The Number of Solid Young Tableaux of Cylindrical Shapes for Floors with at most, 6, cells, and up to height, 20 By Shalosh B. Ekhad, Rutgers University (Email: ShaloshBEkahd at gmail dot com) If the floor has, 1, cells, then it forms one of the following (usual) partitions {[1]} ------------------------------------------------- If the floor is, [1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] If the floor has, 2, cells, then it forms one of the following (usual) partitions {[2], [1, 1]} ------------------------------------------------- If the floor is, [2], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020] If the floor is, [1, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420, 24466267020] If the floor has, 3, cells, then it forms one of the following (usual) partitions {[3], [2, 1], [1, 1, 1]} ------------------------------------------------- If the floor is, [3], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [1, 5, 42, 462, 6006, 87516, 1385670, 23371634, 414315330, 7646001090, 145862174640, 2861142656400, 57468093927120, 1178095925505960, 24584089974896430, 521086299271824330, 11198784501894470250, 243661974372798631650, 5360563436201569896300, 119115896614816702500900, 2670926804331443293626900] If the floor is, [2, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [2, 16, 192, 2816, 46592, 835584, 15876096, 315031552, 6466437120, 136383037440, 2941129850880, 64614360416256, 1442028424527872, 32619677465182208, 746569714888605696, 17262927525017812992, 402801642250415636480, 9474719710174783733760, 224477974671833337692160, 5353056650362464680017920, 128405456141675526975651840] If the floor is, [1, 1, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [1, 5, 42, 462, 6006, 87516, 1385670, 23371634, 414315330, 7646001090, 145862174640, 2861142656400, 57468093927120, 1178095925505960, 24584089974896430, 521086299271824330, 11198784501894470250, 243661974372798631650, 5360563436201569896300, 119115896614816702500900, 2670926804331443293626900] If the floor has, 4, cells, then it forms one of the following (usual) partitions {[4], [2, 2], [3, 1], [2, 1, 1], [1, 1, 1, 1]} ------------------------------------------------- If the floor is, [4], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [1, 14, 462, 24024, 1662804, 140229804, 13672405890, 1489877926680, 177295473274920, 22661585038594320, 3073259571003214320, 438091463242348309440, 65166105157299311029200, 10056663345892631910888600, 1602608179958939072505281850, 262708662267696303439658400600, 44158099880470145320269598879800, 7590544958400815506105289996917200, 1331302722898740433895479758550218000, 237782242855131552280114384701056328000, 43177213944214114930927252531062759828000] If the floor is, [2, 2], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [2, 48, 2452, 183958, 17454844, 1941406508, 242201554680, 32959299267334, 4801233680739724, 738810565910888784, 118929992674840615128, 19880920716640427983476, 3431624482227380273056728, 608880419873586515669564728, 110654016191338341346670548240, 20536574090713344110860752530646, 3882925024331174796857101684510428, 746410931448945012196513727291312844, 145626362670805760264809414243057616552, 28794547473359904233269297596817899967540, 5762931182262926787948946259721346099337448] If the floor is, [3, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [3, 91, 5471, 464836, 48767805, 5900575762, 791402291063, 114754560003596, 17688389169462060, 2864042102057254739, 482894371222455465001, 84225614036198359288620, 15119622005825185224290830, 2782232873996840900804273236, 523114052492282720617167786279, 100231256005025286627952024093564, 19528383010645472628217323778258916, 3861833537008249745004910552954341212, 773955394090091534612088612283700836700, 156986837786461718810856537819327105826896, 32191822675659869353548265077416785090578101] If the floor is, [2, 1, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [3, 91, 5471, 464836, 48767805, 5900575762, 791402291063, 114754560003596, 17688389169462060, 2864042102057254739, 482894371222455465001, 84225614036198359288620, 15119622005825185224290830, 2782232873996840900804273236, 523114052492282720617167786279, 100231256005025286627952024093564, 19528383010645472628217323778258916, 