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

01
01
11
Value of \(\det\Delta\) for \((W,H) = (1,1)\)
012
041
11
Value of \(\det\Delta\) for \((W,H) = (1,2)\)
0123
0961
11
Value of \(\det\Delta\) for \((W,H) = (1,3)\)
012
0281
1282
21
Value of \(\det\Delta\) for \((W,H) = (2,2)\)
01234
0162081
11
Value of \(\det\Delta\) for \((W,H) = (1,4)\)
012345
0255035101
11
Value of \(\det\Delta\) for \((W,H) = (1,5)\)
0123
0393421
114824
21
Value of \(\det\Delta\) for \((W,H) = (2,3)\)
0123456
03610511254121
11
Value of \(\det\Delta\) for \((W,H) = (1,6)\)
01234567
04919629421077141
11
Value of \(\det\Delta\) for \((W,H) = (1,7)\)
01234
04208916561
17641922
21
Value of \(\det\Delta\) for \((W,H) = (2,4)\)
012345678
064336672660352104161
11
Value of \(\det\Delta\) for \((W,H) = (1,8)\)
0123
090902221
1909012723
22223
31
Value of \(\det\Delta\) for \((W,H) = (3,3)\)
0123456789
081540138617821287546135181
11
Value of \(\det\Delta\) for \((W,H) = (1,9)\)
012345
039065144301635701
14004132040
21
Value of \(\det\Delta\) for \((W,H) = (2,5)\)
01234
0401488230602961
118970542288480
2114648
31
Value of \(\det\Delta\) for \((W,H) = (3,4)\)
0123456
0332028186729338082550841
12134084484802
21
Value of \(\det\Delta\) for \((W,H) = (2,6)\)
01234567
026575852117724560742650863661981
111525252024453656
21
Value of \(\det\Delta\) for \((W,H) = (2,7)\)
012345
0150609851621850425153701
1393840911789403264060
26006480
31
Value of \(\det\Delta\) for \((W,H) = (3,5)\)
01234
03130052854173215281
1313005285427200312324
2541732312326
315284
41
Value of \(\det\Delta\) for \((W,H) = (4,4)\)
012345678
020365760218499367969376131496411100849681121
1628732313216373768962
21
Value of \(\det\Delta\) for \((W,H) = (2,8)\)
0123456
0512451432901222264100656674464441
18272379430255588151401694563
23201039123
31
Value of \(\det\Delta\) for \((W,H) = (3,6)\)
0123456789
015100193720996499610100049022461498264167117431864711261
1345658018594002823121082472
21
Value of \(\det\Delta\) for \((W,H) = (2,9)\)
012345
04273859865158780750104195519101
1234614081656533704086631209040
21296642013691201760
3800880
41
Value of \(\det\Delta\) for \((W,H) = (4,5)\)
01234567
01633787169742846584203052357988243130978535181
11758755049742793604568031527648201344
21728782856084
31
Value of \(\det\Delta\) for \((W,H) = (3,7)\)
0123456
052349581590034831239273445197200168811822921
117562335576053274906048159231417652152961344
2318512788499691522035202
342680960
41
Value of \(\det\Delta\) for \((W,H) = (4,6)\)
012345678
049720019795218246175982418931587360770098836144544961337365921
1377647319131776889584018696624004368393619891296
29430981951681344
31
Value of \(\det\Delta\) for \((W,H) = (3,8)\)
012345
011218622003501563640020026220060100101
1112186220035021248119776015662605306399205
2156364002001566260530190012010
32622006063992010
4100105
51
Value of \(\det\Delta\) for \((W,H) = (5,5)\)
0123456789
01461136467112271611331406661025856091395591758536141624398615231399781750956661
181726876246641823049240856441722464200922506417700192297363
251848701278000162363
31
Value of \(\det\Delta\) for \((W,H) = (3,9)\)
01234567
0597387357379056325206151740139207737382946567342248020526741
113240341232848475293985073623094294820816653343841672048112
279812454601651132000149656642912
32305049072
41
Value of \(\det\Delta\) for \((W,H) = (4,7)\)
0123456
0475282996336836132246519745054966657611243488054120121
12891899775161857091903265804011300404452001528078200165120
2157899053235023622019596088476576058360
3681625635326266804560
453350120
51
Value of \(\det\Delta\) for \((W,H) = (5,6)\)
012345678
064852220637303681007691830432400348763048890563872609450601718634144341821630561
11006247821369264410411321263104288459591843843757818705929151979523307524
2202802995236514211276808437882886400006
31257464747524
41
Value of \(\det\Delta\) for \((W,H) = (4,8)\)
01234567
018642674174517124988999509611377966458898463669411271113021064764077140141
17449321112918274522157265529469980621399090241440199177573070079443112027020
