From: Doron Zeilberger zeilberg (AT) math.rutgers.edu Date: Mon, 19 Aug 2013 09:10:35 -0400 (EDT) To: njasloane@gmail.com Subject: Hardin matrices Rigorously derived generating functions for enumerating k by n Hardin matrices for k from 1 to , 11 By Shalosh B. Ekhad Definition: A Hardin matrix has entries from {0,1} with no adjacent 1's horizontally, vertically, or in the NW-SE direction, and whose top-left entry is 1 Definition:For a fixed k, Let f(k)(x) be the generating function whose coefficient of x^n is the number of n by k We have the following (rigorously-derived) facts followed by the first, 20, terms of the enumerating sequence (for the sake of Sloane) f[1](x) = -x/(x^2+x-1) [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765] f[2](x) = -x/(x^3+2*x^2+x-1) [1, 1, 3, 6, 13, 28, 60, 129, 277, 595, 1278, 2745, 5896, 12664, 27201, 58425, 125491, 269542, 578949, 1243524] f[3](x) = -x*(x^3-x-2)/(x^5-4*x^3-5*x^2-x+1) [2, 3, 13, 35, 112, 337, 1034, 3154, 9637, 29431, 89895, 274564, 838609, 2561372, 7823242, 23894643, 72981777, 222909351, 680835436, 2079486057] f[4](x) = x*(2*x^4-7*x^3-x^2+3*x+3)/(x^8-3*x^7+x^6+6*x^5-4*x^4-15*x^3-10*x^2-x+ 1) [3, 6, 35, 133, 587, 2448, 10414, 44024, 186414, 789100, 3340345, 14140347, 59858152, 253389483, 1072638232, 4540650778, 19221306410, 81366888278, 344439152622, 1458066449898] f[5](x) = -x*(x^11-2*x^10-4*x^9+26*x^8-51*x^7+45*x^6-5*x^5-2*x^4-38*x^3-6*x^2+8 *x+5)/(x^13-2*x^12-8*x^11+37*x^10-57*x^9+13*x^8+68*x^7-38*x^6-69*x^5+14*x^4+48* x^3+21*x^2+x-1) [5, 13, 112, 587, 3631, 21166, 126119, 745178, 4416695, 26150120, 154877307, 917205757, 5431915952, 32169045631, 190512481196, 1128258633821, 6681806858103, 39571194265886, 234349700556332, 1387872742075595] f[6](x) = x*(2*x^17+6*x^16-18*x^15-97*x^14+247*x^13-85*x^12-239*x^11-143*x^10+ 1907*x^9-3792*x^8+3365*x^7-999*x^6-425*x^5+112*x^4+241*x^3+27*x^2-20*x-8)/(x^21 +3*x^20-6*x^19-29*x^18+32*x^17+154*x^16-148*x^15-1152*x^14+3926*x^13-5776*x^12+ 3827*x^11+659*x^10-2727*x^9+732*x^8+1199*x^7-449*x^6-478*x^5+70*x^4+147*x^3+42* x^2+x-1) [8, 28, 337, 2448, 21166, 172082, 1428523, 11771298, 97268701, 802886174, 6629901197, 54739811878, 451976078779, 3731849749697, 30812948919061, 254414847888742, 2100639733295629, 17344457600010491, 143208852222784259, 1182439709334842998] f[7](x) = -x*(x^32+5*x^31-9*x^30-145*x^29-524*x^28-870*x^27+93*x^26+3283*x^25+ 2794*x^24-13686*x^23-20581*x^22+52803*x^21+56597*x^20-219345*x^19+64005*x^18+ 577213*x^17-1128820*x^16+695794*x^15+757117*x^14-2151205*x^13+2455581*x^12-\ 1736815*x^11+824071*x^10-300738*x^9+133555*x^8-64864*x^7+9093*x^6+7249*x^5-601* x^4-1336*x^3-131*x^2+47*x+13)/(x^34+5*x^33-14*x^32-173*x^31-545*x^30-561*x^29+ 1405*x^28+5635*x^27+2585*x^26-23292*x^25-27097*x^24+92350*x^23+80904*x^22-\ 365483*x^21+64683*x^20+942033*x^19-1353257*x^18-138325*x^17+2761082*x^16-\ 4149918*x^15+3102284*x^14-897815*x^13-464722*x^12+477721*x^11-37244*x^10-112654 *x^9+27340*x^8+21164*x^7-5096*x^6-3711*x^5+192*x^4+432*x^3+85*x^2+x-1) [13, 60, 1034, 10414, 126119, 1428523, 16566199, 190540884, 2197847780, 25325358687, 291935092921, 3364727410265, 38782728207101, 447011297075966, 5152298718205939, 59385855860517581, 684486816741728022, 7889457806798773244, 90934612740867535991, 1048120674269761559923] f[8](x) = x*(2*x^51-2*x^50-66*x^49+110*x^48+1629*x^47-4813*x^46-67688*x^45-\ 234350*x^44-401990*x^43-658771*x^42-3161057*x^41-14624224*x^40-42871801*x^39-\ 