完美间隔配对排列

aimisiyou

王守恩 发表于 2020-3-1 19:20
_ (ff 1=1)
1, "101"
_ (ff 2=1)

_ (ff 1)
("101")
_ (ff 2)
("12102")
_ (ff 3)
("1312032" "0231213")
_ (ff 4)
("240231413" "141302432" "023421314" "131423024")
_ (ff 5)
("15103425324" "15123425304" "15104235243" "31513420524" "30523421514" "20524131543" "13145320425" "14153042352" "20425314135" "23425314105" "14135243205")
_ (ff 6)
("2612145360435" "2612153460354" "3161345026425" "4260245316135" "4161345236205" "4263245316105" "1612532463504" "2632503461514" "1613504362542" "3062352416154" "1016425324635" "1316435024625" "3046325421615" "2362531416504" "2062514136543" "1016352432654" "2462514136503" "1014635243265" "1314635240265" "1416254230653" "1416354032652" "2042635413165" "2342635410165" "2562141536403" "1015623425364" "1014562342536" "1415604235263" "1413564320526" "1516304532642" "1015364235246" "1415364235206" "1510642532463" "1512642530463" "3151364250246" "1514603542362" "3450364251216" "1510463524326" "2532463514106")
_ (ff 7)
("270234563741516" "372432564701516" "271215364735046" "273205364715146" "370235264715146" "470254263751316" "375131465724026" "375234265741016" "271214635743065" "271215640735463" "237203564171546" "171425624735063" "237243560471516" "027423564371516" "417104562372536" "317135064275246" "317135264275046" "347532462571016" "427524063571316" "527423564371016" "537403564271216" "171306435724625" "274206345731615" "374236245701615" "271216450734653" "237203645171465" "171406354732652" "417104635273265" "427324635171065" "017125623475364" "171356034752642" "121723564307546" "121724560347536" "131743560427526" "420724561317536" "131753064257246" "131753264257046" "141752462357036" "357131652472064" "425724065317136" "520723564317146" "531713564207246" "017126425374635" "270261514736543" "317136052472654" "121723645307465" "131743625427065" "024723645317165" "240275461315736" "257236053471614" "357236250471614" "035723625417164" "131573460524726" "350732652417164" "352732650417164" "352732651417064" "015173465324726" "151742652437063" "245273465301716" "510174562342736" "530473564121726" "372632045761415" "473624325761015" "247261415376035" "237263151476054" "041716425327635" "141706425327635" "141726425307635" "340736415127625" "121726350437654" "023726351417654" "234273645101765" "240257463151376" "151736053427624" "350736151427624" "151706254237643" "121527463504376" "340537462512176" "315137465204276" "151473654302762" "036713145627425" "346731415627025" "034673245261715" "026721514637543" "131673052462754" "141673450362752" "014167345236275" "141367435206275" "014156742352637" "151673054362742" "015167245236473" "121526734503647" "245267415136073" "151267253406374" "253267350416174" "253267351416074" "151467054236273" "151467254236073" "015146735423627" "350436754121627" "362732056417154" "046171435623725" "416174035623725" "360731516427524" "016172452634753" "023627345161475" "041617435263275" "141607435263275" "121627530463574" "131463754206257" "131563724526047" "131563740526427" "015163745326427" "151647053462372" "415164725326037" "250263754316147" "052462754316137" "151463754326027" "026327435614175" "316137425624075" "246207435613175" "161375036425724" "131643752462057" "420624753161357" "121625740365437" "025623745361417" "352632754161047" "052642753461317" "451614752362037" "450634753261217" "261217536403574" "161347536402572" "316134752642057" "264207546131573" "161257236453074" "026325734615147" "316135724625047" "016135743625427" "316135740625427" "161507346532472" "161527426534073" "236253740651417" "256204753641317" "256234753641017" "261215746305437" "161345736425027" "264205746315137" "260253746351417" "264253746351017" "460354736251217" "362532746501417" "461514736523027" "610145763425327" "620425764315137" "630453764251217" "620524763541317")
_ (ff 8)
