Starting 2 Dimensions The first , 30, terms starting at n=1 of the sequence enumerating words with \ 2n letters, where each letter occurs n times such that no run is of length 1 is [0, 2, 2, 6, 14, 34, 84, 208, 518, 1296, 3254, 8196, 20700, 52404, 132942, 337878, 860142, 2192902, 5598144, 14308378, 36610970, 93770358, 240390602, 616787116, 1583765724, 4069672444, 10464501074, 26924530158, 69315481778, 178545361842] ---------------------------- The first , 30, terms starting at n=1 of the sequence enumerating Young table\ aux os shape [n,n] such that no run is of length 1 is [0, 1, 1, 2, 4, 8, 17, 37, 82, 185, 423, 978, 2283, 5373, 12735, 30372, 72832, 175502, 424748, 1032004, 2516347, 6155441, 15101701, 37150472, 91618049, 226460893, 560954047, 1392251012, 3461824644, 8622571758] ---------------------------- The first , 30, terms starting at n=1 of the sequence enumerating words with \ 2n letters, where each letter occurs n times such all runs are odd is [2, 2, 4, 10, 22, 52, 126, 306, 752, 1862, 4634, 11588, 29090, 73262, 185024, 468410, 1188338, 3020384, 7689574, 19605790, 50054592, 127945722, 327402654, 838632660, 2150098022, 5517086402, 14167661296, 36408128806, 93624177502, 240905715952] ---------------------------- The first , 30, terms starting at n=1 of the sequence enumerating Young table\ aux of shape [n,n] such all runs are odd is [1, 1, 2, 4, 8, 17, 37, 82, 185, 423, 978, 2283, 5373, 12735, 30372, 72832, 175502, 424748, 1032004, 2516347, 6155441, 15101701, 37150472, 91618049, 226460893, 560954047, 1392251012, 3461824644, 8622571758, 21511212261] ---------------------------- ----------------------------------------- Starting 3 Dimensions The first , 30, terms starting at n=1 of the sequence enumerating words with \ 3n letters, where each letter occurs n times such that no run is of length 1 is [0, 6, 6, 90, 426, 3300, 21960, 158442, 1131894, 8235648, 60270240, 444642030, 3299324148, 24612560574, 184439101704, 1387685188620, 10477673493366, 79361398607166, 602812514110608, 4590534108044778, 35038732081649490, 268008097446542106, 2053919829710108676, 15768355860254750826, 121254013565740660572, 933810909346132561050, 7201568284545601708980, 55610628919323346004166, 429944845570836533503878, 3327800846672677702010472] ---------------------------- The first , 30, terms starting at n=1 of the sequence enumerating Young table\ aux os shape [n,n,n] such that no run is of length 1 is [0, 1, 1, 5, 15, 69, 304, 1518, 7807, 42314, 236621, 1364570, 8062975, 48680547 , 299388670, 1871463427, 11867447914, 76222340314, 495194637140, 3250470891226, 21536265419473, 143907285718510, 969092434762801, 6572626301369669, 44870249558432379, 308181434318023299, 2128577823365567439, 14778654269169052437, 103106852151899368769, 722616454876660300538] ---------------------------- The first , 30, terms starting at n=1 of the sequence enumerating words with \ 3n letters, where each letter occurs n times such all runs are odd is [6, 30, 240, 2448, 27516, 322548, 3886710, 47817180, 597755424, 7566980070, 96766786200, 1247847200400, 16204766412000, 211697839187250, 2779859867156760, 36666777274321572, 485544883133289570, 6452039588351882460, 86002634241918411816, 1149568827570630100368, 15404530274960534794356, 206895246763204208435550, 2784537385344996044323650, 37547348304572701097051448 , 507179614721264427413172066, 6861849287562234623762454378, 92974690087418560425234136740, 1261495058852688331791497668920, 17138059219894522876524965162424, 233106988466379335650852763008728] ---------------------------- The first , 30, terms starting at n=1 of the sequence enumerating Young table\ aux of shape [n,n,n] such all runs are odd is [1, 2, 9, 46, 306, 2252, 18308, 158872, 1454570, 13888112, 137277741, 1396638636, 14561307281, 155040525128, 1681193493943, 18524184410292, 207015267578400, 2342791950807478, 26814196006326067, 310035656428965874, 3617919491693906525, 42574272856853698850, 