Enumeration
Most of my research into polyominoes has been on enumeration 
counting polyominoes. I've counted 2dimensional and 3dimensional
polyominoes.
2dimensional polyominoes are collections of squares. For a given
size (number of squares), there are 3 main sets of polyominoes.
 The translation polyominoes are the set of polyominoes which are
unique with respect to translation  which means any two polyominoes
in the set can't be made identical by moving them horizontally
and vertically.
 The rotation polyominoes are the set of polyominoes which are
unique with respect to rotation and translation  which means
any two polyominoes can't be made identical by rotating them 90,
180, or 270 degrees and by moving them horizontally and vertically.
These are also referred to as 1sided polyominoes.
 The 3d rotation polyominoes are the set of polyominoes which
are unique with respect to 3d rotation, 2d rotation, and translation.
Any two polyominoes in this set can't be made identical by flipping,
rotating, or translating them.
It is easy to prove that t(n) / r(n) <= 4 for all n, and that
r(n) / p(n) <= 2 for all n.
Here is a list of the number of 2dimensional polyominoes of each
size up to 27.
 p(n) values for n <= 24 were calculated by D.H. Redelmeier in
1981.
 p(n) values for n = 25 thru 28 were calculated by Tomas Oliveira e Silva by using over 20 Pentium machines using a combined total of over
2 years of CPU time (in 1999?). See his results here [external link].
 r(n) for n <= 21 was calculated by Kevin Gong in August/September
1997 on a Power Macintosh 8500/120.
 r(n) for n = 22 was calculated by Kevin Gong in 83 hours in July
1999 on a Power Macintosh G3/400.
 t(n) for n <= 16 was calculated by W. F. Lunnon.
 Lunnon computed t(n) for n = 17 to be 400795860, but the correct
number is 400795844, as computed independently by myself, Uwe
Schult, and Tomas Oliveira e Silva.
 t(n) for n = 17 through 21 was calculated by Kevin Gong in August/September
1997 on a Power Macintosh 8500/120.
 t(n) for n = 22 thru 27 was calculated by Uwe Schult (Switzerland), n = 27 being computed on September 28, 1998 after
running for 64 days on a 266Mhz Apple PowerMac G3.
 t(n) for n = 28 was calculated by Tomas Oliveira e Silva (see above).
 t(n) for n = 29 through 46 was calculated by Tony Guttmann, Iwan Jensen (University of Melbourne) and Ling Heng Wong, possibly published
in "Punctured polygons and polyominoes on the square lattice."
in J. Phys. A 33, 17351764 (2000), but that's just a guess. See
MathSoft's page for more information. The values for 4146 were apparently wrong
when I originally looked at their web page in January 2001, they
had different values listed in November 2001, which are listed
here. Thanks to J K Haugland for pointing this out.
 r(n) for n = 23 was calculated by Kevin Gong in 8 days in
November/December 2004 on a Power Mac G5.
n 
translation t(n)

2d rotation r(n)