3861833537008249745004910552954341212, 773955394090091534612088612283700836700, 156986837786461718810856537819327105826896, 32191822675659869353548265077416785090578101] If the floor is, [1, 1, 1, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [1, 14, 462, 24024, 1662804, 140229804, 13672405890, 1489877926680, 177295473274920, 22661585038594320, 3073259571003214320, 438091463242348309440, 65166105157299311029200, 10056663345892631910888600, 1602608179958939072505281850, 262708662267696303439658400600, 44158099880470145320269598879800, 7590544958400815506105289996917200, 1331302722898740433895479758550218000, 237782242855131552280114384701056328000, 43177213944214114930927252531062759828000] If the floor has, 5, cells, then it forms one of the following (usual) partitions {[5], [3, 2], [4, 1], [2, 2, 1], [3, 1, 1], [2, 1, 1, 1], [1, 1, 1, 1, 1]} ------------------------------------------------- If the floor is, [5], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [1, 42, 6006, 1662804, 701149020, 396499770810, 278607172289160, 231471904322784840, 219738059326729823880, 232553551737813227594400, 269396678720275351794712800, 336839101096824285057473785200, 449620757769949216266129125515200, 635126788746534097077262900550679600, 942715939664950389555756839794925523000, 1461745760642443369719209708512606184967000, 2356257232168216478125801881190294190893911000, 3932398442071055178824599551581089025383141722000, 6771232626478896012919703646045218839071098850582000, 11993913791423922763754292458255748500163329007509160000, 21798231954982180348620739037118447362870358638728088472000] If the floor is, [3, 2], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [5, 707, 268326, 168146839, 143163177336, 149998192424502, 182598353781240533, 249032962712552804432, 371285830572997665257695, 594729699502746726969433566, 1010574132470951359396337494800, 1804193873947216124589237862262262, 3359208139273855304054615931024355421, 6484548542282589186764920463045727068714, 12917026052533071326415441939550737068546683, 26449255599864674591232457723485271005515955748, 55494586452623441708393052242864664929017735937729, 118992243615978642379648717066635972582774227738513580, 260160703163434521325988904861052305443656463568098521127, 578879309842150326491129445329190967995975501993294414829833, 1308708661621023071453865460960933890126538696676685584827666491] If the floor is, [4, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [4, 456, 143164, 75965484, 55824699632, 51274161733160, 55418842406649988, 67819708829687672202, 91539069926354814114556, 133752944758581353219955762, 208673064320580765981337783096, 343997162091593719562479905281938, 594344377404793356460064021706935470, 1069340478354594078729213018151307525060, 1993101681653919805455089744880867309328092, 3832035002061880004175052320417012818381000356, 7573265385291231478795867977666957347340063273188, 15339300137949987791785284608799776088764459511876090, 31762053953324049383349431266288189995908589020437408206, 67090793493094101308242797283669076603902902339988262049542, 144300696905816611147657855653902847800680730845763433226996420] If the floor is, [2, 2, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [5, 707, 268326, 168146839, 143163177336, 149998192424502, 182598353781240533, 249032962712552804432, 371285830572997665257695, 594729699502746726969433566, 1010574132470951359396337494800, 1804193873947216124589237862262262, 3359208139273855304054615931024355421, 6484548542282589186764920463045727068714, 12917026052533071326415441939550737068546683, 26449255599864674591232457723485271005515955748, 55494586452623441708393052242864664929017735937729, 118992243615978642379648717066635972582774227738513580, 260160703163434521325988904861052305443656463568098521127, 578879309842150326491129445329190967995975501993294414829833, 1308708661621023071453865460960933890126538696676685584827666491] If the floor is, [3, 1, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [6, 936, 379366, 249664758, 221005209058, 239143562020194, 299233941746052998, 417999868371999142276, 636568066798406010872120, 1039267652960081699025215774, 1796704965351078502372895796786, 3258764657213579008313421745034602, 6156424558741485934780561870288690162, 12045708723835854761561020892117141334302, 