Probability distribution for exceptional sequences of type $A_n$

TL;DR

This paper derives the probability distribution for relative projective objects in exceptional sequences of type A_n, revealing independence properties and providing a probabilistic interpretation of known combinatorial formulas.

Contribution

It establishes the distribution and independence of events related to projective objects in exceptional sequences of type A_n, offering new probabilistic insights.

Findings

Events are independent of each other and sequence length.

Provides a probabilistic interpretation of the product formula.

Connects exceptional sequences to combinatorial counts.

Abstract

We determine the probability distribution for relative projective objects in an exceptional sequence of type $A_n$ of any length. We show that these events (the $j$-th object in an exceptional sequence of length $k\le n$ being relatively projective) are independent of each other and from the length of the sequence. This gives a probabilistic interpretation of the product formula for the number of exceptional sequences of length $k$ and clusters or partial clusters of size $k$ since the latter numbers are proportional to the number of signed exceptional sequences of length $k$.