POLYOMINO TILINGS

Polyomino Tilings

Select polyominoes for a set (currently 1 or 2), for which tilings should be shown.

Then click "Show" button.

You may also see list of all polyomino sets for which data is available here.


L pentomino and Y2 hexomino

Prime rectangles: 79.

Smallest rectangle tilings

Smallest rectangle (6x7):

Smallest square (10x10):

Rectangle tilings' solutions count (including symmetric)

Blue number - strongly prime rectangle (which cannot be divided into two or more number of rectangles tileable by this set).

Green number - weakly prime rectangle (which cannot be divided into two rectangles tileable by this set, but which can be divided into three or more rectangles).

Purple number - prime rectangle (unknown if weakly or strongly prime).

Red number - composite rectangle (which can be divided into two rectangles tileable by this set).

Gray number - it is unknown whether rectangle is prime or composite.

Question mark (?) - solution count is unknown.

Click on underlined numbers to view picture with one solution.

w \ h
1-3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
N>0
1-3
0
4
0
0
5
0
0
0
6
0
0
0
0
7
0
0
0
8
0
8
0
0
0
0
0
0
9
0
0
0
0
0
40
0
10
0
0
0
0
0
0
0
640
11
0
0
0
2
32
72
40
1704
76176
12
0
0
0
176
64
432
1536
7720
≥1
≥1
13
0
4
0
0
896
160
3776
19148
≥1
≥1
≥1
14
0
0
0
82
64
7112
47714
96726
≥1
≥1
≥1
≥1
15
0
0
0
40
64
3072
9936
492482
≥1
≥1
≥1
≥1
≥1
16
0
0
0
176
14864
21488
176148
2039152
≥1
≥1
≥1
≥1
≥1
≥1
17
0
0
0
3216
1104
45744
623052
7986170
≥1
≥1
≥1
≥1
≥1
≥1
≥1
18
0
32
0
546
22992
73930
2638964
26417068
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
19
0
4
0
2984
77728
595604
5473800
111601544
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
20
0
0
0
1480
10992
572912
9647408
423449096
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
21
0
0
0
7656
271288
2540818
50327592
1.65671783×10¹⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
22
0
0
0
57444
487664
4856260
187451554
5.91897938×10¹⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
23
0
192
0
21400
1529340
12800976
509276948
2.15753364×10¹¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
24
0
32
0
80488
2947804
52199094
1.58150224×10¹⁰
8.04820025×10¹¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
25
0
4
0
49800
3591532
83929116
3.84020472×10¹⁰
2.96519282×10¹²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
26
0
16
0
219624
20565564
275080904
1.52172690×10¹¹
1.08248050×10¹³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
27
0
0
0
1031922
23166408
561652866
4.36757031×10¹¹
3.86435333×10¹³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
28
0
1024
0
638352
54313840
1.59685749×10¹⁰
1.25311302×10¹²
1.39232710×10¹⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
29
0
192
0
1940888
186736604
5.02580412×10¹⁰
3.59333940×10¹²
5.03590492×10¹⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
30
0
32
0
1447728
202820680
1.04018919×10¹¹
1.09641683×10¹³
1.81622825×10¹⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
31
0
196
1152
5531794
614443552
2.95231431×10¹¹
3.60914782×10¹³
6.49420311×10¹⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
32
0
32
0
18948368
1.59833292×10¹⁰
6.68332769×10¹¹
1.04909993×10¹⁴
2.31134771×10¹⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
33
0
5120
4608
16499296
3.03636340×10¹⁰
1.85446892×10¹²
2.98970937×10¹⁴
8.25219159×10¹⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
34
0
1024
1056
44139144
7.21408641×10¹⁰
5.21463238×10¹²
9.34052597×10¹⁴
2.95215596×10¹⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
35
0
192
13824
39363118
1.40536966×10¹¹
1.21718155×10¹³
2.81892943×10¹⁵
1.05319145×10¹⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
36
0
1568
4224
130608662
3.44246344×10¹¹
3.22821962×10¹³