24298404491948466147812393691800122144402100, 50343519594433157122050481389583344534830475422, 106804488936388001231452225884840982718057626558516, 231416715247108833048934975537033953035625552274411698, 510989992385607559712359337180575897940480128980681059558, 1147718550136117228169651730698305302191015527009393891697802, 2618002926994783626202932600912901013149745176477882238396370420] If the floor is, [2, 1, 1, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [4, 456, 143164, 75965484, 55824699632, 51274161733160, 55418842406649988, 67819708829687672202, 91539069926354814114556, 133752944758581353219955762, 208673064320580765981337783096, 343997162091593719562479905281938, 594344377404793356460064021706935470, 1069340478354594078729213018151307525060, 1993101681653919805455089744880867309328092, 3832035002061880004175052320417012818381000356, 7573265385291231478795867977666957347340063273188, 15339300137949987791785284608799776088764459511876090, 31762053953324049383349431266288189995908589020437408206, 67090793493094101308242797283669076603902902339988262049542, 144300696905816611147657855653902847800680730845763433226996420] If the floor is, [1, 1, 1, 1, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [1, 42, 6006, 1662804, 701149020, 396499770810, 278607172289160, 231471904322784840, 219738059326729823880, 232553551737813227594400, 269396678720275351794712800, 336839101096824285057473785200, 449620757769949216266129125515200, 635126788746534097077262900550679600, 942715939664950389555756839794925523000, 1461745760642443369719209708512606184967000, 2356257232168216478125801881190294190893911000, 3932398442071055178824599551581089025383141722000, 6771232626478896012919703646045218839071098850582000, 11993913791423922763754292458255748500163329007509160000, 21798231954982180348620739037118447362870358638728088472000] If the floor has, 6, cells, then it forms one of the following (usual) partitions {[6], [3, 3], [4, 2], [5, 1], [2, 2, 2], [3, 2, 1], [4, 1, 1], [2, 2, 1, 1], [3, 1, 1, 1], [2, 1, 1, 1, 1], [1, 1, 1, 1, 1, 1]} ------------------------------------------------- If the floor is, [6], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [1, 132, 87516, 140229804, 396499770810, 1671643033734960, 9490348077234178440, 67867669180627125604080, 583692803893929928888544400, 5838544419011620940996212276800, 66244124978105851196543024492572800, 836288764382254532915188713779640302400, 11570895443447601081407359451642915869302000, 173350856749047087227868685053324700693497684000, 2784591835268084496051527013236901155628039317067000, 47572076113399769965230407161865228632990820870785062000, 858538265205484953801456866514093556426809869604623359710000, 16274475645063354175139076024244412274999121731816308896110840000, 322467413532568579777476346845777467898829336857344114648508565320000, 6650995938899110343991104100498582529857394689556602949972101398891832000, 142280318416220472363203609001002799916708453455756090713808284175559587\ 660000] If the floor is, [3, 3], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [5, 2452, 4877756, 20071150430, 129586764260850, 1138355914222027660, 12513844842339741519760, 163186564770917385358723138, 2434438822161210367337209525489, 40488679486377745566571570522228550, 736610570835499716578578298705683198672, 14449603409027127772844370973864317813081998, 302224567883797633771115076375643564643240009684, 6680534355232215796359569537846351331799607987626190, 154952783132851780330241522093570392123026650828911658272, 3749373916520735289967200384919530816653374322880254025699170, 94186646363053615161049575244332495349759173669773833907334173189, 2446426685243181084043182009485711296850813221674123925627155156843440, 65478597639156540704193466348938999182815225439291386760836044991715009069, 180061457864625179488388155704735742923701934075107323133621463438291254\ 8278792, 5074621257171088880424423449679054608766474536475721528843619751\ 9334759463453756655] If the floor is, [4, 2], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [9, 6357, 16447321, 83063011917, 634124741200585, 6418271697539486639, 79760647119436009904359, 1158813078073207182208246970, 19039890219067799672109037727333, 345536977359227460016778181115188157, 