Solving "Gaps and Islands" with row_number() and dense_rank()?
So the trick here is a property of two equally incrementing series which produce a difference that can be used to identify islands {11,12,13} - {1,2,3} = {10,10,10}. This property isn't enough to identify islands in and of itself, but it's a crucial step that we can exploit to do so.

Background

Stepping aside from the problem. Let's check this out.. Here we


Define 1 group as 3 rows.
Create one group for each pair/row of offsets in xoffsets.


Here is some code.

SELECT x1, x2, x1-x2 AS difference
FROM (VALUES
  (2,42),
  (13,7),
  (42,2)
)
  AS xoffsets(o1,o2)
CROSS JOIN LATERAL (
  SELECT x+o1,x+o2  -- here we add the offsets above to x
  FROM generate_series(1,3) AS t(x)
) AS t(x1, x2)
ORDER BY x1, x2;

 x1 | x2 | difference 
----+----+------------
  3 | 43 |        -40
  4 | 44 |        -40
  5 | 45 |        -40
 14 |  8 |          6
 15 |  9 |          6
 16 | 10 |          6
 43 |  3 |         40
 44 |  4 |         40
 45 |  5 |         40
(9 rows)


This looks pretty good, and grouping by the difference would be good enough in this example. You can see we have three groups starting at       (2,42), (13,7), and (42,2), corresponding to the groups in xoffsets.  This is essentially the problem in reverse. But we have one major issue because we're demonstrating this with static offsets. If the difference between any two offsets o1-o2 is the same we'll have a problem. Like this,

SELECT x1, x2, x1-x2 AS difference
FROM (VALUES
  (100,90),
  (90,80)
) AS xoffsets(o1,o2)
CROSS JOIN LATERAL (
  SELECT x+o1,x+o2  -- here we add the offsets above to x
  FROM generate_series(1,3) AS t(x)
) AS t(x1, x2)
ORDER BY x1, x2;

 x1  | x2 | difference 
-----+----+------------
  91 | 81 |         10
  92 | 82 |         10
  93 | 83 |         10
 101 | 91 |         10
 102 | 92 |         10
 103 | 93 |         10


We'll we have to find a way define the second offset statically.

SELECT x1, x2, x1-x2 AS difference
FROM (VALUES
  (100,0),
  (90,0)
) AS xoffsets(o1,o2)
CROSS JOIN LATERAL (
  SELECT x+o1,x+o2  -- here we add the offsets above to x
  FROM generate_series(1,3) AS t(x)
) AS t(x1, x2)
ORDER BY x1, x2;

 x1  | x2 | difference 
-----+----+------------
  91 |  1 |         90
  92 |  2 |         90
  93 |  3 |         90
 101 |  1 |        100
 102 |  2 |        100
 103 |  3 |        100
(6 rows)


And, again we're back on track to making groups for each pairs of offsets. This isn't quite what we're doing, but it's pretty close and hopefully it serves to help illustrate how the two sets can be subtracted to create islands.

Application

Now let's revisit the problem above with table foo. We sandwich the variables between two copies of x for display purposes only.

SELECT
  id,
  x,
  dense_rank() OVER (w1) AS lhs,
  dense_rank() OVER (w2) AS rhs,
  dense_rank() OVER (w1)
    - dense_rank() OVER (w2)
    AS diff,
  (
    x,
    dense_rank() OVER (w1)
      - dense_rank() OVER (w2)
  ) AS grp,
  x
FROM foo
WINDOW w1 AS (ORDER BY id),
  w2 AS (PARTITION BY x ORDER BY id)
ORDER BY id
LIMIT 10;


 id | x | lhs | rhs | diff |  grp  | x 
----+---+-----+-----+------+-------+---
  1 | 2 |   1 |   1 |    0 | (2,0) | 2
  2 | 1 |   2 |   1 |    1 | (1,1) | 1
  3 | 2 |   3 |   2 |    1 | (2,1) | 2
  4 | 3 |   4 |   1 |    3 | (3,3) | 3
  5 | 3 |   5 |   2 |    3 | (3,3) | 3
  6 | 2 |   6 |   3 |    3 | (2,3) | 2
  7 | 3 |   7 |   3 |    4 | (3,4) | 3
  8 | 1 |   8 |   2 |    6 | (1,6) | 1
  9 | 3 |   9 |   4 |    5 | (3,5) | 3
 10 | 1 |  10 |   3 |    7 | (1,7) | 1
(10 rows)