73761278*x^38-26604860*x^37+239859895*x^36+713783092*x^35+863260161*x^34-\ 134128185*x^33-1800904161*x^32-1419374681*x^31+1935860639*x^30+2757350404*x^29-\ 2352200353*x^28-3703928419*x^27+3855489166*x^26+3883942773*x^25-6335961004*x^24 -2037479016*x^23+8626282264*x^22-3109079348*x^21-7475315873*x^20+10724160854*x^ 19-5285410006*x^18-736560408*x^17+1717885047*x^16+807074005*x^15-2824314771*x^ 14+2737074476*x^13-1533773855*x^12+524079594*x^11-91693302*x^10+962499*x^9+ 50560*x^8+1401445*x^7-213607*x^6-95198*x^5+4865*x^4+7163*x^3+545*x^2-108*x-21)/ (x^55-x^54-30*x^53+45*x^52+608*x^51-671*x^50-10138*x^49-2782*x^48+87637*x^47-\ 234085*x^46-4214944*x^45-20510652*x^44-59224072*x^43-110007288*x^42-105839886*x ^41+67901250*x^40+405881894*x^39+511379314*x^38-211758176*x^37-1366907685*x^36-\ 877760575*x^35+2023351213*x^34+2807416030*x^33-2482388885*x^32-5396581563*x^31+ 3493944338*x^30+8290578844*x^29-6485990080*x^28-10036097232*x^27+12628340471*x^ 26+6931675953*x^25-19704882159*x^24+5172378119*x^23+18258761486*x^22-\ 21472294755*x^21+2246759550*x^20+18064627845*x^19-23143116063*x^18+15244941445* x^17-5406913431*x^16+225285263*x^15+733571937*x^14-280040412*x^13-22625542*x^12 +40598266*x^11-4142351*x^10-3899000*x^9+677669*x^8+329977*x^7-47829*x^6-25328*x ^5+452*x^4+1243*x^3+170*x^2+x-1) [21, 129, 3154, 44024, 745178, 11771298, 190540884, 3057290265, 49208639399, 791176762937, 12725363193829, 204647839919537, 3291296999329711, 52931964429099939, 851279732514835940, 13690693829364308175, 220180543075683677274, 3541052375099195761698, 56948956408634393389871, 915881298985498469592761] f[9](x) = -x*(x^87-8*x^86-32*x^85+644*x^84-2907*x^83+2151*x^82+28450*x^81-\ 120413*x^80+116834*x^79+1157261*x^78-6031619*x^77+528780*x^76+89023390*x^75-\ 173350222*x^74-1151860848*x^73+2899238663*x^72+16085926699*x^71-24818469003*x^ 70-214721214194*x^69-32166176169*x^68+2188056026979*x^67+4796619392584*x^66-\ 9549922863025*x^65-69086644360435*x^64-122580631673293*x^63+159940777307292*x^ 62+1443424730591592*x^61+3975212268405698*x^60+5403457632838150*x^59-\ 2120616268353442*x^58-32309380383956217*x^57-97892798354412965*x^56-\ 195561955588038608*x^55-292080690551658166*x^54-330897208018026283*x^53-\ 271862382630883591*x^52-143153118189461025*x^51-46790597262700379*x^50-\ 74824158811506368*x^49-191552329531247794*x^48-229734075077442510*x^47-\ 70638447929915341*x^46+170575249373655168*x^45+235562083703165699*x^44+ 52701963398566867*x^43-136611961721162593*x^42-101463611649363881*x^41+ 49806571140121782*x^40+66964999027203092*x^39-31485119275002257*x^38-\ 40080146559677170*x^37+34827698092835986*x^36+25676396868148593*x^35-\ 36115472217148137*x^34-12698940642923466*x^33+31707377773650075*x^32-\ 464132618617674*x^31-21925813281460471*x^30+9354081249342613*x^29+ 9391601184461481*x^28-10350338655011577*x^27+436472375718377*x^26+ 5310023826562667*x^25-3637822592998680*x^24-100789742203460*x^23+ 1782281907825985*x^22-1375752912535871*x^21+444149479690459*x^20+90577465469715 *x^19-194792934082176*x^18+127237623941515*x^17-55330131274873*x^16+ 18527504427376*x^15-5226077139205*x^14+1327892128274*x^13-295461449355*x^12+ 46670718371*x^11-2100803693*x^10-763904766*x^9+31561542*x^8+42827155*x^7-\ 3342239*x^6-1300016*x^5+19620*x^4+37360*x^3+2234*x^2-243*x-34)/(x^89-8*x^88-38* x^87+695*x^86-2879*x^85-105*x^84+39914*x^83-136254*x^82+43847*x^81+1571198*x^80 -6700528*x^79-2570071*x^78+111939979*x^77-181278765*x^76-1496929050*x^75+ 3457565706*x^74+20838665195*x^73-31494932814*x^72-278159240994*x^71-30990252773 *x^70+2847346843155*x^69+6019979527395*x^68-12381088034848*x^67-82879141747172* x^66-127482845730036*x^65+246442107448472*x^64+1638150531308210*x^63+ 3630394365861038*x^62+2148662678861166*x^61-12735538772455692*x^60-\ 53901370391664446*x^59-124871036042872432*x^58-201657150023555253*x^57-\ 226356673050150368*x^56-136563731814790740*x^55+67831170531569559*x^54+ 272395901288262463*x^53+296803477374187509*x^52+64768558699537498*x^51-\ 249495641012276252*x^50-324196672203238169*x^49-44674358779740500*x^48+ 282871338108355230*x^47+247136946296363469*x^46-106868457992460011*x^45-\ 280799538989855391*x^44-34278410467657420*x^43+226961599933750196*x^42+ 95541176582994872*x^41-162855432975425412*x^40-99278700466055506*x^39+ 117589053059603490*x^38+75307132119610574*x^37-89871633733075620*x^36-\ 42509085580493654*x^35+69385711314076868*x^34+11682212847079777*x^33-\ 48126804293022820*x^32+9585671646760248*x^31+24983151120411597*x^30-\ 16816567234540602*x^29-5401968255267581*x^28+11825337408157360*x^27-\ 4230674222649921*x^26-2982940342312306*x^25+3885196495123007*x^24-\ 1464924827758540*x^23-440208193462827*x^22+852626156042858*x^21-524306069845407 *x^20+182611451591524*x^19-29496433268727*x^18-4478480115950*x^17+3500468879427 *x^16-538626934150*x^15-124488945147*x^14+55957856060*x^13-511179629*x^12-\ 3049905478*x^11+272892045*x^10+130485447*x^9-15283855*x^8-5083216*x^7+417526*x^ 6+171909*x^5+334*x^4-3520*x^3-341*x^2-x+1) [34, 277, 9637, 186414, 4416695, 97268701, 2197847780, 49208639399, 1105411581741, 24801939723742, 556713719650007, 12494370307905104, 280426447918993931, 6293836975716291709, 141258681814255369288, 3170396387349296944621, 71156151001790985434648, 1597023012652317575685103, 35843461580050057669914634, 804467875227432424602919960] f[10](x) = x*(2*x^140-8*x^139-166*x^138+530*x^137+10038*x^136-28642*x^135-\ 512099*x^134-6845*x^133+67053158*x^132-671867789*x^131+3505238730*x^130-\ 12918665069*x^129+51936488577*x^128-290508630419*x^127+1571862780978*x^126-\ 6595301771377*x^125+21103600306469*x^124-53062135053784*x^123+104375899947829*x ^122-127819617568919*x^121-55383631860312*x^120+166407931582919*x^119+ 5049487858662493*x^118-35851836130522311*x^117+103516424911961686*x^116-\ 2977150632353102*x^115-1052667066155664532*x^114+3239580422553944906*x^113+ 294587847712018263*x^112-25808743221387948854*x^111+48660876297316198051*x^110+ 92689763761695831432*x^109-444622205366202713259*x^108-26797862361740843700*x^ 107+2616280911930610765245*x^106-1937298339393183135326*x^105-\ 12997405856990006989633*x^104+15722587930333675363488*x^103+ 62609085363188732464212*x^102-83341107539460918630511*x^101-\ 311768494169033347357592*x^100+319519647319106688859559*x^99+ 1560238999075165024152454*x^98-622452933395972617548156*x^97-\ 7161175350075342234119336*x^96-2647169772942112827812228*x^95+ 27027779624096176266125939*x^94+35012462836686750294417288*x^93-\ 69921369974591417277583146*x^92-205645166624731543488730589*x^91+ 25966663649191012404998282*x^90+767658571140573448639801838*x^89+ 838541959131582884728053761*x^88-1690028164835260464910958685*x^87-\ 5399860590803306038139947060*x^86-1557191110726967439392965992*x^85+ 19640394699608523201190588102*x^84+46916563919597007315030664223*x^83+ 