("28324536748501617" "38253246758401617" "48350436758121627" "38013164758246257" "38413164758206257" "48131463758206257" "18130563748526427" "18134563748526027" "48052462758316137" "48151463758326027" "38423624758161057" "18120625748365437" "28021615748365437" "28121605748365437" "18124625748365037" "38453604758261217" "58034653748261217" "28426354738651017" "58236253748601417" "58416154738623027" "38613145768425027" "48620425768315137" "48630453768251217" "38623524768541017" "38013157468254276" "38243257468151076" "28320537468514176" "38023527468514176" "28121367538406574" "38013167528426574" "34813146752802657" "28121567048536473" "02812156734853647" "23824356740851617" "38053267258416174" "38253267058416174" "04815146735823627" "24825346735801617" "41815406735823627" "28520367538416174" "38513167528426074" "42852436753801617" "52812156734803647" "51810456734823627" "18120627538463574" "28021617538463574" "28121607538463574" "02812164753846357" "02832463754816157" "41812462753806357" "42832463750816157" "42832463751816057" "18153647358462072" "28521617538463074" "03852362754816147" "25820365734816147" "35813165724826047" "35813165740826427" "05814165743826327" "31813625724865047" "18146357438652072" "04852642753861317" "34853642752861017" "25821615734863047" "25820635743861417" "35823625740861417" "05824625743861317" "25824605743861317" "25824635743861017" "51812652734863047" "52832653740861417" "03861315746825427" "28621517468534073" "03862352746851417" "45861415736823027" "53862352746801417" "52862354736841017" "62812156734853047" "63825326745801417" "64815146735823027" "18130573468524726" "28023574368541716" "28420574368531716" "28423574368501716" "48530473568121726" "28121375638405764" "38013175628425764" "34813147562802576" "02812157463854376" "18134573648502762" "41813457362852076" "03825327465814176" "23825307465814176" "52812157463804376" "53840357462812176" "02812167345836475" "31813267245806475" "18140673458362752" "23824367045816175" "02842367435816175" "41813467325826075" "03181346752482657" "13180346752482657" "02382436754181657" "34283246750181657" "42082436753181657" "42382436750181657" "01815267245836473" "12182506734583647" "52082356734181647" "28026171458634753" "31813627425864075" "01814637543862572" "12182463754386057" "05181365734286247" "15180365734286247" "35283265714186047" "38613175068425724" "13186325724685047" "05286235743681417" "45086425723681317" "53086352742681417" "46820427635813175" "03681315764285247" "04682542763581317" "34683542762581017" "35683405764281217" "45681415762382037" "45680435763281217" "53682352764181047" "52682453764381017" "63181346752482057" "62382436754181057" "64280246753181357" "62082536743581417" "62382536740581417" "64085346735281217" "18120724568347536" "28021714568347536" "28121704568347536" "18130743568427526" "28421714568307536" "18142752468357036" "28521714568347036" "58042752468317136" "58141753468327026" "02812175364835746" "31813275264805746" "01813475364825726" "31813475064825726" "41813475360825726" "18152742658437063" "23825374065841716" "35813175460824726" "25820475364831716" "25823475364801716" "18120726358437654" "28021716358437654" "28121706358437654" "18123726358407654" "28121746058437653" "02812174635843765" "41812472635803765" "03181347562482576" "13180347562482576" "02382437564181576" "34283247560181576" "42082437563181576" "42382437560181576" "18153706358427624" "38053716158427624" "12182507463584376" "01815374635842762" "12182537463584076" "14183547362582076" "13185307465284276" "42582407563181376" "42582437563181076" "53181357460284276" "28426714158637053" "03181367245286475" "13180367245286475" "13184367045286275" "13184367245286075" "12182367530486574" "23028346751418657" "23028356714158647" "01318536724528647" "13158346750428627" "04258246751318637" "51018456724328637" "02862171456834753" "13186327425684075" "14186247235683075" "14186347530682572" "01418634753268257" "14138643750268257" "14158642752368037" "25328635714168047" "15128625740368437" "51018652742368437