挑战

给定项目数n，在非空的已排序列表中，输出索引， 如果按字典顺序对排列进行排列，则i(n)其
“ 从前到后排列 ”
将位于所有排列的列表中。

结果可能是基于0或1的，只说哪个（即i不是n）。

从后到前的排列

...是通过重复获取向前排序的列表（从左至右）的后（右）然后前（左），直到所有项都移至新列表，来构建项列表的结果，就像这样：

Input being consumed     Output being built
----------------------+----------------------
[1,2,3,4,5,6,7]       |   []
[1,2,3,4,5,6]         |   [7]
  [2,3,4,5,6]         |   [7,1]
  [2,3,4,5]           |   [7,1,6]
    [3,4,5]           |   [7,1,6,2]
    [3,4]             |   [7,1,6,2,5]
      [4]             |   [7,1,6,2,5,3]
       []             |   [7,1,6,2,5,3,4]
----------------------+----------------------
                Result:   [7,1,6,2,5,3,4]

排列指数

如果n为7（如上面的从前到后的示例）7! = 5040，则（不同的）项目可能存在排列。

所有这些排列的按字典顺序排序的列表中的第一个（如果愿意，则为零）项目[1,2,3,4,5,6,7]本身就是它自己。
第二项是[1,2,3,4,5,7,6]。
倒数第二个项目将是[7,6,5,4,3,1,2]。
最后一个项目是[7,6,5,4,3,2,1]。

列表中的某处是[7,1,6,2,5,3,4]-从后到前的排列。
实际上，它位于索引4421（或4420，从0开始）处。

（从1开始）系列的i(n)开头100个术语为n=1：

[1, 2, 5, 20, 101, 620, 4421, 35900, 326981, 3301820, 36614981, 442386620, 5784634181, 81393657020, 1226280710981, 19696509177020, 335990918918981, 6066382786809020, 115578717622022981, 2317323290554617020, 48773618881154822981, 1075227108896452857020, 24776789629988523782981, 595671612103250915577020, 14915538431227735068422981, 388375922695377900515577020, 10500493527722974260252422981, 294387851083990886241251577020, 8547374142655711068302364422981, 256705485669535347568006115577020, 7966133168508387470157556764422981, 255164703765185142697060455395577020, 8428152915046701352821133945884422981, 286804646124557439494797475697635577020, 10046343320261587490171853861825564422981, 361946983469639629977827594289009635577020, 13401806107756705416338151987291892764422981, 509620811358844406343669072112782398435577020, 19888261269838598952296612667790114958364422981, 796027021978059135393314656928325779313635577020, 32656499591185747972776747396512425885838364422981, 1372349618161694150570365858847999144050545635577020, 59042913445212141486784766209665998363213966364422981, 2599228661343236626556841044804949891956424561635577020, 117022992204136957935406320450852765172427309198364422981, 5385599167607951991914899108349402127789224443761635577020, 253237642343560228651049456045262577841408407945358364422981, 12160677950192512442211239591328112460680077946732401635577020, 596121186084075048430040923729967264426872753432477838364422981, 29817972015629302995182567242334801579950768815528034161635577020, 1521300781271752977229060449226968409483308951201458077838364422981, 79136874389672125594431576407176798565806196489681819746161635577020, 4195746409670353438703582176982222851124537591877131904925838364422981, 226647950929571027033389160506045358232154026979930809227362161635577020, 12469755402728704898931711687060471601348167024469505953048477838364422981, 698528832402134746955113935776664478135149811856698952734398562161635577020, 39828390672475082008725487969655657656845234984369903192450082717838364422981, 2310732940610403489820749422545419026172017083196773021228249831522161635577020, 136372385605079432248118270297843987319730859689490659519593045108637838364422981, 8184614727136310712028222912925520393434441746671755292929684651300962161635577020, 499395599150088488088828589263699706832570087241364247806476254829684637838364422981, 30970577661237849037564293765687064381179710710016867944356691992991422562161635577020, 1951637737743202215078582414596211073163593979517251760161922907619738331037838364422981, 124935294448140961888354806920565269729701922195027940438639971467594965899362161635577020, 8122715297634329704834815499864930982456556629150409552483483162921360809076637838364422981, 536222223779808734298894424747977821661836507759648464980376643706749720339339362161635577020, 35934888694408876553950964671857486605505798806289876128721251856561212716604532637838364422981, 2444100653742421723047039453897314094441893402549077796242989486161660232995578763362161635577020, 168678351774398889649421299427375524997828651490971291597405051437095619521145068660637838364422981, 11809893318195492906423362422261723211461109491055454565957957813190913963268700251019362161635577020, 838668695249666824614744281817664287077123498629740781320472805575397766414810317446260637838364422981, 60395789681636420036909326103457008453700968286067588202502542158402987220806878956757899362161635577020, 4409719671831047920854347812021594101623099731996837427616577550212019116846376438060145780637838364422981, 326378824480107593305098680409232188044060152088938133742995349285199216584125189021190726539362161635577020, 24482761986915290498641378436184801472882183734481184704052899163370643460988742220422624697460637838364422981, 1861011939679134964489290882424961756757512351644848150968435083798473400034549180897307347526539362161635577020, 143322080088606734669581493203883323226982866872563510695813139604263517949121870899167900513721460637838364422981, 11180959098117691096787939665528162905504766712615688479353149686064571807285078895345918312663622539362161635577020, 883437253980179837588356231874303489164303450066956218734514913541773418886216781638015892528346553460637838364422981, 70686019792283622457223177491312228676420353892298796358374930144685265836593932061030928974752467526539362161635577020, 5726440000955084363422511054086796876735936890839327162387490119571704913857298124195153605274993472953460637838364422981, 469637893700329090478715695935318149767077357177154001454773443957172289821041850488811978203204173646406539362161635577020, 38985601803506257421418755484185292421669426050466292273769584084412579273175587484390779961900566697260473460637838364422981, 3275254532761847009577968823645945995578996860191583194845076448298646552018541276645494943006816186458917446539362161635577020, 278435156905293180685369975402415213484477637470382623210256836304261379607777392174394791509334107831816205753460637838364422981, 23948660226767439201080153228038844501800392914958999127628507660415900870134672884615069843391985357739844389446539362161635577020, 2083808638152760278012520365471350750727983345146397213195344003554238214857458501196068353393022808146994627392953460637838364422981, 183398833619245678836784325280074933629492985604252949471226236983335323969170740817904072891411479020269638889458246539362161635577020, 16324556327289215402380134937173544376210173250892288905442294470849835710409338998582008497896189183708810744110298553460637838364422981, 1469391408154472281907142598683652193509359788033796478036774569234135557383656537547410122872987870461908423725867813446539362161635577020, 133730761359685823973259426160811489954077506688872881313704960027919535214176338228137873831877461557289259913042140378553460637838364422981, 12304683293281621431502064899712741587623914209186541475526534622910218175769343180214908250005163885795818227069614613285446539362161635577020, 1144467823788359953327703097406527694627129315367226993710615746590336588945697972034988381266839681418043178062317463477466553460637838364422981, 107592147841885948074037582159380073309559674264815645313786758687454863280472229658194120833316575777142822473140067877053221446539362161635577020, 10222386340397173314525664517235347022088186665852557223898463812546839124314230895213571254552107892786139414391086539473362138553460637838364422981, 981455548530552515895045737024658454136095461985415238220477591025945383684777269092475904782448641089288955324574667766166512421446539362161635577020, 95211304133951567337433380212539040258207718457187560919883999728307800228797098229713403270806624010171995234355103499880901319898553460637838364422981, 9331679144749296178288752362844703433551486045621764102574354777566399269794426700653262755936922495813433855354253356929531746247461446539362161635577020, 923930475294692230638703636199822301473608196598194450583355284174609600662504729388761377005628260366723545352917984225582320362921178553460637838364422981, 92402284968649460451060535220066878189242360067783427018009608611042990392567410879552702599150890025886974375474305774025602890553942821446539362161635577020

