--------------------------------------- The first, 20, terms of the Sequences Enumerating Words With r 1's, r 2's, ..., r n's, for r=1 to r=, 4, That do not contain an increasing susequence of length d for d from 3 to, 8 By Shalosh B. Ekhad In this article, dedicated to Neil Sloane (b. Oct. 10, 1939) on the occasion \ of his turning 75-years-old We will find the first, 20, terms of the enumerating sequences described in the title ------------------------------------------------------------------------ ------------------------------------------------------------------------- Fact number, 1 The first, 20, terms of the sequence enumerating words,w, of length, n, with exactly, 1, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 3 [1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786, 208012, 742900, 2674440, 9694845, 35357670, 129644790, 477638700, 1767263190, 6564120420] ------------------------------------------------------------------------- Fact number, 2 The first, 20, terms of the sequence enumerating words,w, of length, 2 n, with exactly, 2, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 3 [1, 6, 43, 352, 3114, 29004, 280221, 2782476, 28221784, 291138856, 3045298326, 32222872906, 344293297768, 3709496350512, 40256666304723, 439645950112788, 4828214610825948, 53286643424088024, 590705976259292856, 6574347641664629388] ------------------------------------------------------------------------- Fact number, 3 The first, 20, terms of the sequence enumerating words,w, of length, 3 n, with exactly, 3, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 3 [1, 20, 374, 8124, 190893, 4727788, 121543500, 3212914524, 86782926068, 2384725558736, 66456350375566, 1873703883228900, 53351152389518550, 1531960347453263112, 44311785923563130392, 1289909841595078198172, 37760636720455988917420, 1110927659386926734186992, 32829851946481173252840184, 974077753606734750485658224] ------------------------------------------------------------------------- Fact number, 4 The first, 20, terms of the sequence enumerating words,w, of length, 4 n, with exactly, 4, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 3 [1, 70, 3199, 173860, 10203181, 631326526, 40553993125, 2678871322640, 180830423671450, 12418980645870820, 864996624914197495, 60957211831578399100, 4338372535640598835279, 311386494956413595138930, 22513820432313175983170649, 1638226907374445245497453464, 119879437233268054455643664540, 8816310717018695747221495985704, 651283473782492244839579091820690, 48305712720059582761530066305796370] ------------------------------------------------------------------------ ------------------------------------------------------------------------- Fact number, 5 The first, 20, terms of the sequence enumerating words,w, of length, n, with exactly, 1, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 4 [1, 2, 6, 23, 103, 513, 2761, 15767, 94359, 586590, 3763290, 24792705, 167078577, 1148208090, 8026793118, 56963722223, 409687815151, 2981863943718, 21937062144834, 162958355218089] ------------------------------------------------------------------------- Fact number, 6 The first, 20, terms of the sequence enumerating words,w, of length, 2 n, with exactly, 2, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 4 [1, 6, 90, 1879, 47024, 1331664, 41250519, 1367533365, 47808569835, 1744233181074, 65905305836049, 2564220925607625, 102277575120518170, 4167486279986250932, 172988069360147449566, 7298137818882637998561, 312349784398279829229533, 13539988681466075755541070, 593697458037818682683254284, 26302053106583451923036304267] ------------------------------------------------------------------------- Fact number, 7 The first, 20, terms of the sequence enumerating words,w, of length, 3 n, with exactly, 3, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 4 [1, 20, 1680, 173891, 21347262, 2977892253, 455912368540, 74876841353159, 12990339123973119, 2354973430941967605, 442587722191655715108, 85717352536181708342445, 17029266882947116165470103, 3457866959157770598680361537, 715559803849259851987691458500, 150551218551130203662620787992583, 32142433864805658006358724788905375, 6952245776854966150188862330300561056, 1521352773314475957928865900786243945640 , 336425217352069009616581283985388825084233] ------------------------------------------------------------------------- Fact number, 8 The first, 20, terms of the sequence enumerating words,w, of length, 4 n, with exactly, 4, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 4 [1, 70, 34650, 16140983, 8854463421, 5532980565456, 3798011394008444, 2798461806432513085, 2179251644112128926809, 1774029308605731224234922, 1497612094060753803137726582, 1303178757814574200714348639251, 1163471249071555286949793002571005, 1061855751542748987658894786275614548, 987782284350242764771833528063298286256, 934347073922599172485027979820932703181401, 896920771124692119469714569436435897896032337, 872344761827372239609582067450327869949604599310, 858443797096732666126199560092160533779016423423858, 853721670141491635889318300245321802861301277299323117] ------------------------------------------------------------------------ ------------------------------------------------------------------------- Fact number, 9 The first, 20, terms of the sequence enumerating words,w, of length, n, with exactly, 1, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 5 [1, 2, 6, 24, 119, 694, 4582, 33324, 261808, 2190688, 19318688, 178108704, 1705985883, 16891621166, 172188608886, 1801013405436, 19274897768196, 210573149141896, 2343553478425816, 26525044132374656] ------------------------------------------------------------------------- Fact number, 10 