使用者:SomeyaMako/MahjongStat

本頁面用來記錄一些立直麻將的數據。

四麻

不同的牌山總數

$\dfrac{136!}{4!^{34}}\approx4.327\times10^{185}$

精確到個位
$\dfrac{136!}{4!^{34}}=432698391742847263810138305033742131882589880014027980942911799590972036509850338890918625906160332246611492154251485203537706583203911528927630102575646500000000000000000000000000000000$

天和時與至少一個其他役滿複合的概率

牌型對應牌山總數約為：$3.534\times10^{176}$

精確到個位
牌型對應牌山總數為：$353406014972536297631832052343631069484377902921518652014708836225337434526124304670483216471795932483372971312613922015609935308620490666741769208875000000000000000000000000000$

故天和時與至少一個其他役滿複合的概率為（精確值）$\dfrac{578572123}{708384171528036000}$，約$\dfrac{1}{1.224366234333824\times10^9}$或$8.167490836956139\times10^{-10}$

牌型對應的牌山種類數如下。SomeyaMako稱兩個牌山的排列是不同的，若且唯若至少存在一個位置使得該位置上的對應牌不同。

國士無雙（十三面）

$13\times\dfrac{14!}{2!\times1!^{12}}\times\dfrac{122!}{4!^{21}\times3!^{12}\times2!}\approx1.332\times10^{176}$。

精確到個位
$13\times\dfrac{14!}{2!\times1!^{12}}\times\dfrac{122!}{4!^{21}\times3!^{12}\times2!}=133223144654716075789437840980460202369023260282862856058197724801190327234446424115018370341681701580244545808433039255539040779957841558924951552000000000000000000000000000000$

四暗刻（單騎）

$5\times\dbinom{34}{5}\times\dfrac{14!}{3!^4\times2!}\times\dfrac{122!}{4!^{29}\times2!\times1!^4}\approx2.176\times10^{176}$

精確到個位
$5\times\dbinom{34}{5}\times\dfrac{14!}{3!^4\times2!}\times\dfrac{122!}{4!^{29}\times2!\times1!^4}=217555937189205911400814443029896275860096484335493157462075254659095269393616451489660126071607053051948701937580658963067106577171614196896017920000000000000000000000000000000$

三元刻子+順子+雀頭

$28\times21\times\dfrac{14!}{3!^3\times2!\times1!^3}\times\dfrac{122!}{4!^{27}\times3!^3\times2!\times1!^3}+3\times21\times\dfrac{14!}{3!^4\times1!^2}\times\dfrac{122!}{4!^{27}\times3!^2\times1!^4}\approx1.497\times10^{174}$

精確到個位
$28\times21\times\dfrac{14!}{3!^3\times2!\times1!^3}\times\dfrac{122!}{4!^{27}\times3!^3\times2!\times1!^3}+3\times21\times\dfrac{14!}{3!^4\times1!^2}\times\dfrac{122!}{4!^{27}\times3!^2\times1!^4}=1497409331352305652042866357925282285116068608760768842976850553889581033086992653861900815983633157973564465639031963537412032504596758056952064000000000000000000000000000000$

三風刻子+順子+風雀頭

$21\times4\times\dfrac{14!}{3!^3\times2!\times1!^3}\times\dfrac{122!}{4!^{27}\times3!^3\times2!\times1!^3}\approx2.102\times10^{173}$

精確到個位
$21\times4\times\dfrac{14!}{3!^3\times2!\times1!^3}\times\dfrac{122!}{4!^{27}\times3!^3\times2!\times1!^3}=210162713172253424848121594094776461068921910001511416558154463703800846749051600542021167155597636206816065352846591373671864211171474815010816000000000000000000000000000000$

大七星

$\dfrac{14!}{2!^7}\times\dfrac{122!}{4!^{27}\times2!^7}\approx2.850\times10^{172}$

精確到個位
$\dfrac{14!}{2!^7}\times\dfrac{122!}{4!^{27}\times2!^7}=28498626841996418659650417056378727700752245608463880705151526049120315714296617707428093537282491963312669576306764343639655693813989051365864000000000000000000000000000000$

