatabase/St000056), i.e., for $\pi \in S\_n$, $\text{Stat19}(\pi) = n - \text{Stat56}(\pi)$. See [A059438](http://oeis.org/A059438). Another related statistic is global ascents [Stat234](http://www.findstat.org/StatisticsDatabase/St000234). 

For Q2, the connection between block numbers of 321-avoiding permutations and Catalan's triangle was established by Adin, Bagno, and Roichman, [arXiv 1611.06979](https://arxiv.org/abs/1611.06979), later *J. Algebraic Combinatorics*. The related [abstract](https://pp2017.github.io/assets/pdf/abstracts/talks/eli_bagno.pdf) for Bagno's talk at the Permutation Patterns 2017 conference is more expository.
For the first part of your question, *support* doesn't appear to be an appropriate word, as in the context of permutations it is normally used for the set of non-fixed points. In general, a mathematical *support* means as an assistance to the main act.

To try to define the term, we can first define the process that arrives at the statistic:

Let $T\_n$ be the set of all non-trivial simple transpositions on a set of size $n$, say $S$.

Let $\pi$ be a permutation of $S$, and let $$\;\_\pi T\_n=\left\{\{t\_1,\dots t\_k\}\in T\_n^k:\pi\prod\_\limits{i=1}^k t\_i=S\right\}$$

i.e. $\;\_\pi T\_n$ is the set of sets of simple transpositions that take $\pi$ to the identity $S$.

Define:

$$\;\_\pi U\_n(t\in \;\_\pi T\_n) =\{u:u\in t\}$$

as the set of unique elements of a set $t$, and

$$\_\pi U\_n=\bigcup\_t \;\_\pi U\_n(t)$$

as the set of sets that contain the unique elements of simple transposition sets from $\;\_\pi T\_n$.

Define $\;\_\pi V\_n=\{|v|:v\in \;\_\pi U\_n\}$ and then the statistic we want is $\min(\;\_\pi V\_n)$.

So, **minimum cardinality of the identifying simple transposition sets for a permutation $\pi$**, or perhaps, **the minimum number of simple transpositions needed to return $\pi$ to $S$**.

The actual statistic can be easily calculated by creating an array of $n-1$, and ticking every cell that is crossed by returning an element in a straight path to it's original position, and counting the number of ticks, max 1 per cell.