Editing Polyomino (section)

===Free, one-sided, and fixed polyominoes===
There are three common ways of distinguishing polyominoes for enumeration:<ref>{{cite journal |last=Redelmeier |first=D. Hugh |year=1981 |title=Counting polyominoes: yet another attack |journal=Discrete Mathematics |volume=36 |pages=191–203 |doi=10.1016/0012-365X(81)90237-5 |issue=2|doi-access=free }}</ref><ref>Golomb, chapter 6</ref>

*''free'' polyominoes are distinct when none is a rigid transformation ([[translation (geometry)|translation]], [[rotation]], [[reflection (mathematics)|reflection]] or [[glide reflection]]) of another (pieces that can be picked up and flipped over). Translating, rotating, reflecting, or glide reflecting a free polyomino does not change its shape.
*''one-sided polyominoes'' are distinct when none is a translation or rotation of another (pieces that cannot be flipped over).  Translating or rotating a one-sided polyomino does not change its shape.
*''fixed'' polyominoes are distinct when none is a translation of another (pieces that can be neither flipped nor rotated). Translating a fixed polyomino will not change its shape.

The following table shows the numbers of polyominoes of various types with ''n'' cells.

{|class=wikitable
!rowspan=2|''n''
!rowspan=2|name
!colspan=3|free 
!rowspan=2|one-sided
!rowspan=2|fixed
|-
!total
!with holes
!without holes
|- align=right
|1 ||align=left |monomino ||1 ||0 ||1 ||1 ||1
|- align=right
|2 ||align=left |[[domino (mathematics)|domino]] ||1 ||0 ||1 ||1 ||2
|- align=right
|3 ||align=left |[[tromino]] ||2 ||0 ||2 ||2 ||6
|- align=right
|4 ||align=left |[[tetromino]] ||5 ||0 ||5 ||7 ||19
|- align=right
|5 ||align=left |[[pentomino]] ||12 ||0 ||12 ||18 ||63
|- align=right
|6 ||align=left |[[hexomino]] ||35 ||0 ||35 ||60 ||216
|- align=right
|7 ||align=left |[[heptomino]] ||108 ||1 ||107 ||196 ||760
|- align=right
|8 ||align=left |[[octomino]] ||369 ||6 ||363 ||704 ||2,725
|- align=right
|9 ||align=left |[[nonomino]] ||1,285 ||37 ||1,248 ||2,500 ||9,910
|- align=right
|10 ||align=left |[[decomino]] ||4,655 ||195 ||4,460 ||9,189 ||36,446
|- align=right
|11 ||align=left |undecomino ||17,073 ||979 ||16,094 ||33,896 ||135,268
|- align=right
|12 ||align=left |dodecomino ||63,600 ||4,663 ||58,937 ||126,759 ||505,861
|- align=right
|colspan=2|[[OEIS]]&nbsp;sequence
|{{OEIS link|id=A000105}}
|{{OEIS link|id=A001419}}
|{{OEIS link|id=A000104}}
|{{OEIS link|id=A000988}}
|{{OEIS link|id=A001168}}
|}

Fixed polyominoes were enumerated in 2004 up to {{nowrap begin}}''n'' = 56{{nowrap end}} by Iwan Jensen,<ref>{{cite web |url=http://www.ms.unimelb.edu.au/~iwan/animals/Animals_ser.html |title=Series for lattice animals or polyominoes |access-date=2007-05-06 |author=Iwan Jensen |archive-url=https://web.archive.org/web/20070612141716/http://www.ms.unimelb.edu.au/~iwan/animals/Animals_ser.html |archive-date=2007-06-12 |url-status=live }}</ref> and in 2024 up to {{nowrap begin}}''n'' = 70{{nowrap end}} by Gill Barequet and Gil Ben-Shachar.<ref>{{cite book |chapter-url=https://epubs.siam.org/doi/10.1137/1.9781611977929.10 |title=2024 Proceedings of the Symposium on Algorithm Engineering and Experiments (ALENEX) - Counting Polyominoes, Revisited|chapter=Counting Polyominoes, Revisited |date=January 2024 |pages=133–143 |publisher=Society for Industrial and Applied Mathematics |doi=10.1137/1.9781611977929.10 |last1=Barequet |first1=Gill |last2=Ben-Shachar |first2=Gil |isbn=978-1-61197-792-9 }}</ref> 

Free polyominoes were enumerated in 2007 up to {{nowrap begin}}''n'' = 28{{nowrap end}} by Tomás Oliveira e Silva,<ref>{{cite web |url=http://www.ieeta.pt/%7Etos/animals/a44.html |title=Animal enumerations on the {4,4} Euclidean tiling |access-date=2007-05-06 |author=Tomás Oliveira e Silva |archive-url=https://web.archive.org/web/20070423213531/http://www.ieeta.pt/%7Etos/animals/a44.html |archive-date=2007-04-23 |url-status=live }}</ref> in 2012 up to {{nowrap begin}}''n'' = 45{{nowrap end}} by Toshihiro Shirakawa,<ref>{{cite web |url=https://www.gathering4gardner.org/g4g10gift/math/Shirakawa_Toshihiro-Harmonic_Magic_Square.pdf |title=Harmonic Magic Square, Enumeration of Polyominoes considering the symmetry}}</ref> and in 2023 up to {{nowrap begin}}''n'' = 50{{nowrap end}} by John Mason.<ref>{{cite web |url=https://oeis.org/A000105/a000105_2.pdf |title=Counting size 50 polyominoes}}</ref>

The above OEIS sequences, with the exception of A001419, include the count of 1 for the number of null-polyominoes; a null-polyomino is one that is formed of zero squares.