Get list of months from one table and counts for each from another

One option uses aggregation and window functions. If you have hires for each month, then you can get the information directly from employees, like so:

select
    date_trunc('month', date_hired) month_hired,
    sum(count(*)) over(order by date_trunc('month', date_hired)) no_employees,
    count(*) hires
from employees
group by date_trunc('month', date_hired)

On the other hand, if there are months without hires, then you could use generate_series() to create the list of months, then bring the employees with a left join, and aggregate:

select
    d.month_hired,
    sum(count(e.id)) over(order by d.month_hired) no_employees,
    count(e.id) hires
from (
    select generate_series(
        date_trunc('month', min(date_hired)), 
        date_trunc('month', max(date_hired)), 
        interval '1' month
    ) month_hired
    from employees
) d
left join employees e
    on  e.date_hired >= d.month_hired
    and e.date_hired <  d.month_hired + interval '1' month
group by d.month_hired

We could run another count for every period distilled from amounts, but that's expensive - unless there are only very few?

For more than a few, compute counts per period for the whole employees table, plus a running total. Then LEFT JOIN to it, should be pretty efficient:

SELECT mon AS period, e.mon_hired AS count, e.all_hired AS hires
FROM  (
   SELECT to_date(payment_period, 'YYYYMM') AS mon
   FROM  (SELECT DISTINCT payment_period FROM  amounts) a0
   )  a
LEFT JOIN (
   SELECT date_trunc('month', date_hired) AS mon
        , count(*) AS mon_hired
        , sum(count(*)) OVER (ORDER BY date_trunc('month', date_hired)) AS all_hired
   FROM   employees e
   GROUP  BY 1
   ) e USING (mon)
ORDER  BY 1;

This assumes we can just count all employees hired so far to get the total number of hires. (Nobody ever gets fired.)

Works just fine as long as there are rows for every period. Else we need to fill in for the gaps. We can compute a complete grid, or default to the latest row in case of a missing month like this:

WITH e AS (
   SELECT date_trunc('month', date_hired) AS mon
        , count(*) AS mon_hired
        , sum(count(*)) OVER (ORDER BY date_trunc('month', date_hired)) AS all_hired
   FROM   employees e
   GROUP  BY 1
   )
SELECT mon AS period, ae.*
FROM  (
   SELECT to_date(payment_period, 'YYYYMM') AS mon
   FROM  (SELECT DISTINCT payment_period FROM  amounts) a0
   )  a
LEFT JOIN LATERAL (
   SELECT CASE WHEN e.mon = a.mon THEN e.mon_hired ELSE 0 END AS count  -- ①
        , e.all_hired AS hires
   FROM   e
   WHERE  e.mon <= a.mon
   ORDER  BY e.mon DESC
   LIMIT  1
   ) ae USING (mon)
ORDER  BY 1;

① If nothing changed for the month, we need to fall back to the last month with data. Take the total count from there, but the monthly count is 0.

We can run a window function over an aggregate on the same query level. See:

• Group and count events per time intervals, plus running total

Related:

• PostgreSQL: running count of rows for a query 'by minute'

Aside: don't omit the AS keyword for a column alias. See:

• Date column arithmetic in PostgreSQL query