Animal enumerations on the {4,4} Euclidean tiling

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Tables: Summary 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
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Introduction

The animals on the {4,4} regular tiling of the Euclidean plane are called polyominoes, a name coined by Solomon Golomb in 1953 [1]. An alternative name would be square animals, since the {4,4} regular tiling is composed of squares. Each polyomino may be equal to its mirror image (achiral) or different from its mirror image (chiral). The vast majority of the polyominoes do not have any kind of symmetry.

Redelmeier [2] developed an algorithm to enumerate the polyominoes of a given area, counting all their possible orientations. He developed another program to enumerate the polyominoes of a given area with a predefined symmetry (axial, rotational, mirror image). By combining the results of the different enumerations it is possible to infer the number of polyominoes with different shapes (ie, disregarding orientation and mirror images). Using 8 months of CPU time, he was able to count the number of polyominoes with areas up to 24. In 2000 there was a spectacular progress in the enumeration of polyominoes (in all possible orientations) of a given area, with Jensen and Guttmann [3] reporting results for areas up to 46 (the current computational limit appears to be area 56; see the appendix of [4] and the notes attached to the POLYNUM and POLYSAVE programs kindly made available by Donald E. Knuth, who was the first to obtain correct results for area 47). The number of different shapes, however, only appears to be known up to area 28 (see results below).

In 1998 we developed a program to count the number of polyominoes with a given area which uses an algorithm similar to the parallel version of the Redelmeier algorithm [5]. With the help of some friends (who donated computational power) we used this program to enumerate the number of polyominoes up to area 28. We modified the algorithm in 1999 to count the number of holes in each polyomino. The new program was only twelve times slower than the old one, which allowed us to obtain results up to area 26 (see results below). In 2003, we decided to use the Redelmeier algorithm to generate all polyominoes of a given area, classifying them according to the areas of their holes (if any), and according to their chirality (see results for areas 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 below). Because we did not use an incremental approach, our new program is slow (we used a simple trick to count each polyomino shape only once).

Tables with counts of the number of polyominoes (June 25, 2007)

The following tables contain counts of the number of polyominoes with distinct shapes. Those interested in one example of a polyomino with a given area, and with holes with given areas, should take a look at this page (be advised that it has hundreds of small gif images).

Summary of the polyomino data

Area	Number of polyominoes
Area	Any number of holes	No holes	One hole	Two holes	Three holes	Four holes	Five holes	Six holes	Seven holes
1	1	1
2	1	1
3	2	2
4	5	5
5	12	12
6	35	35
7	108	107	1
8	369	363	6
9	1285	1248	37
10	4655	4460	195
11	17073	16094	975	4
12	63600	58937	4622	41
13	238591	217117	21128	346
14	901971	805475	94109	2384	3
15	3426576	3001127	410820	14560	69
16	13079255	11230003	1766591	81863	798
17	50107909	42161529	7506179	433172	7021	8
18	192622052	158781106	31596240	2192752	51775	179
19	742624232	599563893	131991619	10726252	339958	2509	1
20	2870671950	2269506062	547992183	51094203	2053872	25609	21
21	11123060678	8609442688	2263477612	238259280	11665593	214932	573
22	43191857688	32725637373	9309386178	1092053068	63192945	1578984	9140
23	168047007728	124621833354	38150082057	4934652298	329798278	10536260	105417	64
24	654999700403	475368834568	155859235424	22034767837	1670466031	65411645	982767	2131
25	2557227044764	1816103345752	635067478628	97407519119	8256762365	383981499	7919375	38022	4
26	9999088822075	6948228104703	2581737704039	426916828181	39992202198	2156114468	57387444	480713	329
27	39153010938487	?	?	?	?	?	?	?	?
28	153511100594603	?	?	?	?	?	?	?	?

Detailed data for polyominoes with area 1

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	1	0	1
0 holes	1	0	1

Detailed data for polyominoes with area 2

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	1	0	1
0 holes	1	0	1

Detailed data for polyominoes with area 3

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	2	0	2
0 holes	2	0	2

Detailed data for polyominoes with area 4

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	5	2	3
0 holes	5	2	3

Detailed data for polyominoes with area 5

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	12	6	6
0 holes	12	6	6

Detailed data for polyominoes with area 6

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	35	25	10
0 holes	35	25	10

Detailed data for polyominoes with area 7

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	108	88	20
0 holes	107	88	19
1 hole with area 1	1	0	1

Detailed data for polyominoes with area 8

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	369	335	34
0 holes	363	330	33
1 hole with area 1	6	5	1

Detailed data for polyominoes with area 9

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	1285	1215	70
0 holes	1248	1184	64
1 hole with area 1	36	30	6
1 hole with area 2	1	1	0

Detailed data for polyominoes with area 10

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	4655	4534	121
0 holes	4460	4348	112
1 hole with area 1	182	174	8
1 hole with area 2	13	12	1

Detailed data for polyominoes with area 11

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	17073	16823	250
0 holes	16094	15874	220
1 hole with area 1	880	855	25
1 hole with area 2	90	89	1
1 hole with area 3	4	3	1
1 hole with area 4	1	0	1
2 holes with areas 1,1	4	2	2

Detailed data for polyominoes with area 12

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	63600	63159	441
0 holes	58937	58550	387
1 hole with area 1	4033	3995	38
1 hole with area 2	531	523	8
1 hole with area 3	50	47	3
1 hole with area 4	8	7	1
2 holes with areas 1,1	41	37	4

Detailed data for polyominoes with area 13

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	238591	237679	912
0 holes	217117	216348	769
1 hole with area 1	17908	17808	100
1 hole with area 2	2747	2739	8
1 hole with area 3	393	384	9
1 hole with area 4	74	67	7
1 hole with area 5	5	5	0
1 hole with area 6	1	1	0
2 holes with areas 1,1	332	313	19
2 holes with areas 2,1	14	14	0