6807037555862486557885055435200136982545, 143659867697958394286469152394909871383781549, 3215034569432119517327981937474950133734723000223, 75681728395113636901288108326935122827080756119192617, 1861732522660248287189629245992453740900864433870603147847, 47604362074262298337591005031934265563041297042663790727579076, 1259679028441453461482138776582091290162235867261865435902820907931, 34367576374115937615518504989289875824239207218252778840566233255460174, 963725891830820179918754748120760958518997394197071560285122907851989884133 , 27702099111668966611113368198673903040383712882416201967930306483835052\ 016128928, 81438041582989841341475978058723634188267722992450507743711407\ 6308674758885048214127] If the floor is, [5, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [5, 2145, 3636776, 12753712037, 70692556053053, 538062084544290042, 5172429435337799442495, 59496753497779282160432712, 789114026962872326337841559201, 11751727385856856418961493392918296, 192672653281429171975863915261887981069, 3425801216108320871923052907182435590314321, 65286814883087788041189662844895960361478416634, 1321133844411057286840297082589727434570482687973793, 28173237507294627294585376411478395160886201033336857591, 629207962318311411973650101341198498009929430671090318982808, 14641094463433804768118266475987847138245262202328275604625755883, 353416897519184421647376661920386779420437105377276409974214726087449, 8817261311212942597857031922534147635184747822371751936980265747873514393, 226642615448278611851995027544701715526140993380175369387970678924798748\ 188187, 59858798275083654291194760841317789551321433759429355148941016480\ 77949141202752446] If the floor is, [2, 2, 2], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [5, 2452, 4877756, 20071150430, 129586764260850, 1138355914222027660, 12513844842339741519760, 163186564770917385358723138, 2434438822161210367337209525489, 40488679486377745566571570522228550, 736610570835499716578578298705683198672, 14449603409027127772844370973864317813081998, 302224567883797633771115076375643564643240009684, 6680534355232215796359569537846351331799607987626190, 154952783132851780330241522093570392123026650828911658272, 3749373916520735289967200384919530816653374322880254025699170, 94186646363053615161049575244332495349759173669773833907334173189, 2446426685243181084043182009485711296850813221674123925627155156843440, 65478597639156540704193466348938999182815225439291386760836044991715009069, 180061457864625179488388155704735742923701934075107323133621463438291254\ 8278792, 5074621257171088880424423449679054608766474536475721528843619751\ 9334759463453756655] If the floor is, [3, 2, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [16, 17086, 61189172, 404233159860, 3880365678824980, 47959061464818182058, 711513280222442751394224, 12121127323153614807021655742, 230127245538294682127207785787376, 4767460278053986542112719904243778834, 106115342273795146740243750912097789131600, 2508221717418822137823694523392722967458869650, 62391202713894548544564463278071633126451060258384, 1621620581169490458873661472700989787258366133388681794, 43787629698511894326692762902872062198845084493067183482184, 1222635373265052638339432167517374369340419931909585212757004690, 35164595470702416620005390036341495807912627130088267078461846550028, 1038419432625650778160934941280004507893869911249175603147987738277295614, 313987404332230296728607119085193970156293576765275524401449790643030267\ 44092, 969868619807724586654094335105389716691936814216822623926941973524\ 847216176837188, 30542615706888574299846357507063357766079568315886084173\ 740284021431727182080422477572] If the floor is, [4, 1, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [10, 7425, 19811341, 102250773030, 793689156114651, 8141496780668441784, 102311405943377364234007, 1500732242959565450608299410, 24864788888918578716074039643936, 454607967463033875318022907946375155, 9015630441456884000942317198866362532841, 191426358715002107500505661821512689159155462, 4307823830252527613182777517524350025780716929549, 101925188553099925839722851949005519570028831423378414, 2519215044950830204603674956552306763625850266078873903062, 64701316761729835898190929297255325322975730670537967005064673, 1719187326356684165357673768179883411543453216743902279027914739746, 