8.71903285×10¹⁵
3.74416821×10¹⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
37
0
388
39040
359277896
7.09861037×10¹¹
7.83098689×10¹³
2.50717475×10¹⁶
1.32973729×10¹⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
38
0
24624
15744
398404380
1.46810644×10¹²
2.10742636×10¹⁴
7.49837548×10¹⁶
4.73063345×10¹⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
39
0
5184
102016
971426850
3.64901300×10¹²
5.64204855×10¹⁴
2.30009748×10¹⁷
1.68366938×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
40
0
1024
87296
1.01099490×10¹⁰
6.87883444×10¹²
1.38924116×10¹⁵
6.95006995×10¹⁷
5.98248016×10²⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
41
0
10432
251904
2.96018812×10¹⁰
1.48989832×10¹³
3.59072176×10¹⁵
2.08105519×10¹⁸
2.12300025×10²¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
42
0
3104
320128
7.03330238×10¹⁰
3.66682049×10¹³
9.02028459×10¹⁵
6.15423322×10¹⁸
7.53406877×10²¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
43
0
115268
684032
9.26665065×10¹⁰
7.50322287×10¹³
2.37727908×10¹⁶
1.85335905×10¹⁹
2.67593868×10²²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
44
0
25664
972928
2.10775607×10¹¹
1.61416021×10¹⁴
6.23178258×10¹⁶
5.65525895×10¹⁹
9.50300175×10²²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
45
0
5312
1824000
2.48614596×10¹¹
3.62041752×10¹⁴
1.56951153×10¹⁷
1.69111303×10²⁰
3.37205139×10²³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
46
0
62464
2804096
6.56791556×10¹¹
7.71062102×10¹⁴
4.02837137×10¹⁷
5.05874455×10²⁰
1.19601802×10²⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
47
0
20672
4801536
1.41522231×10¹²
1.67641790×10¹⁵
1.02755217×10¹⁸
1.50994261×10²¹
4.24307839×10²⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
48
0
528928
7544576
2.10294132×10¹²
3.69480662×10¹⁵
2.67590901×10¹⁸
4.56205169×10²¹
1.50580034×10²⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
49
0
125700
12560384
4.54584869×10¹²
8.00383226×10¹⁵
6.94898544×10¹⁸
1.37691880×10²²
5.34277823×10²⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
50
0
27728
19339776
5.91889934×10¹²
1.71407728×10¹⁶
1.76763825×10¹⁹
4.11987167×10²²
1.89495997×10²⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
51
0
349568
32950528
1.44151337×10¹³
3.71223897×10¹⁶
4.53394005×10¹⁹
1.23374164×10²³
6.72023323×10²⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
52
0
124160
50153600
2.91409347×10¹³
8.25340296×10¹⁶
1.16334683×10²⁰
3.70475257×10²³
2.38365533×10²⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
53
0
2390208
84173056
4.69808171×10¹³
1.79456765×10¹⁷
3.01088225×10²⁰
1.11575433×10²⁴
8.45562982×10²⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
54
0
612384
128866048
9.78126780×10¹³
3.82577121×10¹⁷
7.78278427×10²⁰
3.35576662×10²⁴
2.99905983×10²⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
55
0
146372
213889792
1.37658241×10¹⁴
8.44780570×10¹⁷
1.98942475×10²¹
1.00502375×10²⁵
1.06353248×10²⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
56
0
1865824
323373600
3.14678371×10¹⁴
1.83281642×10¹⁸
5.10708690×10²¹
3.01658473×10²⁵
3.77149733×10²⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
57
0
699008
535062016
6.10662066×10¹⁴
3.93906202×10¹⁸
1.31297573×10²²
9.06302151×10²⁵
1.33757129×10³⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
58
0
10672128
808143616
1.03931931×10¹⁵
8.70574381×10¹⁸
3.38780050×10²²
2.72512822×10²⁶
4.74380156×10³⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
59
0
2973888
1.31446246×10¹⁰
2.10424199×10¹⁵
1.87762683×10¹⁹
8.73571508×10²²
8.18322254×10²⁶
1.68227905×10³¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
60
0
777760
2.00093875×10¹⁰
3.14714394×10¹⁵
4.05006247×10¹⁹
2.23892536×10²³
2.45476960×10²⁷
5.96541576×10³¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
61
0
9614468
3.20707660×10¹⁰
6.85336911×10¹⁵
