Back-To-Front Permutation Index

Question

The Challenge

Given the number of items, n, in a non-empty, sorted list output the index, i(n), at which its
"Back-To-Front Permutation"
would reside in a list of all permutations if said permutations were sorted lexicographically.

Results may be 0 or 1-based, just say which (that is i, not n).

The Back-To-Front Permutation

...is the result of building a list of items by repeatedly taking the back (right) then front (left) of a forward sorted (left-to-right) list until all items have been moved to the new list, like so:

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]

The Permutation Index

If n is 7 (as the above Back-To-Front example) there are 7! = 5040 possible permutations of the (distinct) items.

The first (or zeroth if you prefer) item in the lexicographically sorted list of all those permutations would be [1,2,3,4,5,6,7] itself.
The second item would be [1,2,3,4,5,7,6].
The penultimate item would be [7,6,5,4,3,1,2].
The final item would be [7,6,5,4,3,2,1].

Somewhere in the list is [7,1,6,2,5,3,4] - the Back-To-Front permutation.
In fact it resides at index 4421 (or 4420, 0-based).

The first 100 terms of the (1-based) series of i(n) stating with n=1 are:

[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, but the challenge itself only deals with non-empty lists)

At the time of posting this sequence did not appear in the OEIS.

Output must only need to work in theory (don't worry about overflowing integers, or running out of resources for example).

This is code-golf, so the shortest answer in bytes wins.

However, don't let code-golf languages dissuade you - good solutions should get upvotes too!

• Jonathan Allan: Hope all is OK with this - it was in the sandbox for over a month with no feedback.
• xnor: Related -- Permutation numbering
• xnor: These are alternating factorials with every other entry increased by 1.
• Jonathan Allan: @xnor yes they are - the front-to-back permutation has the previous index to the back-to-front one.

Answer 1

Haskell, 32 bytes

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

Try it online!

Uses the relationship f(n-1) + f(n) = n! + 1. Adjacent members of the sequences add to factorials plus one:

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 

Answer 2

Jelly, 6 bytes

R!ḅ-_Ḃ

0-based. Try it online!

Heavily inspired by @Neil's ES6 answer.

Explanation

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.

But how?

I explain in my ES6 answer a related technique for calculating each number. The formula is this:

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

A realization struck me while reading @Neil's ES6 answer. This formula can be simplified like so:

(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)! + ...

The Jelly code R!ḅ- calculates this formula. However, each odd value of n will have an extra + 0! at the end, which we take care of by subtracting n%2.

• Jonathan Allan: Congratulations you found my solution! (do note that it is 0-based).
• Dennis: Figures that you'd use ḅ- sooner or later... :P Nice work!
• ETHproductions: @JonathanAllan I knew as soon as I saw that you had posted the challenge that there would be a sneaky Jelly answer. It took quite a while for anyone to find it though. Great challenge :-)

Answer 3

JavaScript (ES6), 38 bytes

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

0-indexed. (No explanation because I don't actually know why it works, sorry.)

• ETHproductions: Oh, that's genius. My answer takes (n-1)*(n-1)! + (n-3)*(n-3)! + (n-5)*(n-5)! + ..., which is equivalent to (n! - (n-1)!) + ((n+2)! - (n-3)!) + ((n-4)! - (n-5)!) + ... which is what your answer does.

Answer 4

JavaScript (ES6), 44 bytes

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

0-based. This takes advantage of the fact that the numbers can be represented as sums of factorials in the following pattern:

 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

Why? The permutations can be represented nicely in the factorial base: taking the n^th item out of the remaining list corresponds to a digit of n at that position. We're alternating between taking the last item (highest digit) and the first item (zero); therefore, in the factorial base, these numbers can be represented as:

0
10
200
3010
40200
503010
6040200

and so on.

Answer 5

MATL, 17 bytes

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

Output is 1-indexed.

Try it online!

Explanation

The code appplies the definition: builds the back-to-front permutation, generates all permutations, compares the former with all of the latter, and outputs the index of the matching.

: % 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

Answer 6

Jelly, 9 bytes

RU;\/ỤUŒ¿

Try it online!

Huh, I was trying to FGITW this. Turns out @Dennis posted first, but this is shorter.

Explanation

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

Having Œ¿ as a built-in is fairly handy here letting us convert a permutation to its index, so the other 7 bytes are responsible for constructing the back-to-front permutation.

The way we do this is first to construct a different permutation, via the following pattern:

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

Each time, we're reversing the list we have so far, then appending the next integer. That doesn't produce the back-to-front permutation, but it's clearly related.

The permutation we're trying to get is 7 1 6 2 5 3 4. How is this related? Well, the element in the 7th position of the permutation we have is a 7; the element in the 1st position is a 6; the element in the 6th position is a 5; the element in the 2nd position is a 4, and so on. In other words, it's the inverse of the permutation we have (with the elements in reverse order). As such, after the reduce, we can invert the permutation with Ụ and reverse the result with U to get the back-to-front permutation we want.

It's possible that there are savings here, because it was written in a hurry and feels like it has at least some potential to rearrange things. I'm not sure it's possible to save a whole byte, though.

Answer 7

Jelly, (削除) 10 (削除ここまで) 8 bytes

RṚżRFQŒ¿

Thanks to @ais523 for golfing off 2 bytes and a tremendous speed-up!

Try it online!

How it works

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, ...].

• user62131: Looks like you missed the Œ¿ builtin. Your method for constructing the list is a byte shorter than mine, so if you can replace i@Œ! with that, then you should be able to get this down to 8 bytes, beating my answer.
• Dennis: Completely forgot that was a thing. Thanks!

Answer 8

PHP, 86 bytes

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

Uses the GNU Multiple Precision extension.

This function takes advantage of the fact that i(n) is equal to n! - (n-1)! + (n-2)! - (n-3)! etc

Breakdown

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

Answer 9

Batch, 79 bytes

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

0-indexed.

Answer 10

Pyth, 12 bytes

x.pQ<Q.i_UQU

0-indexed.

Explanation

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