The first, 20, terms of the sequence enumerating words,w, of length, 2 n, with exactly, 2, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 5 [1, 6, 90, 2520, 102011, 5176504, 307027744, 20472135280, 1496594831506, 117857270562568, 9869468603141427, 870255083860881152, 80185525536941657225, 7673807618627318341436, 759049283017632212000140, 77292554293281131959377376, 8075621155990277422800518076, 863336030048719032973236796720, 94217100940352292965547102689050, 10475021245521453871303390607528784] ------------------------------------------------------------------------- Fact number, 11 The first, 20, terms of the sequence enumerating words,w, of length, 3 n, with exactly, 3, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 5 [1, 20, 1680, 369600, 117392909, 46121962742, 21198300356500, 11003612776114008 , 6290031043253973544, 3887357166155963541538, 2562077915376091538040250, 1782153151031487742187453640, 1297781266782084301101836538690, 983066960483171632842827775906144, 770644211485891382797695920817345878, 622570449038167339670802274462606076328, 516508364941647830125638544305941049499400, 438791430572395821810225565686153089930074624, 380771574129580088434904497211211829473230879776, 336812084920518506911361362093988355916953385760640] ------------------------------------------------------------------------- Fact number, 12 The first, 20, terms of the sequence enumerating words,w, of length, 4 n, with exactly, 4, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 5 [1, 70, 34650, 63063000, 142951955371, 389426248416626, 1238402046254860022, 4454056622413300252928, 17668055644599543583018570, 75867559322054514745288107364, 347785237467609520037269752908904, 1684035818793607129226446293560872032, 8542804217282979478103200701687206370545 , 45105324827959031859037805100546610899286870, 246578917399229205506771798883279360292315268706, 1389715912705659388892831639343455579058245441346016, 8046400952479432432038790200252988155815571756562787364, 47719798288541638994722054983774573415857751615377504440440, 289156219701756464055126773095615680482738531994925459144412800, 1786404494536187691424698305111683344784980161248649907215327518400] ------------------------------------------------------------------------ ------------------------------------------------------------------------- Fact number, 13 The first, 20, terms of the sequence enumerating words,w, of length, n, with exactly, 1, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 6 [1, 2, 6, 24, 120, 719, 5003, 39429, 344837, 3291590, 33835114, 370531683, 4285711539, 51990339068, 657723056000, 8636422912277, 117241501095189, 1639974912709122, 23570308719710838, 347217077020664880] ------------------------------------------------------------------------- Fact number, 14 The first, 20, terms of the sequence enumerating words,w, of length, 2 n, with exactly, 2, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 6 [1, 6, 90, 2520, 113400, 7235651, 592616287, 58255807971, 6585311137855, 832218817076725, 115277768079251946, 17248585237921441506, 2756607130342279346898, 466382922740788231293660, 82934291167918369802940155, 15409473946816441390338039549, 2976972014567107396998469900773, 595532772402915680694897608562000, 122931129899096845723882281198534450, 26106165950240392587322865395992247500] ------------------------------------------------------------------------- Fact number, 15 The first, 20, terms of the sequence enumerating words,w, of length, 3 n, with exactly, 3, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 6 [1, 20, 1680, 369600, 168168000, 117108036719, 105795227339731, 115061550940847029, 144608494330313029001, 204355873608295177850585, 318271658632525518202069066, 537979534276664676790895443905, 975168187434299713523227390556343, 1877541644363618408027327954800896687, 3810133810926336264429075840384827236235, 8098053035891794572591600842662004513096437, 17931975963370182861228232413129064037888895645, 41187758324435030408725512162942585687407192986744, 97764001243368868465839965315766602453345651408776564, 239042474512330088501693495145839615588810143418990970275] ------------------------------------------------------------------------- Fact number, 16 The first, 20, terms of the sequence enumerating words,w, of length, 4 n, with exactly, 4, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 6 [1, 70, 34650, 63063000, 305540235000, 2169397142273171, 19353195105158134397, 206267644432668595555251, 2541180244286616138545044804, 35274365988673008260909391558520, 540804251392516657531679467797320185, 9015988636982792548260545203108168061274, 161453119203752650200050770940566785761796679, 3075251306809307177176255368193701344848102769481, 61811351869608086720712436672954706961427923011946280, 1302513687484944643416397043009455287728742536378750400085, 28620505346173460227157309099753521970260102953170869547708611, 652809693483518394647832754893177681041268533266866864400663081210, 15397327882113815443337643669511396057974317351645077713890321659163065, 374310550349331404899116270045826353862311592108231594000571479095975075700] ------------------------------------------------------------------------ ------------------------------------------------------------------------- Fact number, 17 The first, 20, terms of the sequence enumerating words,w, of length, n, with exactly, 1, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 7 [1, 2, 6, 24, 120, 720, 5039, 40270, 361302, 3587916, 38957991, 457647966, 5763075506, 77182248916, 1091842643475, 16219884281650, 251774983140578, 4066273930979460, 68077194367392864, 1177729684507324152] ------------------------------------------------------------------------- Fact