Detailed data for polyominoes with area 14

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	901971	900341	1630
0 holes	805475	804118	1357
1 hole with area 1	77623	77466	157
1 hole with area 2	13355	13314	41
1 hole with area 3	2470	2446	24
1 hole with area 4	564	553	11
1 hole with area 5	80	80	0
1 hole with area 6	17	16	1
2 holes with areas 1,1	2154	2116	38
2 holes with areas 2,1	223	223	0
2 holes with areas 2,2	2	1	1
2 holes with areas 3,1	5	5	0
3 holes with areas 1,1,1	3	3	0

Detailed data for polyominoes with area 15

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	3426576	3423201	3375
0 holes	3001127	2998403	2724
1 hole with area 1	330823	330436	387
1 hole with area 2	61730	61687	43
1 hole with area 3	13602	13551	51
1 hole with area 4	3699	3665	34
1 hole with area 5	766	763	3
1 hole with area 6	180	176	4
1 hole with area 7	15	14	1
1 hole with area 8	4	3	1
1 hole with area 9	1	0	1
2 holes with areas 1,1	12349	12237	112
2 holes with areas 2,1	2030	2029	1
2 holes with areas 2,2	37	33	4
2 holes with areas 3,1	134	132	2
2 holes with areas 4,1	10	8	2
3 holes with areas 1,1,1	69	64	5

Detailed data for polyominoes with area 16

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	13079255	13073163	6092
0 holes	11230003	11225183	4820
1 hole with area 1	1391963	1391347	616
1 hole with area 2	276521	276340	181
1 hole with area 3	69052	68930	122
1 hole with area 4	21481	21414	67
1 hole with area 5	5676	5667	9
1 hole with area 6	1561	1547	14
1 hole with area 7	261	259	2
1 hole with area 8	66	63	3
1 hole with area 9	10	9	1
2 holes with areas 1,1	65337	65113	224
2 holes with areas 2,1	14206	14204	2
2 holes with areas 2,2	454	439	15
2 holes with areas 3,1	1638	1634	4
2 holes with areas 3,2	23	23	0
2 holes with areas 4,1	193	192	1
2 holes with areas 4,2	2	2	0
2 holes with areas 5,1	10	10	0
3 holes with areas 1,1,1	770	759	11
3 holes with areas 2,1,1	28	28	0

Detailed data for polyominoes with area 17

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	50107909	50095285	12624
0 holes	42161529	42151775	9754
1 hole with area 1	5800792	5799329	1463
1 hole with area 2	1208986	1208786	200
1 hole with area 3	331256	331015	241
1 hole with area 4	114194	114042	152
1 hole with area 5	35900	35867	33
1 hole with area 6	11386	11351	35
1 hole with area 7	2729	2718	11
1 hole with area 8	760	750	10
1 hole with area 9	150	140	10
1 hole with area 10	20	20	0
1 hole with area 11	5	5	0
1 hole with area 12	1	1	0
2 holes with areas 1,1	326490	325927	563
2 holes with areas 2,1	85634	85623	11
2 holes with areas 2,2	3918	3875	43
2 holes with areas 3,1	13904	13886	18
2 holes with areas 3,2	519	519	0
2 holes with areas 3,3	4	3	1
2 holes with areas 4,1	2343	2326	17
2 holes with areas 4,2	61	61	0
2 holes with areas 5,1	274	272	2
2 holes with areas 6,1	25	23	2
3 holes with areas 1,1,1	6381	6327	54
3 holes with areas 2,1,1	624	623	1
3 holes with areas 2,2,1	5	5	0
3 holes with areas 3,1,1	11	11	0
4 holes with areas 1,1,1,1	8	5	3

Detailed data for polyominoes with area 18

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	192622052	192599091	22961
0 holes	158781106	158763808	17298
1 hole with area 1	23994287	23991929	2358
1 hole with area 2	5195915	5195169	746
1 hole with area 3	1527732	1527189	543
1 hole with area 4	570240	569917	323
1 hole with area 5	203742	203658	84
1 hole with area 6	73124	73021	103
1 hole with area 7	21930	21907	23
1 hole with area 8	6977	6943	34
1 hole with area 9	1762	1750	12
1 hole with area 10	406	403	3
1 hole with area 11	104	104	0
1 hole with area 12	21	20	1
2 holes with areas 1,1	1567944	1566832	1112
2 holes with areas 2,1	469479	469458	21
2 holes with areas 2,2	27383	27266	117
2 holes with areas 3,1	95534	95496	38
2 holes with areas 3,2	6129	6129	0
2 holes with areas 3,3	129	123	6
2 holes with areas 4,1	20776	20763	13
2 holes with areas 4,2	949	948	1
2 holes with areas 4,3	18	18	0
2 holes with areas 4,4	2	1	1
2 holes with areas 5,1	3736	3732	4
2 holes with areas 5,2	50	50	0
2 holes with areas 6,1	570	569	1
2 holes with areas 6,2	8	8	0
2 holes with areas 7,1	40	40	0
2 holes with areas 8,1	5	5	0
3 holes with areas 1,1,1	43746	43642	104
3 holes with areas 2,1,1	7430	7422	8
3 holes with areas 2,2,1	183	181	2
3 holes with areas 3,1,1	395	394	1
3 holes with areas 4,1,1	21	21	0
4 holes with areas 1,1,1,1	179	175	4

