TITLE: On a statistic for permutations
QUESTION [7 upvotes]: Given a permutation $\pi$ we can write $\pi=s_{i_1} ... s_{i_l}$ as a product of simple transpositions $s_i=(i,i+1)$ in a minimal way. 

Question 1: Is there an "official" name for the permutation statistic given by the cardinality of the set $\{i_1,...,i_l \}$ ? (such as Coxeter length seems to be the official name for $l$)

See http://www.findstat.org/StatisticsDatabase/St000019 for this statistic.

Question 2: Is there a reference that the generating function for this permutation statistic restricted to 321-avoiding permutations is given by Catalan's triangle: http://oeis.org/A009766 ?

REPLY [2 votes]: For Q1, the sequence corresponding to Stat19 is given in 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, i.e., for $\pi \in S_n$, $\text{Stat19}(\pi) = n - \text{Stat56}(\pi)$.  See A059438.  Another related statistic is global ascents Stat234. 
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, later J. Algebraic Combinatorics.  The related abstract for Bagno's talk at the Permutation Patterns 2017 conference is more expository.