number, 18 The first, 20, terms of the sequence enumerating words,w, of length, 2 n, with exactly, 2, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 7 [1, 6, 90, 2520, 113400, 7484400, 674641325, 78029736684, 10993963346208, 1813132907959212, 339903268216009897, 70892623666650500652, 16187362928435847480108, 3996531136680196842744000, 1056418169506540390275872193, 296585690807757153741980060308, 87850244542157405986885148750908, 27301970438876361944257777905735072, 8860297053914292345682542155338134342, 2990502990009784109379982783774590821568 ] ------------------------------------------------------------------------- Fact number, 19 The first, 20, terms of the sequence enumerating words,w, of length, 3 n, with exactly, 3, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 7 [1, 20, 1680, 369600, 168168000, 137225088000, 171248808285596, 286934414708075588, 595536828369938038917, 1459650175209699315273156, 4096453100636188630773909096, 12878611494406064696401518693294, 44608023888114689229061465110211506, 168015473289454445650854465070930763760, 680902136788805954832860677397031730698636, 2943402870959296754918265133523368949117822756, 13474420756635472962682589728462591265970971616116, 64927845320035564506980736573502304749659704223527664, 327624029677125601900657886022355087122825616271517824256, 1723572028773181454332647515708332608938585913806016917099970] ------------------------------------------------------------------------- Fact number, 20 The first, 20, terms of the sequence enumerating words,w, of length, 4 n, with exactly, 4, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 7 [1, 70, 34650, 63063000, 305540235000, 3246670537110000, 54391509124298646609, 1204252024013341582444384, 32724135062879773983583165227, 1048640936549963868454478388428416, 38551196587742599637012937767441006596, 1592130072842374106638425875299040034639720, 72649604049156594389483946044942617209311102214, 3614072175284412864679388115406495186837561715062236, 193882854527336282904010411563170294213302290571810883001, 11116369629223602558267071207975959396859174784850914446231256, 676115114613486276642152530609620942434981241239733114960441902998, 43349199339629709932864307781908555793115979080504417345156792345974992, 2914243361833553741862203644170778477000174020157653185564074934232088197247, 2044892708945262296100792335471258361982454568095211631361126561585729190811216\ 96] ------------------------------------------------------------------------ ------------------------------------------------------------------------- Fact number, 21 The first, 20, terms of the sequence enumerating words,w, of length, n, with exactly, 1, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 8 [1, 2, 6, 24, 120, 720, 5040, 40319, 362815, 3626197, 39832877, 476591309, 6162155981, 85494566892, 1264755621000, 19835792076675, 328115505900675, 5698062006852574, 103455252673577866, 1956590161853191160] ------------------------------------------------------------------------- Fact number, 22 The first, 20, terms of the sequence enumerating words,w, of length, 2 n, with exactly, 2, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 8 [1, 6, 90, 2520, 113400, 7484400, 681080400, 81537026047, 12336753791164, 2275706085836620, 495930917980401095, 124359717079896241982, 35129648277668598362849, 10992465528460933750610244, 3759621205312908005555541311, 1390478693648575626739573917341, 551259673364622477603596786449307, 232577340592058616967783746584300866, 103787059884653553571010147031300217204, 48732599047159430192957756200575142134540] ------------------------------------------------------------------------- Fact number, 23 The first, 20, terms of the sequence enumerating words,w, of length, 3 n, with exactly, 3, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 8 [1, 20, 1680, 369600, 168168000, 137225088000, 182509367040000, 360953073372968159, 971140940173113235912, 3310532091475088258327337, 13602144834440596618535524692, 65098000173380637365846222477475, 354203089405499357148478310430981721, 2151779060779902798220353318652785828733, 14389980769680401335176876811967574752002542, 104733089520393771211188943370694229542260427387, 821828384130230784671715119688532831055514034529515, 6898056931750763282580274458799984808657673062170651060, 61519970824635237580371354378014138876696997520537304883464, 579644378770352158391252466595528997021337533629557953971352695] ------------------------------------------------------------------------- Fact number, 24 The first, 20, terms of the sequence enumerating words,w, of length, 4 n, with exactly, 4, occurrences of the letter i, for i from 1 to n such that you can't find an increasing subsequence of length, 8 [1, 70, 34650, 63063000, 305540235000, 3246670537110000, 66475579247327250000, 2179159866745975811590271, 98380000351748513537890071820, 5611833570326703471700834056086202, 385292217603023358878915390756801760476, 30859318502825645718045972901853345700054830, 2818630306387654091873007359637208248527741891745, 288457237590667497578372982212058466447947172137281340, 32609242809722140052016240755885329465763420491774238258856, 4024749070138892028315447729644574158761342623798584734247979741, 537086473800011410616194434661394870841002770198554834441681265005835, 76858497311104837197860257491747649718766898277114907299215128471050778094, 117\ 12519009246258363364173281197802763188942043436926001595713988666686519173710, 1889384248104430814600625168984702527010119050444821786884232431814058135099911\ 389628] ----------------------------------------- This concludes this article, that took, 13971.605, to generate.