Detailed data for polyominoes with area 19

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	742624232	742576616	47616
0 holes	599563893	599528646	35247
1 hole with area 1	98671032	98665553	5479
1 hole with area 2	22035108	22034248	860
1 hole with area 3	6845126	6844089	1037
1 hole with area 4	2718241	2717598	643
1 hole with area 5	1070321	1070107	214
1 hole with area 6	425091	424889	202
1 hole with area 7	149568	149487	81
1 hole with area 8	53812	53738	74
1 hole with area 9	16618	16553	65
1 hole with area 10	4866	4857	9
1 hole with area 11	1423	1416	7
1 hole with area 12	343	338	5
1 hole with area 13	50	47	3
1 hole with area 14	15	14	1
1 hole with area 15	4	3	1
1 hole with area 16	1	0	1
2 holes with areas 1,1	7309098	7306494	2604
2 holes with areas 2,1	2413180	2413106	74
2 holes with areas 2,2	166733	166448	285
2 holes with areas 3,1	575026	574915	111
2 holes with areas 3,2	52010	52008	2
2 holes with areas 3,3	2024	2000	24
2 holes with areas 4,1	151264	151171	93
2 holes with areas 4,2	10208	10208	0
2 holes with areas 4,3	485	483	2
2 holes with areas 4,4	34	30	4
2 holes with areas 5,1	35873	35853	20
2 holes with areas 5,2	1207	1207	0
2 holes with areas 5,3	15	15	0
2 holes with areas 6,1	7553	7530	23
2 holes with areas 6,2	217	217	0
2 holes with areas 6,3	2	2	0
2 holes with areas 7,1	1106	1104	2
2 holes with areas 7,2	5	5	0
2 holes with areas 8,1	194	192	2
2 holes with areas 9,1	18	16	2
3 holes with areas 1,1,1	265841	265497	344
3 holes with areas 2,1,1	64189	64169	20
3 holes with areas 2,2,1	2943	2929	14
3 holes with areas 2,2,2	15	15	0
3 holes with areas 3,1,1	6163	6151	12
3 holes with areas 3,2,1	157	157	0
3 holes with areas 4,1,1	614	603	11
3 holes with areas 4,2,1	10	10	0
3 holes with areas 5,1,1	25	25	0
3 holes with areas 6,1,1	1	1	0
4 holes with areas 1,1,1,1	2397	2360	37
4 holes with areas 2,1,1,1	112	112	0
5 holes with areas 1,1,1,1,1	1	0	1

Detailed data for polyominoes with area 20

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	2870671950	2870584814	87136
0 holes	2269506062	2269443437	62625
1 hole with area 1	403867921	403859006	8915
1 hole with area 2	92506668	92503708	2960
1 hole with area 3	30019885	30017621	2264
1 hole with area 4	12519762	12518361	1401
1 hole with area 5	5314416	5313925	491
1 hole with area 6	2292717	2292150	567
1 hole with area 7	909661	909484	177
1 hole with area 8	364322	364089	233
1 hole with area 9	130691	130596	95
1 hole with area 10	45072	45021	51
1 hole with area 11	15096	15083	13
1 hole with area 12	4451	4431	20
1 hole with area 13	1094	1089	5
1 hole with area 14	333	331	2
1 hole with area 15	82	79	3
1 hole with area 16	12	11	1
2 holes with areas 1,1	33325720	33320639	5081
2 holes with areas 2,1	11848372	11848233	139
2 holes with areas 2,2	926743	926055	688
2 holes with areas 3,1	3166384	3166160	224
2 holes with areas 3,2	361838	361834	4
2 holes with areas 3,3	21372	21290	82
2 holes with areas 4,1	959865	959772	93
2 holes with areas 4,2	86295	86284	11
2 holes with areas 4,3	6955	6951	4
2 holes with areas 4,4	539	524	15
2 holes with areas 5,1	276992	276949	43
2 holes with areas 5,2	15657	15657	0
2 holes with areas 5,3	570	570	0
2 holes with areas 5,4	44	44	0
2 holes with areas 6,1	73125	73106	19
2 holes with areas 6,2	3324	3323	1
2 holes with areas 6,3	90	90	0
2 holes with areas 6,4	8	8	0
2 holes with areas 7,1	15796	15794	2
2 holes with areas 7,2	296	296	0
2 holes with areas 8,1	3535	3531	4
2 holes with areas 8,2	45	45	0
2 holes with areas 9,1	562	561	1
2 holes with areas 9,2	6	6	0
2 holes with areas 10,1	60	60	0
2 holes with areas 11,1	10	10	0
3 holes with areas 1,1,1	1486410	1485772	638
3 holes with areas 2,1,1	456957	456864	93
3 holes with areas 2,2,1	31478	31433	45
3 holes with areas 2,2,2	375	373	2
3 holes with areas 3,1,1	63863	63822	41
3 holes with areas 3,2,1	4033	4033	0
3 holes with areas 3,2,2	16	16	0
3 holes with areas 3,3,1	20	20	0
3 holes with areas 4,1,1	9354	9338	16
3 holes with areas 4,2,1	380	380	0
3 holes with areas 5,1,1	908	908	0
3 holes with areas 5,2,1	8	8	0
3 holes with areas 6,1,1	68	67	1
3 holes with areas 7,1,1	2	2	0
4 holes with areas 1,1,1,1	22887	22823	64
4 holes with areas 2,1,1,1	2654	2654	0
4 holes with areas 2,2,1,1	34	33	1
4 holes with areas 3,1,1,1	34	34	0
5 holes with areas 1,1,1,1,1	21	20	1