47087118671838361711307360253041365524656470092757509308102789969739029, 132526170995926613122153810359638932407424981123569937936128801108141704\ 9347, 3822705457345431111578832273523762160601819387474303976493478143900\ 8408506986400, 1127506929413944537420952138521733935698284744230286574184\ 868275035579882350609154924] If the floor is, [2, 2, 1, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [9, 6357, 16447321, 83063011917, 634124741200585, 6418271697539486639, 79760647119436009904359, 1158813078073207182208246970, 19039890219067799672109037727333, 345536977359227460016778181115188157, 6807037555862486557885055435200136982545, 143659867697958394286469152394909871383781549, 3215034569432119517327981937474950133734723000223, 75681728395113636901288108326935122827080756119192617, 1861732522660248287189629245992453740900864433870603147847, 47604362074262298337591005031934265563041297042663790727579076, 1259679028441453461482138776582091290162235867261865435902820907931, 34367576374115937615518504989289875824239207218252778840566233255460174, 963725891830820179918754748120760958518997394197071560285122907851989884133 , 27702099111668966611113368198673903040383712882416201967930306483835052\ 016128928, 81438041582989841341475978058723634188267722992450507743711407\ 6308674758885048214127] If the floor is, [3, 1, 1, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [10, 7425, 19811341, 102250773030, 793689156114651, 8141496780668441784, 102311405943377364234007, 1500732242959565450608299410, 24864788888918578716074039643936, 454607967463033875318022907946375155, 9015630441456884000942317198866362532841, 191426358715002107500505661821512689159155462, 4307823830252527613182777517524350025780716929549, 101925188553099925839722851949005519570028831423378414, 2519215044950830204603674956552306763625850266078873903062, 64701316761729835898190929297255325322975730670537967005064673, 1719187326356684165357673768179883411543453216743902279027914739746, 47087118671838361711307360253041365524656470092757509308102789969739029, 132526170995926613122153810359638932407424981123569937936128801108141704\ 9347, 3822705457345431111578832273523762160601819387474303976493478143900\ 8408506986400, 1127506929413944537420952138521733935698284744230286574184\ 868275035579882350609154924] If the floor is, [2, 1, 1, 1, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [5, 2145, 3636776, 12753712037, 70692556053053, 538062084544290042, 5172429435337799442495, 59496753497779282160432712, 789114026962872326337841559201, 11751727385856856418961493392918296, 192672653281429171975863915261887981069, 3425801216108320871923052907182435590314321, 65286814883087788041189662844895960361478416634, 1321133844411057286840297082589727434570482687973793, 28173237507294627294585376411478395160886201033336857591, 629207962318311411973650101341198498009929430671090318982808, 14641094463433804768118266475987847138245262202328275604625755883, 353416897519184421647376661920386779420437105377276409974214726087449, 8817261311212942597857031922534147635184747822371751936980265747873514393, 226642615448278611851995027544701715526140993380175369387970678924798748\ 188187, 59858798275083654291194760841317789551321433759429355148941016480\ 77949141202752446] If the floor is, [1, 1, 1, 1, 1, 1], then for heights n=1 to, 20 the numbers of Solid Young Tableaux are [1, 132, 87516, 140229804, 396499770810, 1671643033734960, 9490348077234178440, 67867669180627125604080, 583692803893929928888544400, 5838544419011620940996212276800, 66244124978105851196543024492572800, 836288764382254532915188713779640302400, 11570895443447601081407359451642915869302000, 173350856749047087227868685053324700693497684000, 2784591835268084496051527013236901155628039317067000, 47572076113399769965230407161865228632990820870785062000, 858538265205484953801456866514093556426809869604623359710000, 16274475645063354175139076024244412274999121731816308896110840000, 322467413532568579777476346845777467898829336857344114648508565320000, 6650995938899110343991104100498582529857394689556602949972101398891832000, 142280318416220472363203609001002799916708453455756090713808284175559587\ 660000] It took, 5148.686, seconds to generate this webbook