Biperiodic forests per homolgoy

Number of periodic forests of period \(W \times H \) raffined by homology classes

The homology classes of such a forest is the number \(k\) of its cycles on the torus multiply by the vector \((i,j)\) counting the number times a cycle cross each side of the fundamental domains of the torus. You can build some examples following this link.

Using a restricted Tutte polynomial definition, we produce \(q,t\) analogues of this number here

	0	1
0		1
1	1

Value of \(\det\Delta\) for \((W,H) = (1,1)\)

	0	1	2
0		4	1
1	1

Value of \(\det\Delta\) for \((W,H) = (1,2)\)

	0	1	2	3
0		9	6	1
1	1

Value of \(\det\Delta\) for \((W,H) = (1,3)\)

	0	1	2
0		28	1
1	28	2
2	1

Value of \(\det\Delta\) for \((W,H) = (2,2)\)

	0	1	2	3	4
0		16	20	8	1
1	1

Value of \(\det\Delta\) for \((W,H) = (1,4)\)

	0	1	2	3	4	5
0		25	50	35	10	1
1	1

Value of \(\det\Delta\) for \((W,H) = (1,5)\)

	0	1	2	3
0		393	42	1
1	148	24
2	1

Value of \(\det\Delta\) for \((W,H) = (2,3)\)

	0	1	2	3	4	5	6
0		36	105	112	54	12	1
1	1

Value of \(\det\Delta\) for \((W,H) = (1,6)\)

	0	1	2	3	4	5	6	7
0		49	196	294	210	77	14	1
1	1

Value of \(\det\Delta\) for \((W,H) = (1,7)\)

	0	1	2	3	4
0		4208	916	56	1
1	764	192	2
2	1

Value of \(\det\Delta\) for \((W,H) = (2,4)\)

	0	1	2	3	4	5	6	7	8
0		64	336	672	660	352	104	16	1
1	1

Value of \(\det\Delta\) for \((W,H) = (1,8)\)

	0	1	2	3
0		9090	222	1
1	9090	1272	3
2	222	3
3	1

Value of \(\det\Delta\) for \((W,H) = (3,3)\)

	0	1	2	3	4	5	6	7	8	9
0		81	540	1386	1782	1287	546	135	18	1
1	1

Value of \(\det\Delta\) for \((W,H) = (1,9)\)

	0	1	2	3	4	5
0		39065	14430	1635	70	1
1	4004	1320	40
2	1

Value of \(\det\Delta\) for \((W,H) = (2,5)\)

	0	1	2	3	4
0		401488	23060	296	1
1	189705	42288	480
2	1146	48
3	1

Value of \(\det\Delta\) for \((W,H) = (3,4)\)

	0	1	2	3	4	5	6
0		332028	186729	33808	2550	84	1
1	21340	8448	480	2
2	1

Value of \(\det\Delta\) for \((W,H) = (2,6)\)

	0	1	2	3	4	5	6	7
0		2657585	2117724	560742	65086	3661	98	1
1	115252	52024	4536	56
2	1

Value of \(\det\Delta\) for \((W,H) = (2,7)\)

	0	1	2	3	4	5
0		15060985	1621850	42515	370	1
1	3938409	1178940	32640	60
2	6006	480
3	1

Value of \(\det\Delta\) for \((W,H) = (3,5)\)

	0	1	2	3	4
0		31300528	541732	1528	1
1	31300528	5427200	31232	4
2	541732	31232	6
3	1528	4
4	1

Value of \(\det\Delta\) for \((W,H) = (4,4)\)

	0	1	2	3	4	5	6	7	8
0		20365760	21849936	7969376	1314964	111008	4968	112	1
1	628732	313216	37376	896	2
2	1

Value of \(\det\Delta\) for \((W,H) = (2,8)\)

	0	1	2	3	4	5	6
0		512451432	90122226	4100656	67446	444	1
1	82723794	30255588	1514016	9456	3
2	32010	3912	3
3	1

Value of \(\det\Delta\) for \((W,H) = (3,6)\)

	0	1	2	3	4	5	6	7	8	9
0		151001937	209964996	101000490	22461498	2641671	174318	6471	126	1
1	3456580	1859400	282312	10824	72
2	1

Value of \(\det\Delta\) for \((W,H) = (2,9)\)