Detailed data for polyominoes with area 21

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	11123060678	11122879867	180811
0 holes	8609442688	8609314346	128342
1 hole with area 1	1646803359	1646782909	20450
1 hole with area 2	385250916	385247381	3535
1 hole with area 3	129516107	129511871	4236
1 hole with area 4	56174393	56171782	2611
1 hole with area 5	25298022	25296911	1111
1 hole with area 6	11669992	11668995	997
1 hole with area 7	5085516	5085029	487
1 hole with area 8	2228684	2228241	443
1 hole with area 9	899746	899382	364
1 hole with area 10	351214	351109	105
1 hole with area 11	132796	132713	83
1 hole with area 12	45810	45754	56
1 hole with area 13	14475	14439	36
1 hole with area 14	4817	4802	15
1 hole with area 15	1367	1354	13
1 hole with area 16	308	297	11
1 hole with area 17	64	64	0
1 hole with area 18	20	20	0
1 hole with area 19	5	5	0
1 hole with area 20	1	1	0
2 holes with areas 1,1	149350574	149339098	11476
2 holes with areas 2,1	56249874	56249467	407
2 holes with areas 2,2	4827076	4825521	1555
2 holes with areas 3,1	16377486	16376911	575
2 holes with areas 3,2	2211943	2211918	25
2 holes with areas 3,3	173931	173696	235
2 holes with areas 4,1	5529529	5529089	440
2 holes with areas 4,2	615779	615776	3
2 holes with areas 4,3	71211	71190	21
2 holes with areas 4,4	6506	6453	53
2 holes with areas 5,1	1845989	1845860	129
2 holes with areas 5,2	146840	146840	0
2 holes with areas 5,3	9940	9940	0
2 holes with areas 5,4	1234	1232	2
2 holes with areas 5,5	22	19	3
2 holes with areas 6,1	578551	578405	146
2 holes with areas 6,2	37044	37043	1
2 holes with areas 6,3	1873	1870	3
2 holes with areas 6,4	237	235	2
2 holes with areas 6,5	6	6	0
2 holes with areas 6,6	2	1	1
2 holes with areas 7,1	160135	160106	29
2 holes with areas 7,2	6017	6017	0
2 holes with areas 7,3	99	99	0
2 holes with areas 7,4	5	5	0
2 holes with areas 8,1	43747	43719	28
2 holes with areas 8,2	1253	1253	0
2 holes with areas 8,3	17	17	0
2 holes with areas 9,1	9818	9792	26
2 holes with areas 9,2	199	199	0
2 holes with areas 9,3	2	2	0
2 holes with areas 10,1	1892	1888	4
2 holes with areas 10,2	10	10	0
2 holes with areas 11,1	392	390	2
2 holes with areas 12,1	44	42	2
2 holes with areas 13,1	3	1	2
3 holes with areas 1,1,1	7834163	7832371	1792
3 holes with areas 2,1,1	2863512	2863341	171
3 holes with areas 2,2,1	262146	261987	159
3 holes with areas 2,2,2	5051	5044	7
3 holes with areas 3,1,1	517826	517683	143
3 holes with areas 3,2,1	55056	55056	0
3 holes with areas 3,2,2	676	675	1
3 holes with areas 3,3,1	1055	1052	3
3 holes with areas 4,1,1	100291	100176	115
3 holes with areas 4,2,1	7311	7311	0
3 holes with areas 4,2,2	49	47	2
3 holes with areas 4,3,1	144	144	0
3 holes with areas 4,4,1	5	5	0
3 holes with areas 5,1,1	15549	15542	7
3 holes with areas 5,2,1	520	520	0
3 holes with areas 6,1,1	1995	1985	10
3 holes with areas 6,2,1	42	42	0
3 holes with areas 7,1,1	183	180	3
3 holes with areas 8,1,1	18	18	0
3 holes with areas 9,1,1	1	1	0
4 holes with areas 1,1,1,1	177493	177201	292
4 holes with areas 2,1,1,1	34602	34602	0
4 holes with areas 2,2,1,1	1203	1191	12
4 holes with areas 3,1,1,1	1533	1527	6
4 holes with areas 3,2,1,1	8	8	0
4 holes with areas 4,1,1,1	93	86	7
5 holes with areas 1,1,1,1,1	568	552	16
5 holes with areas 2,1,1,1,1	5	5	0