（i(0)=i(1)=1，但挑战本身仅处理非空列表）

在发布时，此序列未出现在OEIS中。

输出只需要理论上就可以工作（例如，不必担心整数溢出或资源用完）。

这是代码高尔夫球，因此最短的答案以字节为单位。

但是，请不要让代码高尔夫球的语言迷惑您-好的解决方案也应该受到欢迎！

Haskell，32个字节

f 1=1
f n=product[1..n]+1-f(n-1)

在线尝试！

使用关系f(n-1) + f(n) = n! + 1。序列的相邻成员加到阶乘加一：

1,   2,   5,   20,   101,   620,   4421, ...
  3     7    25    121    721   5041  ...
 2!+1  3!+1  4!+1  5!+1   6!+1  7!+1 

果冻，6个字节

R!ḅ-_Ḃ

从0开始。在线尝试！

受@Neil的ES6答案的启发。

说明

R!ḅ-_Ḃ
R       Create the range [1..N].
 !      Take the factorial of each.
  ḅ-    Convert from base -1; that is, sum, but alternate between adding and subtracting.
    _Ḃ  Subtract N%2.

但是如何？

我在ES6答案中解释了一种用于计算每个数字的相关技术。公式是这样的：

(n-1)(n-1)! + (n-3)(n-3)! + (n-5)(n-5)! + ...

在阅读@Neil的ES6答案时，一个实现感动了我。可以像下面这样简化该公式：

(n-1)(n-1)!        + (n-3)(n-3)!            + (n-5)(n-5)!            + ...
(n(n-1!) - (n-1)!) + ((n-2)(n-3!) - (n-3)!) + ((n-4)(n-5)! - (n-5)!) + ...
(n!      - (n-1)!) + ((n-2)!      - (n-3)!) + ((n-4)!      - (n-5)!) + ...
n! - (n-1)! + (n-2)! - (n-3)! + (n-4)! - (n-5)! + ...