504851656580095984499, 6028848893702426438978, 72463203293780666418142, 876191903478439803085372, 10653422082750134560355983, 130201818054458732103309128, 1598932943599407005732609201, 19723825338202705632692393924] ---------------------------- ----------------------------------------- Starting 4 Dimensions The first , 30, terms starting at n=1 of the sequence enumerating words with \ 4n letters, where each letter occurs n times such that no run is of length 1 is [0, 24, 24, 2520, 24024, 784944, 14736960, 378512544, 8631684984, 214014091032, 5204923108344, 130016957183256, 3253718984636736, 82382126720169624, 2096843001254641560, 53743233444240168936, 1384392963470144157816, 35840320086055158728232, 931792904704594369498176, 24320659148445738524208696, 637018098153746380965679992, 16738799275817435195108909856, 441129689479807312892989763352, 11656699867093173612064874231232, 308786347911624245773885554631488, 8198461394140367951653964816636472, 218134848130901049790002666552589464, 5815286296742605744167165734347048176, 155314460315238714391522129624668955032, 4155226966768264455302220143926123800312] ---------------------------- The first , 30, terms starting at n=1 of the sequence enumerating Young table\ aux os shape [n,n,n,n] such that no run is of length 1 is [0, 1, 1, 14, 60, 753, 6444, 84376, 1012780, 14475831, 208179355, 3249823356, 52169799087, 877622543352, 15193337937435, 271480649350677, 4972078598219127, 93252248028246469, 1785055377251093097, 34822164050096777346, 690882917465524716531, 13921933498515663868206, 284538104890718607310033, 5891689048522258097317217, 123465984285526827823445114, 2616216819476940590563952759, 56010188657139146793750992840, 1210639469646307421331266137441, 26401897011943526113069427385073, 580596389588611724440299547729109] ---------------------------- The first , 30, terms starting at n=1 of the sequence enumerating words with \ 4n letters, where each letter occurs n times such all runs are odd is [24, 864, 51264, 3824784, 321138264, 28883125632, 2717677024152, 264089024306352, 26294507104750272, 2668421875692485184, 274985306622234034392, 28698005087215927961472, 3026853305178557340375192, 322136313498919029255923712 , 34550338504221024362779857984, 3730710035419633077490136404656, 405228425487254219292749155243896, 44246811779965842082358280753948576, 4853896337020287430459172859221179416, 534707786390920751853798079744640040624, 59126456258371695818411056165274175320064, 6560445919710713901057946918763861441891424, 730194238562784334813332695854902230998967544, 81504456434502163846194550312784297016750743424, 9121385672903274214890947730447953366918390780664, 1023263233969447705649620568816743356952409271320352, 115048708414928126898350022779196913133794688562484992, 12962040294315478118476945878083332143353994283844700912, 1463184526880656116411351934955855244596980561617103451352, 165463518752427383902331393237797211350374475184451299984352] ---------------------------- The first , 30, terms starting at n=1 of the sequence enumerating Young table\ aux of shape [n,n,n,n] such all runs are odd is [1, 6, 88, 2078, 66273, 2583073, 116483831, 5873821655, 323629818611, 19162146193716, 1204354453972083, 79595998308918644, 5491147756259198388, 393124976294383738037, 29069697714186707661680, 2211612876314498746152823, 172559427912599155962293001, 13770705515415050813209049284, 1121417891791133401804460705258, 93008519766903841954858705636769, 7843136294662183442481541738268344, 671480274580510243849160243075438766, 58290584494859588512779003712004168801, 5125046834896559222416612164949008632220, 455933071630813196266743221436529576814538, 41003989984750769549460021840115639496599016, 3725051130532659001636714311819173978715370521, 341597277352219909495960069576362325729978002217, 31600821139406259515481230975310248022361361075057, 2947383853025232075909827422830216058590935151155098] ----------------------------