8.88154013×10¹⁹
5.75232401×10²³
7.37178213×10²⁷
2.11539221×10³²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
62
0
3766384
4.88416268×10¹⁰
1.29605310×10¹⁶
1.92633683×10²⁰
1.47963302×10²⁴
2.21469334×10²⁸
7.50168868×10³²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
63
0
47196096
7.78323993×10¹⁰
2.28491636×10¹⁶
4.19652595×10²⁰
3.81204791×10²⁴
6.65255938×10²⁸
2.66025395×10³³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
64
0
14405120
1.18545198×10¹¹
4.53071481×10¹⁶
9.09749871×10²⁰
9.81685850×10²⁴
1.99756448×10²⁹
9.43337457×10³³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
65
0
4132032
1.87053240×10¹¹
7.10537094×10¹⁶
1.97587239×10²¹
2.51982611×10²⁵
5.99628132×10²⁹
3.34504103×10³⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
66
0
48210976
2.85905725×10¹¹
1.49119787×10¹⁷
4.31048533×10²¹
6.47740687×10²⁵
1.80095159×10³⁰
1.18615824×10³⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
67
0
19666244
4.47146022×10¹¹
2.77536660×10¹⁷
9.32519065×10²¹
1.66627005×10²⁶
5.40882924×10³⁰
4.20620640×10³⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
68
0
207117440
6.82412015×10¹¹
5.00407082×10¹⁷
2.03405694×10²²
4.28954828×10²⁶
1.62436587×10³¹
1.49153501×10³⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
69
0
69591360
1.06374308×10¹²
9.76728513×10¹⁷
4.42223752×10²²
1.10390348×10²⁷
4.87769768×10³¹
5.28891768×10³⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
70
0
21833728
1.61936412×10¹²
1.58952701×10¹⁸
9.56416312×10²²
2.83602312×10²⁷
1.46463243×10³²
1.87541698×10³⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
71
0
236553408
2.51304816×10¹²
3.24362764×10¹⁸
2.08776658×10²³
7.29224634×10²⁷
4.39849726×10³²
6.65017507×10³⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
72
0
100313632
3.82607950×10¹²
5.97936823×10¹⁸
4.53575905×10²³
1.87581945×10²⁸
1.32089158×10³³
2.35814364×10³⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
73
0
903290372
5.91924991×10¹²
1.09337658×10¹⁹
9.84391179×10²³
4.82698092×10²⁸
3.96665584×10³³
8.36187777×10³⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
74
0
335156368
8.98667563×10¹²
2.10846943×10¹⁹
2.14282314×10²⁴
1.24179262×10²⁹
1.19119665×10³⁴
2.96505875×10³⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
75
0
114371328
1.38942878×10¹³
3.53187918×10¹⁹
4.65109145×10²⁴
3.19187228×10²⁹
3.57702734×10³⁴
1.05138741×10⁴⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
76
0
1.14030361×10¹⁰
2.10496236×10¹³
7.05501346×10¹⁹
1.01149448×10²⁵
8.20835589×10²⁹
1.07418588×10³⁵
3.72815861×10⁴⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
77
0
502584512
3.24572606×10¹³
1.29346766×10²⁰
2.19857832×10²⁵
2.11137643×10³⁰
3.22574801×10³⁵
1.32198394×10⁴¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
78
0
3.91977372×10¹⁰
4.91719218×10¹³
2.38568612×10²⁰
4.77669414×10²⁵
5.43187379×10³⁰
9.68707209×10³⁵
4.68765755×10⁴¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
79
0
1.60828998×10¹⁰
7.56244572×10¹³
4.55757338×10²⁰
1.03882095×10²⁶
1.39718883×10³¹
2.90905344×10³⁶
1.66220287×10⁴²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
80
0
592941216
1.14459822×10¹⁴
7.80846702×10²⁰
2.25529169×10²⁶
3.59230617×10³¹
8.73568921×10³⁶
5.89403387×10⁴²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
81
0
5.41672064×10¹⁰
1.75768385×10¹⁴
1.53450110×10²¹
4.90426449×10²⁶
9.23871745×10³¹
2.62329429×10³⁷
2.08998014×10⁴³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
82
0
2.48279347×10¹⁰
2.65991760×10¹⁴
2.80557250×10²¹
1.06650252×10²⁷
2.37631994×10³²
7.87774390×10³⁷
7.41090804×10⁴³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
83
0
1.69415016×10¹¹
4.07347021×10¹⁴
5.20115983×10²¹
2.31588358×10²⁷
6.11269429×10³²
2.36571440×10³⁸
2.62784416×10⁴⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
84
0
7.68598582×10¹⁰
6.17190489×10¹⁴