果冻代码将R!ḅ-计算此公式。但是，每个的奇数值在末尾n都会有一个额外+ 0!的字符，我们通过减去来解决n%2。

JavaScript（ES6），38个字节

f=(n,x=n%2,y=1)=>n-x&&f(n,++x,y*=-x)+y

0索引。（没有解释，因为我实际上不知道为什么会起作用，抱歉。）

JavaScript（ES6），44个字节

f=(x,n=0,g=1)=>x-n&&(x-n&1)*g*n+f(x,++n,g*n)

从0开始。这利用了以下事实：数字可以按以下模式表示为阶乘和。

       1   2   6  24 120 720
   0:                       
   1:  1
   4:      2
  19:  1       3
 100:      2       4
 619:  1       3       5
4420:      2       4       6

为什么？排列可以很好地表示为阶乘：将第n ^个项目从其余列表中取出对应于该位置的n位。我们在获取最后一项（最高位数）和第一项（零）之间进行交替；因此，在阶乘基础上，这些数字可以表示为：

0
10
200
3010
40200
503010
6040200

等等。

MATL，17个字节

:t"&0)P]vG:Y@!=Af

输出为1索引。

在线尝试！

说明

该代码适用于定义：构建从前到后的排列，生成所有排列，将前者与所有后者进行比较，并输出匹配项的索引。

:        % Input n implicitly. Push [1 2 ... n]
t        % Duplicate
"        % For each: do the following n times
  &0)    %   Push the last element and then the rest of the array
  P      %   Reverse
]        % End
v        % Concatenate the whole stack vertically. This produces into a column vector
         % with the back-to-front permutation
G:       % Push [1 2 ... n] again
Y@!      % Permutations of [1 2 ... n]. Gives a matrix. Each column is a permutation
=        % Test for equality, element-wise with broadcast
A        % All: true for columns that have all entries equal to true. Gives a row vector
f        % Find: index of non-zero value. Implicitly display

果冻，9字节

RU;¥/ỤUŒ¿

在线尝试！

was，我正在尝试FGITW。原来@Dennis首先发布，但这要短一些。

说明

RU;¥/ỤUŒ¿
R           List of numbers from 1 to {the input}
   ¥/       Left-fold the list by
 U;         prepending the reverse of the list to the next element
     Ụ      Invert permutation
      U     Reverse the list
       Œ¿   Find index of permutation

具有Œ¿内置功能在这里非常方便，让我们将排列转换为其索引，因此其他7个字节负责构造从后到前的排列。

我们这样做的方法是首先通过以下模式构造不同的排列：

1
1 2
2 1 3
3 1 2 4
4 2 1 3 5
5 3 1 2 4 6
6 4 2 1 3 5 7

每次，我们都会反转到目前为止的列表，然后追加下一个整数。这不会产生从后到前的排列，但显然是相关的。

我们试图获得的排列是7 1 6 2 5 3 4。这有什么关系？好吧，我们在排列的第7位的元素是7；位于第一位置的元素是6；位于第六位置的元素是5；位于第二位置的元素是4，依此类推。换句话说，这是我们拥有的排列的倒数（元素的顺序相反）。这样，在reduce之后，我们可以将的排列反转，Ụ并用的结果反转，U以获得我们想要的从后到前排列。

可能有节省的地方，因为它是匆忙编写的，并且感觉至少有一些重新布置事情的潜力。不过，我不确定是否可以保存整个字节。

果冻，10 8字节

RṚżRFQŒ¿

感谢@ ais523打高尔夫球2个字节，并大大提高了速度！

在线尝试！

怎么运行的

RṚżRFQŒ¿  Main link. Argument: n

R         Range; yield [1, ..., n].
 Ṛ        Reverse; yield [n, ..., 1].
   R      Range; yield [1, ..., n] again.
  ż       Zip; yield [[n, 1], ..., [1, n]].
    F     Flatten.
     Q    Unique; deduplicate the results.
      Œ¿  Compute the permutation index of [n, 1, n-1, 2, ...].

PHP，86字节

for($i=$argv[1];$i>0;$i--)$o+=gmp_strval(gmp_fact($i))*($i%2==$argv[1]%2?1:-1);echo$o;

使用GNU多精度扩展。

该功能利用了i(n)等于n! - (n-1)! + (n-2)! - (n-3)! etc

分解

for($i=$argv[1];$i>0;$i--) {        // Simple decreasing for loop (added { for readability)
    $o+=                            //  increment output with
        gmp_strval(gmp_fact($i))    //      $i!
    * ($i%2 == $argv[1]%2 ? 1 : -1) //      multiplied by -1 if ($i is odd when the input is even) or (if $i is even when the input is odd), else by 1
    ;
}
echo $o;                            // echoes output

批处理，79字节

@set/ax=%1%%2-1,y=z=1
@for /l %%i in (-%1,1,%x%)do @set/az+=y*=x-=1
@echo %z%

0索引。

Pyth，12个字节

x.pQ<Q.i_UQU

0索引。

说明

x.pQ<Q.i_UQU
      .i       Interleave
        _UQUQ  Reversed range and range
    <Q         Take first n
x              Find the index
 .pQ           In the list of permutations