" "26820374635841715" "46830473625821715" "03681317465284275" "34683247265181075" "03681317562482574" "13168304756248257" "04268243756318157" "15168345736428027" "45068425726318137" "45368435716128027" "53468354716128027" "36283207645181475" "46081417635283275" "46181407635283275" "16180537643582472" "16184507643582372" "16184537643582072" "41618452762358037" "16185347635482072" "05618145763428327" "05628425764318137" "25628405764318137" "45628425760318137" "45628425761318037" "51618453764328027" "61218246753408357" "61318436754208257" "61018356734258247" "61218256740358437" "61318536724528047" "61018536743528427" "61318536740528427" "64258246750318137" "64258246751318037" "48027423568371516" "48527423568371016" "48537403568271216" "58247253468371016" "18127325638475064" "18137503648572462" "25821715364837046" "05823725364817146" "25823705364817146" "35823725064817146" "05841715463827326" "51812752364837046" "52832753064817146" "31813724625847065" "03181375264285746" "13180375264285746" "12182475064385736" "42082475161385736" "24825741615837063" "28527306538471614" "03485374265281716" "25821715640837463" "03582372564181746" "02582473564381716" "42582473560381716" "05283275364181746" "35083275264181746" "35283275064181746" "35083475264281716" "45181475062382736" "18137263258476054" "24820746151837653" "02482374635181765" "12182375630485764" "23028347561418576" "13148357460258276" "41018457263258376" "02852716154837643" "15180375632482764" "23528347560418176" "23528347561418076" "41518437563208276" "51018457364238276" "53048357461218276" "01318637245268475" "01415864725326837" "01518627523468374" "15186375034682742" "12152863754306847" "12152864750346837" "02452864751316837" "25024865741316837" "02682171465384735" "42682473065381715" "13168307526428574" "13168327526408574" "01316834752642857" "14136843752602857" "01316835724625847" "35683275260481714" "03568327526418174" "15126825734603847" "51416854730623827" "36083171652482754" "16180473654382752" "23628347065418175" "04618147365238275" "41618407365238275" "03628327561418574" "23628307561418574" "02362834756141857" "01618257263458374" "23628357161458074" "04161845726325837" "14160845726325837" "34063845716125827" "24628547161538073" "05161845736423827" "15160845736423827" "35063845726421817" "05364835746121827" "26028347635418175" "31613845760425827" "24620845761315837" "16128527640538473" "02632853764151847" "16158407654238273" "24625843765301817" "04635843765121827" "35623825764101847" "52632853764101847" "51610854762342837" "61014856742352837" "61015846735423827" "62352836754101847" "64151846752302837" "38273204568741516" "48372432568701516" "38473254268751016" "58273254368741016" "23827315164875046" "45827425063871316" "53847352462871016" "24827141635873065" "12182703564387546" "04181742562387536" "14180742562387536" "14182742560387536" "15183705364287246" "35083715164287246" "52482754161387036" "03847326425871615" "24827346035871615" "12182703645387465" "34083741615287265" "15180725623487364" "01518473564328726" "41518472562308736" "41518473560328726" "05248275461318736" "12182736053487654" "02382736151487654" "23028374615148765" "23428374615108765" "41018473625328765" "13158372652408764" "24025847365131876" "25028375631418764" "25028475614138763" "31513847562402876" "12152867350436874" "31513867520426874" "46837423625871015" "01316837425624875" "03568371516428724" "14156847052632873" "14156847350632872" "01415684735263287" "15168375036428724" "02452684753161387" "25024685743161387" "16182702645387435" "16182702653487354" "16182732653487054" "01618274265348735" "41618473065328725" "12162857036453874" "23026385741615487" "41016485723625387" "13156384752642087" "03456384752612187" "25026485734613187" "46038473615128725" "24620847365131875" "24623847365101875" "26028571614538743" "01613857362452874" "31613857062452874" "13164385746205287" "01615847365432872" "12162584736543087" "24625847065