# Description of an Increasing Maximum Value in a Sequence of Integers

I came across this sequence when visiting the Longest increasing subsequence problem. In particular, this implementation. I will demonstrate the observation below:

Given

Here we have a sequence of integers:

[0, 1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 5, 2, 3, 7, 6]


Notice from left to right, at each position, the next number is either:

2. a number larger than the maximum value observed

Thus the following pattern is invalid:

[0, 5, 1, ...]


as the value after 5 must be observed prior or larger than 5.

Question

What is the mathematical term of such a sequence? If no exact term is defined, how might this be formally described? The sequence is not strictly monotonic, but the maximum appears to be monotonically increasing.

I don't know of any name for it, but you could formally define such a sequence as: $(a_n)$ where $a_i \in \mathbb{Z}$ and $a_i \geq a_j$ whenever $i > j$. Of course you could specify bounds on the indices if the sequence it finite. Also it follows from this that the first element is the smallest element.