Detailed data for polyominoes with area 22

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	43191857688	43191525139	332549
0 holes	32725637373	32725408958	228415
1 hole with area 1	6694052374	6694018850	33524
1 hole with area 2	1594200534	1594189034	11500
1 hole with area 3	551730538	551721469	9069
1 hole with area 4	247036147	247030408	5739
1 hole with area 5	116629832	116627428	2404
1 hole with area 6	56815289	56812592	2697
1 hole with area 7	26674410	26673349	1061
1 hole with area 8	12606474	12605178	1296
1 hole with area 9	5595309	5594712	597
1 hole with area 10	2414244	2413810	434
1 hole with area 11	1010403	1010245	158
1 hole with area 12	394910	394719	191
1 hole with area 13	146014	145945	69
1 hole with area 14	54064	54023	41
1 hole with area 15	17999	17963	36
1 hole with area 16	5375	5356	19
1 hole with area 17	1603	1603	0
1 hole with area 18	506	503	3
1 hole with area 19	128	128	0
1 hole with area 20	25	24	1
2 holes with areas 1,1	660327206	660305036	22170
2 holes with areas 2,1	260355556	260354796	760
2 holes with areas 2,2	23991302	23987785	3517
2 holes with areas 3,1	80933112	80932001	1111
2 holes with areas 3,2	12369714	12369666	48
2 holes with areas 3,3	1194912	1194278	634
2 holes with areas 4,1	29689608	29689102	506
2 holes with areas 4,2	3888906	3888830	76
2 holes with areas 4,3	585185	585141	44
2 holes with areas 4,4	63038	62893	145
2 holes with areas 5,1	11077385	11077110	275
2 holes with areas 5,2	1123665	1123663	2
2 holes with areas 5,3	114642	114636	6
2 holes with areas 5,4	18713	18709	4
2 holes with areas 5,5	750	736	14
2 holes with areas 6,1	3965527	3965371	156
2 holes with areas 6,2	331234	331219	15
2 holes with areas 6,3	25725	25719	6
2 holes with areas 6,4	3980	3978	2
2 holes with areas 6,5	261	261	0
2 holes with areas 6,6	37	33	4
2 holes with areas 7,1	1308427	1308386	41
2 holes with areas 7,2	75949	75949	0
2 holes with areas 7,3	3089	3089	0
2 holes with areas 7,4	296	296	0
2 holes with areas 7,5	4	4	0
2 holes with areas 8,1	420441	420390	51
2 holes with areas 8,2	19339	19337	2
2 holes with areas 8,3	621	621	0
2 holes with areas 8,4	46	46	0
2 holes with areas 9,1	119064	119041	23
2 holes with areas 9,2	3960	3960	0
2 holes with areas 9,3	81	81	0
2 holes with areas 9,4	6	6	0
2 holes with areas 10,1	30962	30957	5
2 holes with areas 10,2	566	566	0
2 holes with areas 11,1	7866	7864	2
2 holes with areas 11,2	90	90	0
2 holes with areas 12,1	1515	1511	4
2 holes with areas 12,2	16	16	0
2 holes with areas 13,1	227	226	1
2 holes with areas 14,1	40	40	0
2 holes with areas 15,1	5	5	0
3 holes with areas 1,1,1	39535456	39532193	3263
3 holes with areas 2,1,1	16432933	16432290	643
3 holes with areas 2,2,1	1847469	1847085	384
3 holes with areas 2,2,2	49438	49394	44
3 holes with areas 3,1,1	3573376	3572983	393
3 holes with areas 3,2,1	538606	538606	0
3 holes with areas 3,2,2	12016	12004	12
3 holes with areas 3,3,1	19995	19978	17
3 holes with areas 3,3,2	272	272	0
3 holes with areas 4,1,1	851689	851490	199
3 holes with areas 4,2,1	93907	93907	0
3 holes with areas 4,2,2	1390	1383	7
3 holes with areas 4,3,1	4375	4375	0
3 holes with areas 4,3,2	43	43	0
3 holes with areas 4,4,1	211	209	2
3 holes with areas 5,1,1	179164	179139	25
3 holes with areas 5,2,1	11834	11834	0
3 holes with areas 5,2,2	61	61	0
3 holes with areas 5,3,1	177	177	0
3 holes with areas 5,4,1	9	9	0
3 holes with areas 6,1,1	33060	33028	32
3 holes with areas 6,2,1	1524	1524	0
3 holes with areas 6,2,2	5	5	0
3 holes with areas 6,3,1	16	16	0
3 holes with areas 7,1,1	5034	5029	5
3 holes with areas 7,2,1	76	76	0
3 holes with areas 8,1,1	737	737	0
3 holes with areas 8,2,1	10	10	0
3 holes with areas 9,1,1	60	59	1
3 holes with areas 10,1,1	2	2	0
4 holes with areas 1,1,1,1	1196571	1196021	550
4 holes with areas 2,1,1,1	330338	330334	4
4 holes with areas 2,2,1,1	20450	20414	36
4 holes with areas 2,2,2,1	187	187	0
4 holes with areas 3,1,1,1	27335	27320	15
4 holes with areas 3,2,1,1	1156	1156	0
4 holes with areas 4,1,1,1	2746	2741	5
4 holes with areas 4,2,1,1	74	74	0
4 holes with areas 5,1,1,1	118	118	0
4 holes with areas 6,1,1,1	9	9	0
5 holes with areas 1,1,1,1,1	8577	8543	34
5 holes with areas 2,1,1,1,1	561	561	0
5 holes with areas 2,2,1,1,1	2	2	0

