# Non-continuous subsequences

(Redirected from Non Continuous Subsequences)
Non-continuous subsequences
You are encouraged to solve this task according to the task description, using any language you may know.

Consider some sequence of elements. (It differs from a mere set of elements by having an ordering among members.)

A subsequence contains some subset of the elements of this sequence, in the same order.

A continuous subsequence is one in which no elements are missing between the first and last elements of the subsequence.

Note: Subsequences are defined structurally, not by their contents. So a sequence a,b,c,d will always have the same subsequences and continuous subsequences, no matter which values are substituted; it may even be the same value.

Task: Find all non-continuous subsequences for a given sequence.

Example: For the sequence   1,2,3,4,   there are five non-continuous subsequences, namely:

•   1,3
•   1,4
•   2,4
•   1,3,4
•   1,2,4

Goal: There are different ways to calculate those subsequences. Demonstrate algorithm(s) that are natural for the language.

### Recursive

`with Ada.Text_IO;  use Ada.Text_IO; procedure Test_Non_Continuous is   type Sequence is array (Positive range <>) of Integer;   procedure Put_NCS             (  Tail : Sequence;                -- To generate subsequences of                Head : Sequence := (1..0 => 1); -- Already generated                Contiguous : Boolean := True    -- It is still continuous             )  is   begin      if not Contiguous and then Head'Length > 1 then         for I in Head'Range loop            Put (Integer'Image (Head (I)));         end loop;         New_Line;      end if;      if Tail'Length /= 0 then          declare            New_Head : Sequence (Head'First..Head'Last + 1);         begin            New_Head (Head'Range) := Head;            for I in Tail'Range loop               New_Head (New_Head'Last) := Tail (I);               Put_NCS               (  Tail => Tail (I + 1..Tail'Last),                  Head => New_Head,                  Contiguous => Contiguous and then (I = Tail'First or else Head'Length = 0)               );            end loop;         end;      end if;   end Put_NCS;begin   Put_NCS ((1,2,3));     New_Line;   Put_NCS ((1,2,3,4));   New_Line;   Put_NCS ((1,2,3,4,5)); New_Line;end Test_Non_Continuous;`
Output:
``` 1 3

1 2 4
1 3
1 3 4
1 4
2 4

1 2 3 5
1 2 4
1 2 4 5
1 2 5
1 3
1 3 4
1 3 4 5
1 3 5
1 4
1 4 5
1 5
2 3 5
2 4
2 4 5
2 5
3 5```

## ALGOL 68

### Recursive

- note: This specimen retains the original Ada coding style.
Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny
Works with: ELLA ALGOL 68 version Any (with appropriate job cards) - tested with release 1.8-8d
`PROC test non continuous = VOID: BEGIN   MODE SEQMODE = CHAR;   MODE SEQ = [1:0]SEQMODE;   MODE YIELDSEQ = PROC(SEQ)VOID;    PROC gen ncs =             (  SEQ tail,       # To generate subsequences of #                SEQ head,       #           Already generated #                BOOL contiguous,#      It is still continuous #                YIELDSEQ yield             )  VOID:   BEGIN      IF NOT contiguous ANDTH UPB head > 1 THEN         yield (head)      FI;      IF UPB tail /= 0 THEN             [UPB head+1]SEQMODE new head;            new head [:UPB head] := head;            FOR i TO UPB tail DO               new head [UPB new head] := tail [i];               gen ncs               (  tail[i + 1:UPB tail],                  new head,                  contiguous ANDTH (i = LWB tail OREL UPB head = 0),                  yield               )            OD      FI   END # put ncs #;  # FOR SEQ seq IN # gen ncs(("a","e","i","o","u"), (), TRUE, # ) DO ( # ##   (SEQ seq)VOID:      print((seq, new line)) # OD # )END; test non continuous`
Output:
```aeiu
aeo
aeou
aeu
ai
aio
aiou
aiu
ao
aou
au
eiu
eo
eou
eu
iu
```

### Iterative

Translation of: C
- note: This specimen retains the original C coding style.