	0	1	2	3	4	5
0		4273859865	158780750	1041955	1910	1
1	2346140816	565337040	8663120	9040
2	12966420	1369120	1760
3	8008	80
4	1

Value of \(\det\Delta\) for \((W,H) = (4,5)\)

	0	1	2	3	4	5	6	7
0		16337871697	4284658420	305235798	8243130	97853	518	1
1	1758755049	742793604	56803152	764820	1344
2	172878	28560	84
3	1

Value of \(\det\Delta\) for \((W,H) = (3,7)\)

	0	1	2	3	4	5	6
0		523495815900	34831239273	445197200	1688118	2292	1
1	175623355760	53274906048	1592314176	5215296	1344
2	318512788	49969152	203520	2
3	42680	960
4	1

Value of \(\det\Delta\) for \((W,H) = (4,6)\)

	0	1	2	3	4	5	6	7	8
0		497200197952	182461759824	18931587360	770098836	14454496	133736	592	1
1	37764731913	17768895840	1869662400	43683936	198912	96
2	943098	195168	1344
3	1

Value of \(\det\Delta\) for \((W,H) = (3,8)\)

	0	1	2	3	4	5
0		1121862200350	15636400200	26220060	10010	1
1	1121862200350	212481197760	1566260530	639920	5
2	15636400200	1566260530	1900120	10
3	26220060	639920	10
4	10010	5
5	1

Value of \(\det\Delta\) for \((W,H) = (5,5)\)

	0	1	2	3	4	5	6	7	8	9
0		14611364671122	7161133140666	1025856091395	59175853614	1624398615	23139978	175095	666	1
1	817268762466	418230492408	56441722464	2009225064	17700192	29736	3
2	5184870	1278000	16236	3
3	1

Value of \(\det\Delta\) for \((W,H) = (3,9)\)

	0	1	2	3	4	5	6	7
0		59738735737905	6325206151740	139207737382	946567342	2480205	2674	1
1	13240341232848	4752939850736	230942948208	1665334384	1672048	112
2	7981245460	1651132000	14965664	2912
3	230504	9072
4	1

Value of \(\det\Delta\) for \((W,H) = (4,7)\)

	0	1	2	3	4	5	6
0		475282996336836	13224651974505	49666576112	43488054	12012	1
1	289189977516185	70919032658040	1130040445200	1528078200	165120
2	1578990532350	236220195960	884765760	58360
3	681625635	32626680	4560
4	53350	120
5	1

Value of \(\det\Delta\) for \((W,H) = (5,6)\)

	0	1	2	3	4	5	6	7	8
0		6485222063730368	1007691830432400	34876304889056	387260945060	1718634144	3418216	3056	1
1	1006247821369264	410411321263104	28845959184384	375781870592	915197952	330752	4
2	202802995236	51421127680	843788288	640000	6
3	1257464	74752	4
4	1

Value of \(\det\Delta\) for \((W,H) = (4,8)\)

	0	1	2	3	4	5	6	7
0		186426741745171249	8899950961137796	64588984636694	112711130210	64764077	14014	1
1	74493211129182745	22157265529469980	621399090241440	1991775730700	794431120	27020
2	162792917064170	32268033889840	267856285900	81846800	140
3	18178392435	1393631820	763840
4	288130	1680
5	1

Value of \(\det\Delta\) for \((W,H) = (5,7)\)

	0	1	2	3	4	5	6
0		416066003938094040	5113415904319770	5773239179248	1159480218	64020	1
1	416066003938094040	82295173251366912	664991166673668	334469718618	12083616	6
2	5113415904319770	664991166673668	1689388167936	83403840	15
3	5773239179248	334469718618	83403840	20
4	1159480218	12083616	15
5	64020	6
6	1

Value of \(\det\Delta\) for \((W,H) = (6,6)\)

	0	1	2	3	4	5	6	7	8	9
0		678396861427313745	145950270185292516	7418600240193642	125460128544138	870216838407	2817140574	4502151	3438	1
1	77021959665782032	34710866650590480	3259547319314448	67499080470672	317583703440	312505488	42384
2	5204401136660	1542185366688	40071475296	75944352	5472
3	6913160	564624	144
4	1

Value of \(\det\Delta\) for \((W,H) = (4,9)\)