Detailed data for polyominoes with area 23

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	168047007728	168046317330	690398
0 holes	124621833354	124621362987	470367
1 hole with area 1	27139947540	27139871257	76283
1 hole with area 2	6562772229	6562758087	14142
1 hole with area 3	2326872375	2326855582	16793
1 hole with area 4	1069436317	1069425982	10335
1 hole with area 5	524556447	524551320	5127
1 hole with area 6	267162721	267158220	4501
1 hole with area 7	133196710	133194171	2539
1 hole with area 8	67041503	67039218	2285
1 hole with area 9	32157058	32155181	1877
1 hole with area 10	15067394	15066628	766
1 hole with area 11	6870358	6869754	604
1 hole with area 12	2968895	2968477	418
1 hole with area 13	1232715	1232427	288
1 hole with area 14	504044	503890	154
1 hole with area 15	191571	191449	122
1 hole with area 16	68407	68317	90
1 hole with area 17	24104	24085	19
1 hole with area 18	8285	8270	15
1 hole with area 19	2506	2497	9
1 hole with area 20	664	657	7
1 hole with area 21	144	140	4
1 hole with area 22	50	47	3
1 hole with area 23	15	14	1
1 hole with area 24	4	3	1
1 hole with area 25	1	0	1
2 holes with areas 1,1	2887818753	2887769641	49112
2 holes with areas 2,1	1181696446	1181694444	2002
2 holes with areas 2,2	115120600	115112943	7657
2 holes with areas 3,1	386549096	386546420	2676
2 holes with areas 3,2	64896148	64895962	186
2 holes with areas 3,3	7295991	7294408	1583
2 holes with areas 4,1	151302131	151300190	1941
2 holes with areas 4,2	22450553	22450514	39
2 holes with areas 4,3	4119875	4119734	141
2 holes with areas 4,4	512624	512191	433
2 holes with areas 5,1	61542836	61542131	705
2 holes with areas 5,2	7480070	7480061	9
2 holes with areas 5,3	1020632	1020618	14
2 holes with areas 5,4	204373	204351	22
2 holes with areas 5,5	13447	13387	60
2 holes with areas 6,1	24467078	24466321	757
2 holes with areas 6,2	2516309	2516296	13
2 holes with areas 6,3	269458	269422	36
2 holes with areas 6,4	49420	49395	25
2 holes with areas 6,5	5415	5415	0
2 holes with areas 6,6	641	624	17
2 holes with areas 7,1	9215353	9215127	226
2 holes with areas 7,2	724006	724004	2
2 holes with areas 7,3	50217	50214	3
2 holes with areas 7,4	6767	6763	4
2 holes with areas 7,5	328	328	0
2 holes with areas 7,6	49	49	0
2 holes with areas 8,1	3379821	3379602	219
2 holes with areas 8,2	216874	216874	0
2 holes with areas 8,3	11918	11916	2
2 holes with areas 8,4	1424	1420	4
2 holes with areas 8,5	47	47	0
2 holes with areas 8,6	12	12	0
2 holes with areas 9,1	1135104	1134915	189
2 holes with areas 9,2	55779	55778	1
2 holes with areas 9,3	2033	2030	3
2 holes with areas 9,4	217	215	2
2 holes with areas 9,5	6	6	0
2 holes with areas 9,6	2	2	0
2 holes with areas 10,1	358829	358775	54
2 holes with areas 10,2	12352	12352	0
2 holes with areas 10,3	196	196	0
2 holes with areas 10,4	10	10	0
2 holes with areas 11,1	108804	108770	34
2 holes with areas 11,2	2791	2791	0
2 holes with areas 11,3	38	38	0
2 holes with areas 12,1	28314	28285	29
2 holes with areas 12,2	568	568	0
2 holes with areas 12,3	8	8	0
2 holes with areas 13,1	6581	6555	26
2 holes with areas 13,2	45	45	0
2 holes with areas 14,1	1566	1564	2
2 holes with areas 14,2	5	5	0
2 holes with areas 15,1	298	296	2
2 holes with areas 16,1	36	34	2
2 holes with areas 17,1	4	4	0
3 holes with areas 1,1,1	193109652	193101226	8426
3 holes with areas 2,1,1	88519223	88518156	1067
3 holes with areas 2,2,1	11592142	11591054	1088
3 holes with areas 2,2,2	393245	393149	96
3 holes with areas 3,1,1	22053063	22052051	1012
3 holes with areas 3,2,1	4265309	4265309	0
3 holes with areas 3,2,2	140339	140290	49
3 holes with areas 3,3,1	239660	239570	90
3 holes with areas 3,3,2	7693	7691	2
3 holes with areas 3,3,3	31	31	0
3 holes with areas 4,1,1	6155496	6154737	759
3 holes with areas 4,2,1	920640	920640	0
3 holes with areas 4,2,2	22014	21976	38
3 holes with areas 4,3,1	71981	71975	6
3 holes with areas 4,3,2	1675	1675	0
3 holes with areas 4,3,3	2	2	0
3 holes with areas 4,4,1	4566	4553	13
3 holes with areas 4,4,2	59	59	0
3 holes with areas 5,1,1	1608199	1608075	124
3 holes with areas 5,2,1	166035	166035	0
3 holes with areas 5,2,2	2114	2114	0
3 holes with areas 5,3,1	6635	6635	0
3 holes with areas 5,3,2	33	33	0
3 holes with areas 5,4,1	549	549	0
3 holes with areas 5,5,1	2	2	0
3 holes with areas 6,1,1	383901	383755	146
3 holes with areas 6,2,1	29215	29215	0
3 holes with areas 6,2,2	269	268	1
3 holes with areas 6,3,1	854	854	0
3 holes with areas 6,4,1	47	47	0
3 holes with areas 7,1,1	80971	80920	51
3 holes with areas 7,2,1	3306	3306	0
3 holes with areas 7,2,2	5	5	0
3 holes with areas 7,3,1	12	12	0
3 holes with areas 8,1,1	16050	16034	16
3 holes with areas 8,2,1	523	523	0
3 holes with areas 9,1,1	2379	2360	19
3 holes with areas 9,2,1	36	36	0
3 holes with areas 10,1,1	309	306	3
3 holes with areas 11,1,1	40	40	0
3 holes with areas 12,1,1	4	4	0
4 holes with areas 1,1,1,1	7306331	7304492	1839
4 holes with areas 2,1,1,1	2588530	2588508	22
4 holes with areas 2,2,1,1	235175	234997	178
4 holes with areas 2,2,2,1	5036	5036	0
4 holes with areas 2,2,2,2	13	12	1
4 holes with areas 3,1,1,1	317735	317660	75
4 holes with areas 3,2,1,1	29638	29638	0
4 holes with areas 3,2,2,1	239	237	2
4 holes with areas 3,3,1,1	211	205	6
4 holes with areas 4,1,1,1	45770	45693	77
4 holes with areas 4,2,1,1	2747	2747	0
4 holes with areas 4,2,2,1	10	7	3
4 holes with areas 4,3,1,1	15	13	2
4 holes with areas 4,4,1,1	1	0	1
4 holes with areas 5,1,1,1	4295	4295	0
4 holes with areas 5,2,1,1	63	63	0
4 holes with areas 6,1,1,1	447	446	1
4 holes with areas 7,1,1,1	4	4	0
5 holes with areas 1,1,1,1,1	91601	91416	185
5 holes with areas 2,1,1,1,1	13402	13398	4
5 holes with areas 2,2,1,1,1	221	218	3
5 holes with areas 3,1,1,1,1	188	186	2
5 holes with areas 4,1,1,1,1	5	3	2
6 holes with areas 1,1,1,1,1,1	64	58	6