9.86339640×10²¹
5.03562085×10²⁷
1.57220310×10³³
7.10425649×10³⁸
9.31809848×10⁴⁴
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
85
0
3.04068390×10¹⁰
9.43224523×10¹⁴
1.71991669×10²²
1.09446631×10²⁸
4.04290492×10³³
2.13336680×10³⁹
3.30411659×10⁴⁵
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
86
0
2.54154999×10¹¹
1.42887525×10¹⁵
3.33769614×10²²
2.37779464×10²⁸
1.03978748×10³⁴
6.40644672×10³⁹
1.17161190×10⁴⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
87
0
1.21273055×10¹¹
2.18194743×10¹⁵
6.09600793×10²²
5.16991292×10²⁸
2.67439959×10³⁴
1.92385930×10⁴⁰
4.15443432×10⁴⁶
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
88
0
7.29909145×10¹¹
3.30504478×10¹⁵
1.13338136×10²³
1.12347850×10²⁹
6.87896076×10³⁴
5.77738455×10⁴⁰
1.47312472×10⁴⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
89
0
3.65675644×10¹¹
5.03961772×10¹⁵
2.13691742×10²³
2.44144898×10²⁹
1.76924186×10³⁵
1.73494052×10⁴¹
5.22356535×10⁴⁷
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
90
0
1.54258995×10¹¹
7.63720039×10¹⁵
3.77781116×10²³
5.30660838×10²⁹
4.54996346×10³⁵
5.20995932×10⁴¹
1.85222939×10⁴⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
91
0
1.18010013×10¹²
1.16351870×10¹⁶
7.26003438×10²³
1.15324580×10³⁰
1.17021256×10³⁶
1.56454365×10⁴²
6.56784057×10⁴⁸
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
92
0
5.86909994×10¹¹
1.76240782×10¹⁶
1.32603357×10²⁴
2.50676454×10³⁰
3.00980740×10³⁶
4.69833188×10⁴²
2.32889675×10⁴⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
93
0
3.13711423×10¹²
2.68501588×10¹⁶
2.46901996×10²⁴
5.44773749×10³⁰
7.74137672×10³⁶
1.41090905×10⁴³
8.25805287×10⁴⁹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
94
0
1.73162048×10¹²
4.06593409×10¹⁶
4.63400194×10²⁴
1.18391612×10³¹
1.99103386×10³⁷
4.23693445×10⁴³
2.92822934×10⁵⁰
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
95
0
7.74583932×10¹¹
6.19028066×10¹⁶
8.28068294×10²⁴
2.57343922×10³¹
5.12057086×10³⁷
1.27234110×10⁴⁴
1.03832337×10⁵¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
96
0
5.43069225×10¹²
9.37786348×10¹⁶
1.57922919×10²⁵
5.59236364×10³¹
1.31697587×10³⁸
3.82083065×10⁴⁴
3.68179985×10⁵¹
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
97
0
2.81855955×10¹²
1.42697370×10¹⁷
2.88646498×10²⁵
1.21546198×10³²
3.38724563×10³⁸
1.14739454×10⁴⁵
1.30553226×10⁵²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
98
0
1.34590968×10¹³
2.16149660×10¹⁷
5.37766428×10²⁵
2.64187933×10³²
8.71198451×10³⁸
3.44561607×10⁴⁵
4.62929613×10⁵²
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
99
0
8.16057959×10¹²
3.28889124×10¹⁷
1.00570828×10²⁶
5.74102422×10³²
2.24066332×10³⁹
1.03471280×10⁴⁶
1.64150550×10⁵³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
100
0
3.85234585×10¹²
4.98173437×10¹⁷
1.81220997×10²⁶
1.24780054×10³³
5.76270493×10³⁹
3.10722654×10⁴⁶
5.82062705×10⁵³
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
≥1
?
N>0
x
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all
all
all
all
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Formulas

$N(w; h)$ - number of ways to tile $w\times h$ rectangle (including symmetric solutions)

$T(w; h) = \begin{cases} 1, & N(w; h) \geq 1 \\ 0, & \text{else} \end{cases}$ - tileability function, $1$ if tiles rectangle, $0$ otherwise

$A(w; h) = \left(N(w; h)\right)^{\frac{1}{wh}}$ - average number of ways to tile cell in $w\times h$ rectangle (including symmetric solutions)

$G(T; x; y) = \sum_{w=1}^{\infty}\sum _{h=1}^{\infty}T(w; h)x^wy^h$ - bivariate generating function of $T(w; h)$

$G(A; x; y) = \sum_{w=1}^{\infty}\sum _{h=1}^{\infty}A(w; h)x^wy^h$ - bivariate generating function of $A(w; h)$

$G(N(4); x) = \frac{4x^{13}}{1 - 8x^5 - x^6 + 16x^{10} + 8x^{11} - 4x^{13} - 16x^{16} + 16x^{18}} \tag{1}$

See Also

L pentomino and Y1 hexominoL pentomino and Z1 hexomino