You can see here all of the variables at play, in addition to x and id


The lhs is simple. We're just generating a unique sequential identifier with that (because we're feeding it a unique sequential identifier: id -- though never forget that ids are seldom gapless)
The rhs is slightly more complex. We partition by x and generate a sequential identifier with that amongst the different x values. Observe how rhs increments over the set each time it sees a row with a value that it has already seen. The property of rhs is how many times the value was seen.
The diff is the simple result of subtraction but it's not too useful to think of it like that. Think of more like subtracting a sequence than digits (though they're digits for any single row). We have a sequence increasing by one for the amount of times a value is seen. And, we have a sequence increasing by one for every distinct id (every time). Subtracting these two sequences will return the same number for repeating values, just like in our example above. (11,12,13) - (1,2,3) = (10,10,10). This is the same principle in the first section of this answer


diff doesn't independently mark the group by itself
diff forces all identical islands of x into a group (which may have false-positives, ex. the three cases where diff=3 above and their corresponding x values)



The group grp is a function of (x, diff). It serves as the grouping id, albeit in a slightly weird format. This serves to reduce false positives that would happen if we just grouped by diff.

Simple Unoptimized Query

So now we have our simple unoptimized query 

SELECT x, diff, count(*)
FROM (
  SELECT
    id,
    x,
    dense_rank() OVER (ORDER BY id)
      - dense_rank() OVER (PARTITION BY x ORDER BY id)
    AS diff
  FROM foo
) AS t
GROUP BY x, diff;


Optimizations

On the matter of replacing dense_rank() with something else, like row_number(). @ypercube commented


  It also works with ROW_NUMBER() but I think it may give different results with different data. Depends if there are duplicate data in the table.


So let's review it, here is a query that shows


row_number() and dense_rank()
the respective differences computed.
over a result set that includes, ids, 1,2,3,4,5,6,7,8,6,7,8, and two different "islands" of x vals.


SQL Query,

SELECT
  id,
  x,
  dense_rank() OVER (w1) AS dr_w1,
  dense_rank() OVER (w2) AS dr_w2,
  (
    x,
    dense_rank() OVER (w1)
    - dense_rank() OVER (w2)
  ) AS dense_diffgrp,
  row_number() OVER (w1) AS rn_w1,
  row_number() OVER (w2) AS rn_w2,
  (
    x,
    row_number() OVER (w1)
    - row_number() OVER (w2)
  ) AS rn_diffgrp,
  x
FROM (
  SELECT id,
  CASE WHEN id&lt;4
    THEN 1
    ELSE 0
  END
  FROM
  (
    SELECT * FROM generate_series(1,8)
    UNION ALL
    SELECT * FROM generate_series(6,8)
  ) AS t(id)
) AS t(id,x)
WINDOW w1 AS (ORDER BY id),
  w2 AS (PARTITION BY x ORDER BY id)
ORDER BY id;


Result set, dr dense_rank, rn row_number

 id | x | dr_w1 | dr_w2 | dense_diffgrp | rn_w1 | rn_w2 | rn_diffgrp | x 
----+---+-------+-------+---------------+-------+-------+------------+---
  1 | 1 |     1 |     1 | (1,0)         |     1 |     1 | (1,0)      | 1
  2 | 1 |     2 |     2 | (1,0)         |     2 |     2 | (1,0)      | 1
  3 | 1 |     3 |     3 | (1,0)         |     3 |     3 | (1,0)      | 1
  4 | 0 |     4 |     1 | (0,3)         |     4 |     1 | (0,3)      | 0
  5 | 0 |     5 |     2 | (0,3)         |     5 |     2 | (0,3)      | 0
  6 | 0 |     6 |     3 | (0,3)         |     7 |     3 | (0,4)      | 0
  6 | 0 |     6 |     3 | (0,3)         |     6 |     4 | (0,2)      | 0
  7 | 0 |     7 |     4 | (0,3)         |     8 |     6 | (0,2)      | 0
  7 | 0 |     7 |     4 | (0,3)         |     9 |     5 | (0,4)      | 0
  8 | 0 |     8 |     5 | (0,3)         |    10 |     8 | (0,2)      | 0
  8 | 0 |     8 |     5 | (0,3)         |    11 |     7 | (0,4)      | 0


You'll see here, that when the column you're ordering by has duplicates you can't use this method because you can't ensure an ordering for a query that has duplicate in the ORDER BY clause. It's very much just the effect of the ordering of the duplicates that throws off the difference: the relationship of the global incremeneting value to the incrementing value over the partition. However, when you have an unique id column, or a series of columns that define uniqueness, by all means use row_number() instead of dense_rank().