	0	1	2	3	4	5	6	7	8
0		69314023502407590464	5126266495305830736	64160637003271072	201793219815060	212800736352	90048104	16016	1
1	19272699503868339225	6650429751255214400	287206109973207360	1785441015992480	1779917735680	264601760	2880
2	17030200047569870	4154079131553440	62564215261440	55447818400	1655040
3	493910758435	53512691360	77701440	160
4	1571830	18080
5	1

Value of \(\det\Delta\) for \((W,H) = (5,8)\)

	0	1	2	3	4	5	6	7
0		550489244757792589361	12615805741616922012	31271512812390438	14083068084190	1751198029	74690	1
1	360572876155002045396	88395295010309703696	1372411378879446696	1799953578520104	246132141864	2959656
2	2014888401187313001	353006634500484720	2077923052894416	436473443952	1838592
3	693125611225264	62421093645736	53712672552	226856
4	31772979894	673022784	9744
5	345756	168
6	1

Value of \(\det\Delta\) for \((W,H) = (6,7)\)

	0	1	2	3	4	5	6	7	8	9
0		24775954679408996311281	2641587905005880458140	52258744391256516186	273235467624902982	487866261381687	357959418546	119340135	18018	1
1	5013420134858082407545	1945072498735741686840	118019123761899520800	1242524061566889780	2511480342155040	990982532340	61562640	180
2	1799753450201632850	514859055136896240	12328175163345660	24320736477600	3088964340	9360
3	13602129220835	1918266586740	5843626560	130740
4	8641450	166320
5	1

Value of \(\det\Delta\) for \((W,H) = (5,9)\)

	0	1	2	3	4	5	6	7	8
0		687753319554083785962944	25919992828223739020112	122697333908236995680	111244328012184724	27731401957280	2456764776	85360	1
1	312275989074940801701852	90538571476099679704320	2317779794278012144128	6416463950439905664	2333443541041152	116501127552	514560
2	805309474193175380073	176065068556616444544	1950762702325353216	1171440351179136	31325313024	6528
3	85119239067066256	10581626520324736	22454139627008	572432000	2
4	888785817654	31816159872	2204160
5	1886196	2688
6	1

Value of \(\det\Delta\) for \((W,H) = (6,8)\)

	0	1	2	3	4	5	6	7
0		1593131195595008417600420	18155778048027774740796	15676595841974806354	1828747178469932	48691089230	403382	1
1	1593131195595008417600420	322884633472287356809472	2717293477634088919279	1395291429728381080	60453745573011	219163112	7
2	18155778048027774740796	2717293477634088919279	10163866420200651200	1142014733490421	3095311996	21
3	15676595841974806354	1395291429728381080	1142014733490421	6863171504	35
4	1828747178469932	60453745573011	3095311996	35
5	48691089230	219163112	21
6	403382	7
7	1

Value of \(\det\Delta\) for \((W,H) = (7,7)\)

	0	1	2	3	4	5	6	7	8	9
0		824081555491379390311959906	46799844232257542420196318	379726926710928609745203	631290929955439803930	291458344527516231	47933155870350	3276180423	96030	1
1	271192471353959960518593480	89893280910761664249084336	3414671694744742934137296	17220868233285138890436	13583687132514740688	1848447349995096	39442427736	63576
2	325156007451688759656690	84339613313207989144992	1531190703599004774816	2065941206140995936	194496846180480	759489120	3
3	10616749263940462960	1685992016974876632	7154640889552080	662903654352	277200
4	25251373657206	1352044929600	291340080	3
5	10369740	32472
6	1

Value of \(\det\Delta\) for \((W,H) = (6,9)\)

	0	1	2	3	4	5	6	7	8
0		6619626249094663354617541440	132344263688874690123403600	242885219913990862352928	64179440935375547028	3765816942733792	68929982568	461008	1
1	4581069791005351230978205745	1116582990025769968095887808	16644007992355013616615360	19790405112473429973728	2467099851683323136	37319608019680	52369856
2	26472208858238089977468732	5032940445363689481672672	37293401230064175936768	12469292601388231392	184911874447104	56179424
3	8050901244407482603814	986002226327492778208	1982137369664274368	60162202086112	10506496
4	244256830486960942	12900822020983008	2485230633536	670432
5	1382799344717	12882372832	18368
6	2200562	224
7	1

