2D partitions

[hv at crypt.org]

Earlier I wrote:
:Consider the ways in which we may partition an a x b rectangle into
:regions, defined for this purpose as orthogonally connected sets of
:unit squares.
:
:A region is defined by the squares comprising it (rather than the
:connections).
:
:I looked at two functions:
:  f(a, b) counts the number of partitions
:  g(a, b) counts the same, after discarding mirror/rotate symmetries
[...]
:I understand a bit better how to extend this: we can find matrices
:such that f(a, b) = F_a * P_a ^ (b - 1) * U_a:

That done; f(a, b) for all a <= 6, b <= 20 are recorded below.

I've submitted the table via the webform (up to a+b = 9). I'm not sure
if there is value in adding the individual rows as well (for n >= 3),
or f(n, n) (which I can calculate up to around 7 or 8 with my current
program).

The matrix size may also be of interest. Without removing impossible
configurations (such as "abab with both a's and b's connected") or
symmetries, that gives:
  1, 2, 6, 24, 119, 696, 4686, 35623, 301304, 2801967, ...
.. which is not in OEIS. But it would be a tricky sequence to describe.

Still hoping to progress on g(a, b) - I had a suggestion of using Polya
(aka Burnside) counting, but I suspect it isn't applicable for this
problem.

Hugo
---
f(1, n) = <1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192,
  16384, 32768, 65536, 131072, 262144, 524288>
f(2, n) = <2, 12, 74, 456, 2810, 17316, 106706, 657552, 4052018, 24969660,
  153869978, 948189528, 5843007146, 36006232404, 221880401570, 1367288641824,
  8425612252514, 51920962156908, 319951385193962, 1971629273320680>
f(3, n) = <4, 74, 1434, 27780, 538150, 10424872, 201947094, 3912050356,
  75782907270, 1468040672696, 28438383992230 550898690444420, 10671821831261942,
  206730898391393192, 4004720564629102582, 77578083032366404308,
  1502816206487087179878, 29112043791259796460440, 563948598668183331934022,
  10924620209429992515333924>
f(4, n) = <8, 456, 27780, 1691690, 103015508, 6273056950, 381992581548,
  23261112447444, 1416465537909008, 86254456382365422, 5252391281433321840,
  319839870650458477666, 19476375116903366115612, 1185997189541765630381252,
  72220283556802506132778356, 4397792341346509099717980618,
  267799799794395524429805470412, 16307439552264147863293668460374,
  993027571174138531222370457667420, 60469563842417968456270619096702476>
f(5, n) = <16, 2810, 538150, 103015508, 19719299768, 3774627281350,
  722529289997430, 138304601846374768, 26473890580295999056,
  5067560038024745010778, 970018539928219042359646, 185678306810346544211634436,
  35542035744053670003787644728, 6803359673650785124011930619958,
  1302280577914868906982520391258694, 249279001106260657212587710461398560,
  47716307412056046176493501774605976216,
  9133725596370184196611926199534920900674,
  1748352875451333443165347701972469440044846,
  334664945300493661432157903306911568781965444>
f(6, n) = <32, 17316, 10424872, 6273056950, 3774627281350, 2271230282824746,
  1366616876858117336, 822303473245589500576, 494786023682933392539098,
  297716369442290419344830218, 179138116466289549298988916728,
  107788714513732641294426969380866, 64857257653980142202422375104026116,
  39025086153927457182347402657874978462,
  23481679682513609746829457441240385452638,
  14129098358350061404959611693696056668583908,
  8501581791381817334324164744137238178639736178,
  5115463925752907633553673176166286697283584448740,
  3078012047382312970733108949764930341009394142422144,
  1852062354723032716839437582095526151895561581318868080>