21627929170641703226803388984026785628590081846800140
3181783924351393631820763840
42881301680
51
Value of \(\det\Delta\) for \((W,H) = (5,7)\)
0123456
0416066003938094040511341590431977057732391792481159480218640201
141606600393809404082295173251366912664991166673668334469718618120836166
2511341590431977066499116667366816893881679368340384015
357732391792483344697186188340384020
411594802181208361615
5640206
61
Value of \(\det\Delta\) for \((W,H) = (6,6)\)
0123456789
067839686142731374514595027018529251674186002401936421254601285441388702168384072817140574450215134381
1770219596657820323471086665059048032595473193144486749908047067231758370344031250548842384
25204401136660154218536668840071475296759443525472
36913160564624144
41
Value of \(\det\Delta\) for \((W,H) = (4,9)\)
012345678
06931402350240759046451262664953058307366416063700327107220179321981506021280073635290048104160161
1192726995038683392256650429751255214400287206109973207360178544101599248017799177356802646017602880
217030200047569870415407913155344062564215261440554478184001655040
34939107584355351269136077701440160
4157183018080
51
Value of \(\det\Delta\) for \((W,H) = (5,8)\)
01234567
05504892447577925893611261580574161692201231271512812390438140830680841901751198029746901
136057287615500204539688395295010309703696137241137887944669617999535785201042461321418642959656
2201488840118731300135300663450048472020779230528944164364734439521838592
36931256112252646242109364573653712672552226856
4317729798946730227849744
5345756168
61
Value of \(\det\Delta\) for \((W,H) = (6,7)\)
0123456789
024775954679408996311281264158790500588045814052258744391256516186273235467624902982487866261381687357959418546119340135180181
1501342013485808240754519450724987357416868401180191237618995208001242524061566889780251148034215504099098253234061562640180
21799753450201632850514859055136896240123281751633456602432073647760030889643409360
31360212922083519182665867405843626560130740
48641450166320
51
Value of \(\det\Delta\) for \((W,H) = (5,9)\)
012345678
068775331955408378596294425919992828223739020112122697333908236995680111244328012184724277314019572802456764776853601
131227598907494080170185290538571476099679704320231777979427801214412864164639504399056642333443541041152116501127552514560
280530947419317538007317606506855661644454419507627023253532161171440351179136313253130246528
38511923906706625610581626520324736224541396270085724320002
4888785817654318161598722204160
518861962688
61
Value of \(\det\Delta\) for \((W,H) = (6,8)\)
01234567
0159313119559500841760042018155778048027774740796156765958419748063541828747178469932486910892304033821
1159313119559500841760042032288463347228735680947227172934776340889192791395291429728381080604537455730112191631127
2181557780480277747407962717293477634088919279101638664202006512001142014733490421309531199621
31567659584197480635413952914297283810801142014733490421686317150435
4182874717846993260453745573011309531199635
54869108923021916311221
64033827
71
Value of \(\det\Delta\) for \((W,H) = (7,7)\)
0123456789
082408155549137939031195990646799844232257542420196318379726926710928609745203631290929955439803930291458344527516231479331558703503276180423960301
1271192471353959960518593480898932809107616642490843363414671694744742934137296172208682332851388904361358368713251474068818484473499950963944242773663576
232515600745168875965669084339613313207989144992153119070359900477481620659412061409959361944968461804807594891203
31061674926394046296016859920169748766327154640889552080662903654352277200
42525137365720613520449296002913400803
51036974032472
61
Value of \(\det\Delta\) for \((W,H) = (6,9)\)
012345678
06619626249094663354617541440132344263688874690123403600242885219913990862352928641794409353755470283765816942733792689299825684610081
145810697910053512309782057451116582990025769968095887808166440079923550136166153601979040511247342997372824670998516833231363731960801968052369856
2264722088582380899774687325032940445363689481672672372934012300641759367681246929260138823139218491187444710456179424