Detailed data for polyominoes with area 24

Number of holes and their respective areas	Number of polyominoes	Number of chiral polyominoes	Number of achiral polyominoes
any number of holes	654999700403	654998425237	1275166
0 holes	475368834568	475367996161	838407
1 hole with area 1	109794444006	109794318181	125825
1 hole with area 2	26901821224	26901777061	44163
1 hole with area 3	9734764192	9734728684	35508
1 hole with area 4	4572395579	4572372838	22741
1 hole with area 5	2314024441	2314013709	10732
1 hole with area 6	1222503229	1222491419	11810
1 hole with area 7	640034047	640028634	5413
1 hole with area 8	339548503	339542138	6365
1 hole with area 9	173655378	173652116	3262
1 hole with area 10	87176556	87173855	2701
1 hole with area 11	42742127	42740948	1179
1 hole with area 12	20067584	20066247	1337
1 hole with area 13	9130583	9129993	590
1 hole with area 14	4071639	4071194	445
1 hole with area 15	1721373	1721079	294
1 hole with area 16	697003	696787	216
1 hole with area 17	276491	276451	40
1 hole with area 18	105760	105694	66
1 hole with area 19	37517	37498	19
1 hole with area 20	12428	12403	25
1 hole with area 21	3901	3892	9
1 hole with area 22	1346	1341	5
1 hole with area 23	405	403	2
1 hole with area 24	98	95	3
1 hole with area 25	14	13	1
2 holes with areas 1,1	12517151509	12517057406	94103
2 holes with areas 2,1	5281261363	5281257636	3727
2 holes with areas 2,2	537818544	537801945	16599
2 holes with areas 3,1	1798745775	1798740716	5059
2 holes with areas 3,2	324784239	324783888	351
2 holes with areas 3,3	40961841	40958036	3805
2 holes with areas 4,1	741137656	741135242	2414
2 holes with areas 4,2	121194237	121193807	430
2 holes with areas 4,3	25901544	25901256	288
2 holes with areas 4,4	3644404	3643337	1067
2 holes with areas 5,1	322678017	322676564	1453
2 holes with areas 5,2	45032075	45032039	36
2 holes with areas 5,3	7622677	7622601	76
2 holes with areas 5,4	1797015	1796968	47
2 holes with areas 5,5	164378	164177	201
2 holes with areas 6,1	139458738	139457810	928
2 holes with areas 6,2	16901703	16901581	122
2 holes with areas 6,3	2320401	2320326	75
2 holes with areas 6,4	494215	494179	36
2 holes with areas 6,5	76045	76043	2
2 holes with areas 6,6	9034	8977	57
2 holes with areas 7,1	58242870	58242506	364
2 holes with areas 7,2	5725433	5725429	4
2 holes with areas 7,3	572219	572209	10
2 holes with areas 7,4	97343	97337	6
2 holes with areas 7,5	9077	9077	0
2 holes with areas 7,6	1704	1704	0
2 holes with areas 7,7	30	27	3
2 holes with areas 8,1	23797051	23796670	381
2 holes with areas 8,2	1964657	1964629	28
2 holes with areas 8,3	158404	158394	10
2 holes with areas 8,4	24349	24341	8
2 holes with areas 8,5	1822	1822	0
2 holes with areas 8,6	419	419	0
2 holes with areas 8,7	10	10	0
2 holes with areas 8,8	2	1	1
2 holes with areas 9,1	9111135	9110916	219
2 holes with areas 9,2	606028	606026	2
2 holes with areas 9,3	34888	34882	6
2 holes with areas 9,4	4654	4653	1
2 holes with areas 9,5	268	268	0
2 holes with areas 9,6	67	67	0
2 holes with areas 10,1	3327231	3327151	80
2 holes with areas 10,2	172064	172061	3
2 holes with areas 10,3	6385	6385	0
2 holes with areas 10,4	605	605	0
2 holes with areas 10,5	9	9	0
2 holes with areas 11,1	1166337	1166298	39
2 holes with areas 11,2	47971	47971	0
2 holes with areas 11,3	1431	1431	0
2 holes with areas 11,4	96	96	0
2 holes with areas 12,1	370767	370714	53
2 holes with areas 12,2	11919	11918	1
2 holes with areas 12,3	286	286	0
2 holes with areas 12,4	18	18	0
2 holes with areas 13,1	110139	110120	19
2 holes with areas 13,2	2130	2130	0
2 holes with areas 13,3	5	5	0
2 holes with areas 14,1	31926	31924	2
2 holes with areas 14,2	448	448	0
2 holes with areas 15,1	8043	8039	4
2 holes with areas 15,2	81	81	0
2 holes with areas 16,1	1675	1674	1
2 holes with areas 16,2	10	10	0
2 holes with areas 17,1	351	351	0
2 holes with areas 18,1	60	60	0
2 holes with areas 19,1	10	10	0
3 holes with areas 1,1,1	919848367	919833202	15165
3 holes with areas 2,1,1	455025279	455021718	3561
3 holes with areas 2,2,1	66910972	66908622	2350
3 holes with areas 2,2,2	2710707	2710290	417
3 holes with areas 3,1,1	125612012	125609513	2499
3 holes with areas 3,2,1	29218840	29218834	6
3 holes with areas 3,2,2	1269019	1268836	183
3 holes with areas 3,3,1	2189061	2188792	269