131873" "02462584736513187" "35263285746101487" "45161485736203287" "26121847635403875" "26324837645101875" "26121857630453874" "31613485764205287" "16125827635403874" "31613584762542087" "16150847652432873" "23625384765141087" "27821415674835063" "03782342567481516" "42782403567381516" "42782415167385036" "35783245267481016" "45783405367281216" "52782151467384036" "37813106475824625" "47820426375831615" "47823426375801615" "04578141563728326" "05378235264718146" "45078415163728326" "45378435161728026" "53478354161728026" "37843161475826025" "27820161574836543" "13178324625748065" "03478324625718165" "01317835264275846" "24027845161375836" "15178425624738063" "24527843560371816" "25327835064171846" "15127825460374836" "15147805462372836" "51017852462374836" "51417854062372836" "14178346035728625" "14178246250738653" "14157843650372862" "01415784365237286" "15127825630473864" "31513784560247286" "35043785461217286" "12152786350437684" "31513786520427684" "14167843056372852" "01516782542637483" "12152678450364738" "25026785141637483" "01514678542362738" "46378431615728025" "24627843065371815" "04637843265271815" "23026378451614758" "02462784516137583" "13146378452602758" "41016478352632758" "25627835063471814" "13156378450624728" "15163785326427084" "15160785246237483" "15164785046237283" "15164785246237083" "01516478534623728" "41516478530623728" "16137803652472854" "16137843650472852" "01613784365247285" "31613784065247285" "31613784265247085" "02642783465317185" "02632783561417584" "31613782562407584" "42062478356131758" "16157803654372842" "25623785360417184" "25623785361417084" "03564378546121728" "26121783645307485" "26024783645317185" "16134783645207285" "16135783625427084" "31613578460524728" "26425784615137083" "16152782653407384" "36053782652417184" "03645378465121728" "26121578634503748" "36253278654101748" "36053478654121728" "46151478652302738" "47382432675181065" "24728141567308536" "04718143567328526" "41718403567328526" "45728425067318136" "51718254267348036" "37083246275481615" "37283246075481615" "47380436275281615" "17183506374582642" "23728351617458064" "12172805364735846" "14173845360725826" "34073845161725826" "24728541617538063" "25728305637418164" "02572834563741816" "34573842562701816" "05171835463724826" "15170835463724826" "52472854360731816" "23728316157408654" "01718246257438653" "12172804635743865" "14172842635703865" "23427384560171586" "41017483562372586" "13147385460275286" "41017485263275386" "03758316157428624" "15174805642732863" "15174835643702862" "41517483562372086" "05247285463171386" "23027386415174685" "02472864151736853" "41017486235273685" "02572861514736843" "13157386052472684" "01517386532472684" "12152748653407368" "34053748651217268" "26728141657438053" "04672842365731815" "02672815164735843" "01316738524627548" "14156748053627328" "15167285236473084" "01516738543627428" "36073824625741815" "36473804625721815" "36473824625701815" "16170834653742852" "16170825624735843" "01617483564372582" "12162748356437058" "04161748356237258" "14160748356237258" "14163748356207258" "34063748256217158" "01617285263475384" "23627385161475084" "04161748526327538" "14160748526327538" "34063748516127528" "12162758046357438" "34563748516127028" "26427384635171085" "31613748560427528" "24620748561317538" "16137583640572482" "01613758364257248" "31613758064257248" "31613758264257048" "16157483654372082" "24625748065317138" "51610758364237248" "51613758364207248" "52642758460317138" "52642758461317038" "26121748653407358" "26121758640357438" "16125728645307438" "06151748653427328" "16150748653427328" "47028423675318165" "27028151673458364" "01713845367425826" "31713845067425826" "01712852467354836" "02742853467351816" "25721815467304836" "25720845367431816" "25723845367401816" "35743805467121826" "51712852467304836" "51710853467324826" "52742853467301816" "31713805627425864" "31713825627405864" "03171384562742586" "13170384