234s+三組綠刻子+綠雀頭

總數約$3.320\times10^{171}$

精確到個位
總數為$3319831037513943231635802827105160859344745689343304004468847395339697271673570757027428163047203871326396586660392322695506016286844789398436875000000000000000000000000000$

公式

$3\times\dfrac{14!}{3!^4\times1!^2}\times\dfrac{122!}{4!^{28}\times3!^2\times1!^4}+3\times\dbinom{3}{3}\times3\times\dfrac{14!}{4!\times3!^2\times2!\times1!^2}\times\dfrac{122!}{4!^{28}\times3!^2\times2!\times1!^2}+3\times2\times\dbinom{3}{2}\times\dfrac{14!}{4!\times3!^3\times1!}\times\dfrac{122!}{4!^{29}\times3!\times1!^3}+\dbinom{3}{2}\times2\times\dbinom{3}{2}\times\dfrac{14!}{4!^2\times3!\times2!\times1!}\times\dfrac{122!}{4!^{29}\times3!\times2!\times1!}+3\times\dbinom{3}{3}\times\dbinom{3}{1}\times\dfrac{14!}{4!^2\times3!^2}\times\dfrac{122!}{4!^{30}\times1!^2}+\dbinom{3}{1}\times\dfrac{14!}{4!^3\times2!}\times\dfrac{122!}{4!^{30}\times2!}$

細分如下

刻子中有零組為🀑🀒🀓：$3\times\dfrac{14!}{3!^4\times1!^2}\times\dfrac{122!}{4!^{28}\times3!^2\times1!^4}\approx1.251\times10^{171}$ 刻子中有一組為🀑🀒🀓： 雀頭不為🀑🀒🀓：$3\times\dbinom{3}{3}\times3\times\dfrac{14!}{4!\times3!^2\times2!\times1!^2}\times\dfrac{122!}{4!^{28}\times3!^2\times2!\times1!^2}\approx1.407\times10^{171}$ 雀頭為🀑🀒🀓：$3\times2\times\dbinom{3}{2}\times\dfrac{14!}{4!\times3!^3\times1!}\times\dfrac{122!}{4!^{29}\times3!\times1!^3}\approx4.691\times10^{170}$ 刻子中有兩組為🀑🀒🀓： 雀頭不為🀑🀒🀓：$\dbinom{3}{2}\times2\times\dbinom{3}{2}\times\dfrac{14!}{4!^2\times3!\times2!\times1!}\times\dfrac{122!}{4!^{29}\times3!\times2!\times1!}\approx1.759\times10^{170}$ 雀頭為🀑🀒🀓：$3\times\dbinom{3}{3}\times\dbinom{3}{1}\times\dfrac{14!}{4!^2\times3!^2}\times\dfrac{122!}{4!^{30}\times1!^2}\approx1.466\times10^{169}$ 刻子中有三組為🀑🀒🀓：$\dbinom{3}{1}\times\dfrac{14!}{4!^3\times2!}\times\dfrac{122!}{4!^{30}\times2!}\approx1.832\times10^{168}$ 精確到個位 刻子中有零組為🀑🀒🀓：$3\times\dfrac{14!}{3!^4\times1!^2}\times\dfrac{122!}{4!^{28}\times3!^2\times1!^4}=1250968530787222766953104726754621792076916130961377479512824188713100278268164288940602185449985929802476579481229710557570620304592111994112000000000000000000000000000000$ 刻子中有一組為🀑🀒🀓： 雀頭不為🀑🀒🀓：$3\times\dbinom{3}{3}\times3\times\dfrac{14!}{4!\times3!^2\times2!\times1!^2}\times\dfrac{122!}{4!^{28}\times3!^2\times2!\times1!^2}=1407339597135625612822242817598949516086530647331549664451927212302237813051684825058177458631234171027786151916383424377266947842666125993376000000000000000000000000000000$ 雀頭為🀑🀒🀓：$3\times2\times\dbinom{3}{2}\times\dfrac{14!}{4!\times3!^3\times1!}\times\dfrac{122!}{4!^{29}\times3!\times1!^3}=469113199045208537607414272532983172028843549110516554817309070767412604350561608352725819543744723675928717305461141459088982614222041997792000000000000000000000000000000$ 刻子中有兩組為🀑🀒🀓： 雀頭不為🀑🀒🀓：$\dbinom{3}{2}\times2\times\dbinom{3}{2}\times\dfrac{14!}{4!^2\times3!\times2!\times1!}\times\dfrac{122!}{4!^{29}\times3!\times2!\times1!}=175917449641953201602780352199868689510816330916443708056490901537779726631460603132272182328904271378473268989547928047158368480333265749172000000000000000000000000000000$ 雀頭為🀑🀒🀓：$3\times\dbinom{3}{3}\times\dbinom{3}{1}\times\dfrac{14!}{4!^2\times3!^2}\times\dfrac{122!}{4!^{30}\times1!^2}=14659787470162766800231696016655724125901360909703642338040908461481643885955050261022681860742022614872772415795660670596530706694438812431000000000000000000000000000000$ 刻子中有三組為🀑🀒🀓：$\dbinom{3}{1}\times\dfrac{14!}{4!^3\times2!}\times\dfrac{122!}{4!^{30}\times2!}=1832473433770345850028962002081965515737670113712955292255113557685205485744381282627835232592752826859096551974457583824566338336804851553875000000000000000000000000000$