3d rotation p(n) 
t(n) / t(n1) 
t(n) / r(n) 
r(n) / p(n) 
1

1

1

1



1.00

1.00

2

2

1

1

2.00

2.00

1.00

3

6

2

2

3.00

3.00

1.00

4

19

7

5

3.17

2.71

1.40

5

63

18

12

3.32

3.50

1.50

6

216

60

35

3.43

3.60

1.71

7

760

196

108

3.52

3.88

1.81

8

2725

704

369

3.59

3.59

1.91

9

9910

2500

1285

3.64

3.96

1.95

10

36446

9189

4655

3.68

3.97

1.97

11

135268

33896

17073

3.71

3.99

1.99

12

505861

126759

63600

3.74

3.99

1.99

13

1903890

476270

238591

3.76

4.00

2.00

14

7204874

1802312

901971

3.78

4.00

2.00

15

27394666

6849777

3426576

3.80

4.00

2.00

16

104592937

26152418

13079255

3.82

4.00

2.00

17

400795844

100203194

50107909

3.83

4.00

2.00

18

1540820542

385221143

192622052

3.84

4.00

2.00

19

5940738676

1485200848

742624232

3.86

4.00

2.00

20

22964779660

5741256764

2870671950

3.87

4.00

2.00

21

88983512783

22245940545

11123060678

3.87

4.00

2.00

22

345532572678

86383382827

43191857688

3.883





23

1344372335524

336093325058

168047007728

3.890





24

5239988770268



654999700403

3.897





25

20457802016011



2557227044764

3.904





26

79992676367108



9999088822075

3.910





27

313224032098244



39153010938487

3.916





28

1228088671826973



153511100594603

3.921





n 
translation t(n)

t(n) / t(n1) 
29

4820975409710116

3.926

30

18946775782611174

3.930

31

74541651404935148

3.934

32

293560133910477776

3.938

33

1157186142148293638

3.942

34

4565553929115769162

3.945

35

18027932215016128134

3.949

36

71242712815411950635

3.952

37

281746550485032531911

3.955

38

1115021869572604692100

3.958

39

4415695134978868448596

3.960

40

17498111172838312982542

3.963

41

69381900728932743048483

3.965

42

275265412856343074274146

3.967

43

1092687308874612006972082

3.970

44

4339784013643393384603906

3.972

45

17244800728846724289191074

3.974

46

68557762666345165410168738

3.976

Counting 3dimensional polyominoes is similar. Again, for a given
size (number of cubes), there are 3 main sets of polyominoes.
 The translation polyominoes are the set of polyominoes which are
unique with respect to translation  which means any two polyominoes
in the set can't be made identical by moving them horizontally
and vertically.
 The 3d rotation polyominoes are the set of polyominoes which
are unique with respect to rotation and translation  which means
any two polyominoes can't be made identical by rotating them in
3 dimensions and by translating them.
 The 4d rotation polyominoes are the set of polyominoes which
are unique with respect to 4d rotation, 3d rotation, and translation.
It may sound strange to think of 4d rotation, but think of it
as just flipping. A good example is your hands. You have a right
hand and a left hand; they would be distinct elements in the translation
set and the 3d rotation set. But they would be considered the
same element in the 4d rotation set. You can flip a left hand
through 4d space to create a right hand.
It is easy to prove that t(n) / r(n) <= 24 for all n, and that
r(n) / p(n) <= 2 for all n.
Here is a list of the number of 3dimensional polyominoes of each
size up to 12
 t(n), r(n), and p(n) for n <= 6 were calculated by Lunnon (by
hand!) in 1972.
 t(n), r(n), and p(n) for n = 7 through 9 were calculated by Kevin
Gong in 1992, on a 12processor Sequent computer.
 t(n), r(n), and p(n) for n = 10 through 15 were computed by Kevin
Gong in August/September 1997, on a Power Macintosh 8500/120.
 t(n), r(n), and p(n) for n = 16 was computed by Kevin
Gong in 11 days in November/December 2004, on a Power Mac G5.
n 
translation t(n) 
3d rotation r(n) 
4d rotation p(n) 
t(n) / t(n1)

1

1

1

1



2

3

1

1

3.00

3

15

2

2

5.00

4

86

8

7

5.73

5

534

29

23

6.21

6

3481

166

112

6.52

7

23502

1023

607

6.75

8

162913

6922

3811

6.93

9

1152870

48311

25413

7.08

10

8294738

346543

178083

7.19

11

60494549

2522522

1279537

7.29

12

446205905

18598427

9371094

7.38

13

3322769321

138462649

69513546

7.45

14

24946773111

1039496297

520878101

7.51

15

188625900446

7859514470

3934285874

7.56

16

1435074454755

59795121480

29915913663

7.61

It appears plausible that t(n) / t(n1) > t(n1) / t(n  2) for
all n > 2. I don't know of any known proof, however.