3805090124440748260381498600222632749277820819821373696642743686016220208611210506496
4244256830486960942129008220209830082485230633536670432
513827993447171288237283218368
62200562224
71
Value of \(\det\Delta\) for \((W,H) = (7,8)\)
0123456789
026313391925014204604820264252561831692116690374405067270919244284991573284136113939310719415317040071221918576175341833250285443840049356695923412813082924896318315186341
113164264379175914103701863288433372412320262937067625108273303287195743686659131394852061584204295484213964176219224332597273584510707161059522586924885435557436165799267878729491580
238983983965325971496151385956892405039910300961840332913611291384736711537604249294888504260599626359600000449954224752751513211452266013104255612
342023278294553910698782626563176180776173364096602641257873097654314672265786513572529404411549981696252
433295262288213969778251271105726469852813057709822844122671112640
5399131741555336520661121245370624
6120980304032
71
Value of \(\det\Delta\) for \((W,H) = (7,9)\)
012345678
06292073066741554219377182710700868326064869741251672445897243249275410716657984679028553638250027341158899028772527422903000251296197564597879225149281
16292073066741554219377182710700812946793096657348677298718507008111253153456106097146597153024556519124529562903511709922290798209374437318656986915465009584038902251528
26832606486974125167244589724321112531534561060971465971530245237883368021481746501468168822933743859866477363259251437165199404810427835187228
34927541071665798467902855365565191245295629035117099288229337438598664773632206010553535260262444158472192056
438250027341158899028772229079820937443731865659251437165199404844158472192070
5527422903000251296986915465009584010427835187256
61975645978792389022515228
725149288
81
Value of \(\det\Delta\) for \((W,H) = (8,8)\)
0123456789
082456459727953762940536125072707796914902668521181435259980270056023524219133362357765391021214229947943847611968376582743622741258119767293788143487322375965852199540230622266731864487128292941
15954074946561303808485361871591803521440824398631812352922621283266238082061551955583599999804509577820704220557151526601187254131957404823183265372563796103517212830441511026775249369925422688090112288917969184
235605971826159678702273640970366247093387039265219411438825245921925978277078278937199154826815232250073150149245859922094086459028982864549553727936711255340918108801682337216
310164291003342043163749654601281473821078453905725700667576644780131106114202763229980483670553920104638173161654344492162164512465975072
42331883889251992529250984419556539658499977253158401300646035943989496064558984268918963242156288
5755948096574553823682303655156541048608980114176100161660704
65756298297764023666631840031680
713826320288
81
Value of \(\det\Delta\) for \((W,H) = (8,9)\)
0123456789
02562076195453999304225472810496497871858226924359255433098391841411740496663081617135607496851273564017018751768235298972266566003648738029964824408042730572780571954618392414276055771864993537678199794215064155546101
1256207619545399930422547281049649787185825325315788220567789196153826463570499584463172633567527962294860165112605666042239096870887792726111497825125881683018229972867406757760514862432013045852866438804640001507242131040163140681831714249
22692435925543309839184141174049666308164631726335675279622948601651126056660425298579605575740962844996864298316853677076581148201931542042561976539226800365809764662656264260843699734402416372330919923672036
31713560749685127356401701875176823522239096870887792726111497825125881653677076581148201931542042561976265612505760817550550497164825757973434487195751502446938187920084
49897226656600364873802996482440830182299728674067577605148624539226800365809764662656264257579734344871957515045866396984472126
580427305727805719546183924320130458528664388046400026084369973440241637224469381879200126
61427605577186499353761507242131040163140330919923672084
7781997942150646818317142436
8155546109
91
Value of \(\det\Delta\) for \((W,H) = (9,9)\)