223344s+兩組綠刻子+綠雀頭

$3\times\dbinom{3}{2}\times\dfrac{14!}{4!\times3!^2\times2!^2}\times\dfrac{122!}{4!^{29}\times2!^2\times1!^2}+3\times\dbinom{3}{3}\times\dfrac{14!}{3!^2\times2!^4}\times\dfrac{122!}{4!^{28}\times2!^4\times1!^2}\approx6.861\times10^{171}$

精確到個位
$3\times\dbinom{3}{2}\times\dfrac{14!}{4!\times3!^2\times2!^2}\times\dfrac{122!}{4!^{29}\times2!^2\times1!^2}+3\times\dbinom{3}{3}\times\dfrac{14!}{3!^2\times2!^4}\times\dfrac{122!}{4!^{28}\times2!^4\times1!^2}=6860780536036174862508433735794878890921836905741304614203145159973409338626963522158615110827266583760457490592369193839176370732997364217708000000000000000000000000000000$

（純正）九蓮寶燈

$3\times\left(2\times\dfrac{14!}{4!\times3!\times1!^7}\times\dfrac{122!}{4!^{25}\times3!^7\times1!}+7\times\dfrac{14!}{3!^2\times2!\times1!^6}\times\dfrac{122!}{4!^{25}\times3!^6\times2!\times1!^2}\right)\approx8.807\times10^{173}$

精確到個位
$3\times\left(2\times\dfrac{14!}{4!\times3!\times1!^7}\times\dfrac{122!}{4!^{25}\times3!^7\times1!}+7\times\dfrac{14!}{3!^2\times2!\times1!^6}\times\dfrac{122!}{4!^{25}\times3!^6\times2!\times1!^2}\right)=880681845674204827934985727635253741622148956196809745577028228854022595900787659414183938556790094580943511954785716232529716694432846843854848000000000000000000000000000000$

天鳳規則天和四倍役滿概率

待補

雀魂規則天和六倍役滿概率

天和六倍役滿牌山數：$3\times\dfrac{14!}{3!^4\times2!}\times\dfrac{122!}{4!^{29}\times2!\times1!^4}\approx4.691\times10^{170}$

精確到個位
$3\times\dfrac{14!}{3!^4\times2!}\times\dfrac{122!}{4!^{29}\times2!\times1!^4}=469113199045208537607414272532983172028843549110516554817309070767412604350561608352725819543744723675928717305461141459088982614222041997792000000000000000000000000000000$

故天和六倍役滿概率（精確值）：$P=\dfrac{3\times\dfrac{14!}{3!^4\times2!}\times\dfrac{122!}{4!^{29}\times2!\times1!^4}}{\dfrac{136!}{4!^{34}}}=\dfrac{8}{7379001786750375}=\dfrac{1}{922375223343796.875}\approx1.084157\times10^{-15}$，略高於$\dfrac{1}{10^{15}}$。

三麻

待補