562742586" "24720845617135863" "17128526074538643" "27028516174538643" "01712852637453864" "01714853647352862" "12172485364735086" "25721815637403864" "25724805647131863" "35273285460714186" "35273285461714086" "45171485362732086" "45073485362712186" "27428614175368035" "01712862357436854" "12172386435704685" "03171386425724685" "13170386425724685" "01317538642572468" "13157348652472068" "04257248653171368" "51017458634273268" "13147368450276258" "41017468253276358" "23027368514176548" "13157368054276248" "13157368254276048" "15173685340762482" "01517368534276248" "23527368514176048" "15174685043762382" "03672382465714185" "13167308456274258" "04567348536271218" "53067358246271418" "46072482635713185" "03627328564171548" "23627308564171548" "23627348560471518" "04617148562372538" "41617408562372538" "16175384635742082" "24627548063571318" "53647358460271218" "26327438654171058" "46027428653171358" "16127528634573048" "26027538643571418" "26327538640571418" "06257248653471318" "06357438654271218" "46057438653271218" "46357438650271218" "56017158634273248" "56017158642372438" "27320834675141865" "37023824675141865" "37423824675101865" "27121835673405864" "27320835671415864" "37023825671415864" "27121845670435863" "23720384567141586" "42732483567101586" "17130853672452864" "41710485367235286" "41713485367205286" "02732583467514186" "17135843675402862" "27425804675131863" "35713185467204286" "27121864075346835" "17142862475306835" "17140863475326825" "17135863075426824" "03171358642752468" "13170358642752468" "01714586347532682" "12172458634753068" "13174358642752068" "04275248635713168" "53171358642702468" "52072458634713168" "53074358642712168" "17136803475624825" "27326834075641815" "27426804375631815" "17126825074635843" "27026815174635843" "01712682537463584" "01714683547362582" "12172468354736058" "17136853074652842" "12172368534706548" "03171368524726548" "13170368524726548" "13174368540726528" "13175368045726428" "02572368534716148" "17163805376425824" "37063815176425824" "12172608453764358" "01716384537642582" "12172638453764058" "14173648352762058" "01716285247635483" "12172608534763548" "02572638543761418" "26720385637415184" "36713185627425084" "42672438563701518" "42672458063751318" "52672458364731018" "46370438652712158" "36273258640751418" "06274258643751318" "46375438625721018" "56270258634713148" "56171458634723028" "56274258643701318" "37243286475101685" "27121586470534683" "23720358647151468" "17140586472532683" "23724358647151068" "41710458637253268" "24720548637513168" "37513186572402684" "27520486574131683" "27121683475364085" "17130683475264285" "27423684375161085" "17130683572462584" "41710468357236258" "41713468357206258" "27121685073465384" "41710468527326538" "01713568347526428" "31713568047526428" "17145683475362082" "42752468057316138" "31713608457264258" "17136483574602582" "24723648357161058" "04753648357261218" "34753648057261218" "34753648257261018" "03726328457614158" "23726308457614158" "37063285276415184" "37263285076415184" "04716148537623528" "24726348537601518" "41716408537623528" "17162582476534083" "23726358047651418" "17163584376542082" "51716358437624028" "53746358427621018" "36723528647501418" "27121468574306538" "27121568374536048" "27023568374516148" "07141568473526328" "17140568473526328" "07245268475316138" "37513168574206248" "27520468574316138" "27320638574161548" "37023628574161548" "17125628475364038" "07345638475261218" "37523628574161048" "27520648573461318" "57141658472362038" "27326438574601518" "07246258473651318" "47526428573601318" "57016158473624328" "57246258473601318" "37263248576401518" "47360438576121528" "47063458376251218" "57260258476314138" "71014568743526328" "71215268745306438" "73151368754206248" "72052468754316138" "72352638754161048" "72052648753461318" "74151648752362038" "71216258734653048" "74256248753601318" "73064358746251218" "72362538746501418" "74161548736523028")
_