24021580551441864487681090494*x^82-144537522852998096942645414457*x^81-\ 542185061141251679488656090757*x^80-1178670860512984229582727998747*x^79-\ 1992417396805655572420337868132*x^78-2953677803279215843396245811225*x^77-\ 4189715316965415828143835994294*x^76-5957402831496549082282848388125*x^75-\ 8343337807559899022953746844400*x^74-10805046600478011461594406217677*x^73-\ 11927196713099871015003606128668*x^72-9800583642061466966883467726012*x^71-\ 3091114325122292780035155433332*x^70+7709052537594223363979137927392*x^69+ 19751076088264736728041114171629*x^68+28713064287130768677560473368810*x^67+ 30921960848150604198289102031504*x^66+25522373541084601609295105251194*x^65+ 15170483550429893852984030907542*x^64+4570716155898167405163305861717*x^63-\ 2268075682608984936188738264057*x^62-4139648209123703322368310381270*x^61-\ 2670320092844874900332205350351*x^60-520907844355396519679810339666*x^59+ 590548140031187954610620337831*x^58+551027297412087543161722583419*x^57+ 120705467021083129608290324160*x^56-104701454739301104785311965428*x^55-\ 73564531665536079768955087754*x^54+6079823064198573711834412629*x^53+ 19933248156426030398405496800*x^52-454136038060919919356539247*x^51-\ 6589801411534661488313849385*x^50+382393325552974530338644857*x^49+ 2884723623219040312856507795*x^48-2057341430344632270414344*x^47-\ 1210084991573563308168708830*x^46-76701064871627823887678250*x^45+ 453608060153057943906721778*x^44+35592000363941526351469954*x^43-\ 156569231723286739219032797*x^42-4610494752273247474202955*x^41+ 50388130751016310762811305*x^40-4293946563270159799255087*x^39-\ 14547440271961853747966549*x^38+3877587253186390546481027*x^37+ 3345312888532805633967207*x^36-1890861373374053196682004*x^35-\ 397596132586806008949183*x^34+617807706690723576116001*x^33-\ 98271946990017304751155*x^32-118730846855512230482679*x^31+ 67436016832386560570402*x^30+848786759719627104258*x^29-15141424289321106905249 *x^28+6288183106820427835265*x^27+209561996272466937685*x^26-\ 1262356983194346790448*x^25+559516957091134544002*x^24-78228184981483286290*x^ 23-29470828074758170416*x^22+14735772701453512492*x^21+966675479771951445*x^20-\ 3488733657959634422*x^19+1856212832344850072*x^18-598803720718764665*x^17+ 134012959989303766*x^16-21102471437243425*x^15+2268339616688237*x^14-\ 174235502412321*x^13+17585725456002*x^12-2632415804625*x^11+97024371950*x^10+ 39122234872*x^9-1437421859*x^8-1027677550*x^7+47458152*x^6+15453997*x^5-24417*x ^4-188806*x^3-8674*x^2+540*x+55)/(x^144-4*x^143-80*x^142+268*x^141+4626*x^140-\ 16797*x^139-194370*x^138+942568*x^137+6173741*x^136-51966904*x^135+30823622*x^ 134-173482730*x^133+19084417569*x^132-229388278225*x^131+1515048277124*x^130-\ 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[89, 1278, 89895, 3340345, 154877307, 6629901197, 291935092921, 12725363193829, 556713719650007, 24323608858322906, 1063228770141145384, 46467826999734230731, 2030974556422845366545, 88766094139582331605246, 3879654488735271050141403, 169565642639954148329642958, 7411106619606782377852950977, 323912805472254569571065583672, 14157065183943434981087679558140, 618754431478169665879994284792309] The first, 11, terms of the enumerating sequence for square Hardin matrices is [1, 1, 13, 133, 3631, 172082, 16566199, 3057290265, 1105411581741, 776531523355217, 1063228770141145384] This took , 189.104, seconds, to generate