Value of \(\det\Delta\) for \((W,H) = (7,8)\)

	0	1	2	3	4	5	6	7	8	9
0		26313391925014204604820264252561	831692116690374405067270919244	2849915732841361139393107194	1531704007122191857617534	183325028544384004935	6695923412813082	92489631831	518634	1
1	13164264379175914103701863288433	3724123202629370676251082733032	87195743686659131394852061584	204295484213964176219224332	59727358451070716105952	2586924885435557436	16579926787872	9491580
2	38983983965325971496151385956	8924050399103009618403329136	112913847367115376042492948	88504260599626359600000	4499542247527515132	11452266013104	255612
3	4202327829455391069878262	656317618077617336409660	2641257873097654314672	265786513572529404	411549981696	252
4	33295262288213969778	2512711057264698528	1305770982284412	2671112640
5	39913174155533	652066112124	5370624
6	12098030	4032
7	1

Value of \(\det\Delta\) for \((W,H) = (7,9)\)

	0	1	2	3	4	5	6	7	8
0		62920730667415542193771827107008	683260648697412516724458972432	492754107166579846790285536	38250027341158899028772	527422903000251296	1975645978792	2514928	1
1	62920730667415542193771827107008	12946793096657348677298718507008	111253153456106097146597153024	55651912452956290351170992	2290798209374437318656	9869154650095840	3890225152	8
2	683260648697412516724458972432	111253153456106097146597153024	523788336802148174650146816	88229337438598664773632	592514371651994048	104278351872	28
3	492754107166579846790285536	55651912452956290351170992	88229337438598664773632	2060105535352602624	441584721920	56
4	38250027341158899028772	2290798209374437318656	592514371651994048	441584721920	70
5	527422903000251296	9869154650095840	104278351872	56
6	1975645978792	3890225152	28
7	2514928	8
8	1

Value of \(\det\Delta\) for \((W,H) = (8,8)\)

	0	1	2	3	4	5	6	7	8	9
0		824564597279537629405361250727077969	14902668521181435259980270056023524	21913336235776539102121422994794	3847611968376582743622741258	119767293788143487322375	965852199540230622	2667318644871	2829294	1
1	595407494656130380848536187159180352	144082439863181235292262128326623808	2061551955583599999804509577820704	2205571515266011872541319574048	231832653725637961035172128	3044151102677524936992	5422688090112288	917969184
2	3560597182615967870227364097036624	709338703926521941143882524592192	5978277078278937199154826815232	2500731501492458599220940864	59028982864549553727936	71125534091810880	1682337216
3	1016429100334204316374965460128	147382107845390572570066757664	478013110611420276322998048	36705539201046381731616	54344492162164512	465975072
4	23318838892519925292509844	1955653965849997725315840	1300646035943989496064	5589842689189632	42156288
5	75594809657455382368	2303655156541048608	98011417610016	1660704
6	57562982977640	236666318400	31680
7	13826320	288
8	1

Value of \(\det\Delta\) for \((W,H) = (8,9)\)

	0	1	2	3	4	5	6	7	8	9
0		25620761954539993042254728104964978718582	269243592554330983918414117404966630816	171356074968512735640170187517682352	9897226656600364873802996482440	80427305727805719546183924	142760557718649935376	78199794215064	15554610	1
1	25620761954539993042254728104964978718582	5325315788220567789196153826463570499584	46317263356752796229486016511260566604	22390968708877927261114978251258816	830182299728674067577605148624	3201304585286643880464000	1507242131040163140	68183171424	9
2	269243592554330983918414117404966630816	46317263356752796229486016511260566604	252985796055757409628449968642983168	53677076581148201931542042561976	539226800365809764662656264	260843699734402416372	3309199236720	36
3	171356074968512735640170187517682352	22390968708877927261114978251258816	53677076581148201931542042561976	2656125057608175505504971648	2575797343448719575150	24469381879200	84
4	9897226656600364873802996482440	830182299728674067577605148624	539226800365809764662656264	2575797343448719575150	45866396984472	126
5	80427305727805719546183924	3201304585286643880464000	260843699734402416372	24469381879200	126
6	142760557718649935376	1507242131040163140	3309199236720	84
7	78199794215064	68183171424	36
8	15554610	9
9	1

Value of \(\det\Delta\) for \((W,H) = (9,9)\)