aimisiyou

_ (length (ff 1))
1
_ (length (ff 2))
1
_ (length (ff 3))
2
_ (length (ff 4))
4
_ (length (ff 5))
11
_ (length (ff 6))
38
_ (length (ff 7))
156
_ (length (ff 8))
788
_ (length (ff 9))
4158

dlpg070

aimisiyou 发表于 2020-3-1 17:54
由于之前算法有瑕疵，导致结果有误。经修改，不考虑首尾限制的情况，得到的结果如下：
_ (length (ff 1))
...

不考虑首尾限制的结果:
      1. 2. 6.10. 22. 76. 364.1876. 8316
考虑首尾限制的结果:
      1, 1, 2, 4, 11, 38, 156, 788, 4158
我的结果      
      1, 1, 2, 4, 11, 38, 156, 788, 4158, 23384   
谢谢验证了 n=9 4158 的结果
希望你能验证 n=10  23384
甚至         n=20  ?

王守恩

dlpg070 发表于 2020-3-1 22:19
不考虑首尾限制的结果:
      1. 2. 6.10. 22. 76. 364.1876. 8316
考虑首尾限制的结果:

这些数字串还是互相有联系的。
(ff 01)=000001：000001×1+000000×4=000001
(ff 02)=000001：000001×2+000000×4=000002
(ff 03)=000002：000001×2+000001×4=000006
(ff 04)=000004：000003×2+000001×4=000010
(ff 05)=000011：000011×2+000000×4=000022
(ff 06)=000038：000038×2+000000×4=000076
(ff 07)=000156：000130×2+000026×4=000364
(ff 08)=000788：000638×2+000150×4=001876
(ff 09)=004158：004158×2+000000×4=008316
(ff 10)=023384：023384×2+000000×4=046768
(ff 11)=142312：124520×2+017792×4=320208   
(ff 12)=    ？   ：    ？   ×2+108144×4=   ？
说明。
第1列数=考虑首尾限制的结果=(首不是0)+(首是0)，(首不是0)×2+(首是0)×4=不考虑首尾限制的结果

aimisiyou

如果能找到通项公式，一切都不是问题。
初步想了下，可以看成1到（2n+1）的一个排列，满足1跟n+2之间隔1个数，2跟n+3之间隔2个数……n跟（2n+1）之间隔n个数。
然后用(2n+1)!-不符合要求的，最后再去重。难点在于不符合要求的不是很好求，二是结果去重不像简单除以2那么容易。

dlpg070

aimisiyou 发表于 2020-3-1 18:26
如果不包含0。则
_ (defun ff (n)
(setq num (+ n n 1))

好,这个代码和结果都很重要,找到了王守恩题问题根源,请做下去,看看n能做到多大
这就是 王守恩在1#所说的 三十年前玩的原题  2n数2n个位置(朗福德配对问题,已经经研究的较为成熟)
王守恩在1#所说的改进,实质是放宽条件, 2n数2n+1个位置,多一个空位( "0")确保对任意n都有解,改得好
但是求解数难度较大,相信会成功的

dlpg070

王守恩 发表于 2020-3-2 10:25
这些数字串还是互相有联系的。
0001：0001×1+000000×4=0001
0001：0001×2+000000×4=0002

现在有点信心,把我的计算结果n=1,2,---,10,数很大
放于附件,请参考,审查

aimisiyou

dlpg070 发表于 2020-3-2 12:05
现在有点信心,把我的计算结果n=1,2,---,10,数很大
放于附件,请参考,审查

你把所有对应的个数对出来吧。比如f(1)=1，f(2)=1……
lisp局限性和效率低，如果换其他高级语言，可获得更大n的结果。对于两位数（如10），可取对应的单节字符，如10=":"，11=";"，12="<"，13="="，14=">"。然后统计符合要求的个数。

王守恩

aimisiyou 发表于 2020-3-2 11:10
如果能找到通项公式，一切都不是问题。
初步想了下，可以看成1到（2n+1）的一个排列，满足1跟n+2之间隔1个 ...

找这样的 B 也可以，B=首尾没有0，中间必须有0，不去重。
1, 1, 2, 4, 11, 38, 156, 788,.......=B/2+A014552
通项不太好找，这串数长得太快了，当然我在找。

aimisiyou

跑了近20分钟，得出无首尾比较规定情况下f (10）
_ (length (ff 9))
8316
_ (length (ff 10))
46768
_$