Window function with order by returns strange results Here is example:

drop table if exists tst;

create table tst (
num integer not null
);

insert into tst values (1), (2), (3);

-- window functions WITH order by clause
select *, max(num) over (partition by true order by num asc), array_agg(num) over (partition by true order by num asc) as test
from tst;

-- window functions WITHOUT order by clause
select *, max(num) over (partition by true), array_agg(num) over (partition by true) as test
from tst;


The result is as follows:



Why order by clause has impact on aggregation function?
That's expected.

An aggregate function used as a window function together with an order by acts as a "rolling" aggregate, i.e. it only considers values up to the "current rows" based on the order by expression. 

Quote from the manual


  There is another important concept associated with window functions: for each row, there is a set of rows within its partition called its window frame. Some window functions act only on the rows of the window frame, rather than of the whole partition. By default, if ORDER BY is supplied then the frame consists of all rows from the start of the partition up through the current row, plus any following rows that are equal to the current row according to the ORDER BY clause. When ORDER BY is omitted the default frame consists of all rows in the partition. 


(Emphasis mine)

For a more detailed explanation see this answer