3 holes with areas 3,3,2	114472	114455	17
3 holes with areas 3,3,3	1447	1444	3
3 holes with areas 4,1,1	39603761	39602308	1453
3 holes with areas 4,2,1	7463061	7463061	0
3 holes with areas 4,2,2	249772	249670	102
3 holes with areas 4,3,1	834995	834983	12
3 holes with areas 4,3,2	32888	32888	0
3 holes with areas 4,3,3	347	346	1
3 holes with areas 4,4,1	66945	66902	43
3 holes with areas 4,4,2	1887	1885	2
3 holes with areas 4,4,3	28	28	0
3 holes with areas 5,1,1	12162371	12162014	357
3 holes with areas 5,2,1	1728690	1728690	0
3 holes with areas 5,2,2	37316	37311	5
3 holes with areas 5,3,1	121380	121380	0
3 holes with areas 5,3,2	2183	2183	0
3 holes with areas 5,4,1	14204	14204	0
3 holes with areas 5,4,2	154	154	0
3 holes with areas 5,5,1	355	354	1
3 holes with areas 6,1,1	3516001	3515615	386
3 holes with areas 6,2,1	385933	385933	0
3 holes with areas 6,2,2	6348	6340	8
3 holes with areas 6,3,1	20218	20218	0
3 holes with areas 6,3,2	245	245	0
3 holes with areas 6,4,1	1918	1918	0
3 holes with areas 6,4,2	20	20	0
3 holes with areas 6,5,1	85	85	0
3 holes with areas 6,6,1	3	3	0
3 holes with areas 7,1,1	934159	934062	97
3 holes with areas 7,2,1	67660	67660	0
3 holes with areas 7,2,2	507	506	1
3 holes with areas 7,3,1	1434	1434	0
3 holes with areas 7,4,1	85	85	0
3 holes with areas 8,1,1	233868	233806	62
3 holes with areas 8,2,1	13336	13336	0
3 holes with areas 8,2,2	66	66	0
3 holes with areas 8,3,1	209	209	0
3 holes with areas 8,4,1	13	13	0
3 holes with areas 9,1,1	49708	49673	35
3 holes with areas 9,2,1	1809	1809	0
3 holes with areas 9,2,2	5	5	0
3 holes with areas 9,3,1	16	16	0
3 holes with areas 10,1,1	9712	9704	8
3 holes with areas 10,2,1	170	170	0
3 holes with areas 11,1,1	1738	1738	0
3 holes with areas 11,2,1	30	30	0
3 holes with areas 12,1,1	209	208	1
3 holes with areas 12,2,1	2	2	0
3 holes with areas 13,1,1	11	11	0
4 holes with areas 1,1,1,1	41516763	41513200	3563
4 holes with areas 2,1,1,1	17705835	17705755	80
4 holes with areas 2,2,1,1	2122455	2121998	457
4 holes with areas 2,2,2,1	73164	73164	0
4 holes with areas 2,2,2,2	480	474	6
4 holes with areas 3,1,1,1	2857182	2857000	182
4 holes with areas 3,2,1,1	428010	428009	1
4 holes with areas 3,2,2,1	10040	10034	6
4 holes with areas 3,2,2,2	7	7	0
4 holes with areas 3,3,1,1	8971	8941	30
4 holes with areas 3,3,2,1	49	49	0
4 holes with areas 4,1,1,1	536065	535992	73
4 holes with areas 4,2,1,1	55206	55206	0
4 holes with areas 4,2,2,1	771	768	3
4 holes with areas 4,3,1,1	1329	1325	4
4 holes with areas 4,3,2,1	3	3	0
4 holes with areas 4,4,1,1	54	51	3
4 holes with areas 5,1,1,1	78548	78546	2
4 holes with areas 5,2,1,1	4023	4023	0
4 holes with areas 5,2,2,1	3	3	0
4 holes with areas 5,3,1,1	12	12	0
4 holes with areas 6,1,1,1	11302	11299	3
4 holes with areas 6,2,1,1	326	326	0
4 holes with areas 7,1,1,1	925	925	0
4 holes with areas 8,1,1,1	113	113	0
4 holes with areas 9,1,1,1	9	9	0
5 holes with areas 1,1,1,1,1	781287	780908	379
5 holes with areas 2,1,1,1,1	184796	184777	19
5 holes with areas 2,2,1,1,1	8144	8132	12
5 holes with areas 2,2,2,1,1	18	18	0
5 holes with areas 3,1,1,1,1	7868	7862	6
5 holes with areas 3,2,1,1,1	89	89	0
5 holes with areas 4,1,1,1,1	556	554	2
5 holes with areas 4,2,1,1,1	5	5	0
5 holes with areas 5,1,1,1,1	4	4	0
6 holes with areas 1,1,1,1,1,1	2077	2065	12
6 holes with areas 2,1,1,1,1,1	54	54	0

References

[1]	S. Golomb, Polyominoes: puzzles, patterns, problems, and packings, Princeton University Press, Princeton, NJ, second edition, 1994.
[2]	D. H. Redelmeier, Counting polyominoes: yet another attack, Discrete mathematics, vol. 36, no. 2, pp. 191-203, 1981.
[3]	Iwan Jensen and Anthony J. Guttmann, Statistics of lattice animals (polyominoes) and polygons, Journal of Physics A: Mathematical and General, vol. 33, pp. L257-L263, 2000.
[4]	Anthony J. Guttmann (editor), Polygons, polyominoes and polycubes, Lecture notes in Physics, vol. 775, Springer, 2009.
[5]	S. Mertens, Counting lattice animals: a parallel attack, Journal of Statistical Physics, vol. 66, no. 1/2, pp. 669-678, Jan. 1992.

Additional links

The geometry junkyard polyomino entry.
The poly pages.

Tomás Oliveira e Silva Departamento de Eletrónica, Telecomunicações e Informática Universidade de Aveiro 3810-193 AVEIRO PORTUGAL		January 21, 2012

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