Works with: ALGOL 68 version Revision 1 - no extensions to language used
Works with: ALGOL 68G version Any - tested with release 1.18.0-9h.tiny
Works with: ELLA ALGOL 68 version Any (with appropriate job cards) - tested with release 1.8-8d

Note: This specimen can only handle sequences of length less than bits width of bits.

`MODE SEQMODE = STRING;MODE SEQ = [1:0]SEQMODE;MODE YIELDSEQ = PROC(SEQ)VOID; PROC gen ncs = (SEQ seq, YIELDSEQ yield)VOID:BEGIN  IF UPB seq - 1 > bits width THEN stop FI;  [UPB seq]SEQMODE out;  INT upb out;   BITS lim := 16r1 SHL UPB seq;  BITS upb k := lim SHR 1;  # assert(lim); #   BITS empty = 16r000000000; # const #   FOR j TO ABS lim-1 DO    INT state := 1;    BITS k1 := upb k;     WHILE k1 NE empty DO      BITS b := BIN j AND k1;      CASE state IN        # state 1 # IF b NE empty THEN state +:= 1 FI,        # state 2 # IF b EQ empty THEN state +:= 1 FI,        # state 3 #           BEGIN            IF b EQ empty THEN GO TO continue k1 FI;            upb out := 0;             BITS k2 := upb k; FOR i WHILE k2 NE empty DO              IF (BIN j AND k2) NE empty THEN out[upb out +:= 1] := seq[i] FI;              k2 := k2 SHR 1            OD;            yield(out[:upb out]);            k1 := empty # empty: ending containing loop #         END      ESAC;      continue k1: k1 := k1 SHR 1    OD  ODEND; main:(  []STRING seqs = ("a","e","i","o","u");# FOR SEQ seq IN # gen ncs(seqs, # ) DO ( ###   (SEQ seq)VOID:    print((seq, new line))# OD # ))`
Output:
```iu
eu
eo
eou
eiu
au
ao
aou
ai
aiu
aio
aiou
aeu
aeo
aeou
aeiu
```

## AutoHotkey

using filtered templates ahk forum: discussion

`MsgBox % noncontinuous("a,b,c,d,e", ",")MsgBox % noncontinuous("1,2,3,4", ",") noncontinuous(list, delimiter){stringsplit, seq, list, %delimiter%n := seq0                                            ; sequence lengthLoop % x := (1<<n) - 1 {                                  ; try all 0-1 candidate sequences   If !RegExMatch(b:=ToBin(A_Index,n),"^0*1*0*\$") {  ; drop continuous subsequences      Loop Parse, b         t .= A_LoopField ? seq%A_Index% " " : ""         ; position -> number		 t .= "`n"                                   ; new sequences in new lines   }}return t} ToBin(n,W=16) {  ; LS W-bits of Binary representation of n   Return W=1 ? n&1 : ToBin(n>>1,W-1) . n&1}`

## BBC BASIC

`      DIM list1\$(3)      list1\$() = "1", "2", "3", "4"      PRINT "For [1, 2, 3, 4] non-continuous subsequences are:"      PROCnon_continuous_subsequences(list1\$())      DIM list2\$(4)      list2\$() = "1", "2", "3", "4", "5"      PRINT "For [1, 2, 3, 4, 5] non-continuous subsequences are:"      PROCnon_continuous_subsequences(list2\$())      END       DEF PROCnon_continuous_subsequences(l\$())      LOCAL i%, j%, g%, n%, r%, s%, w%, a\$, b\$      n% = DIM(l\$(),1)      FOR s% = 0 TO n%-2        FOR g% = s%+1 TO n%-1          a\$ = "["          FOR i% = s% TO g%-1            a\$ += l\$(i%) + ", "          NEXT          FOR w% = 1 TO n%-g%            r% = n%+1-g%-w%            FOR i% = 1 TO 2^r%-1 STEP 2              b\$ = a\$              FOR j% = 0 TO r%-1                IF i% AND 2^j% b\$ += l\$(g%+w%+j%) + ", "              NEXT              PRINT LEFT\$(LEFT\$(b\$)) + "]"            NEXT i%          NEXT w%        NEXT g%      NEXT s%      ENDPROC`
Output:
```For [1, 2, 3, 4] non-continuous subsequences are:
[1, 3]
[1, 3, 4]
[1, 4]
[1, 2, 4]
[2, 4]
For [1, 2, 3, 4, 5] non-continuous subsequences are:
[1, 3]
[1, 3, 4]
[1, 3, 5]
[1, 3, 4, 5]
[1, 4]
[1, 4, 5]
[1, 5]
[1, 2, 4]
[1, 2, 4, 5]
[1, 2, 5]
[1, 2, 3, 5]
[2, 4]
[2, 4, 5]
[2, 5]
[2, 3, 5]
[3, 5]
```

## Bracmat

`( ( noncontinuous  =   sub    .     ( sub          =   su a nc            .   !arg:(?su.?nc)              &   !su                :   %                    %?a                    ( %:[%(sub\$(!sjt.!nc !a))                    |   ?                      & !nc:~                      & out\$(!nc !a)                      & ~                    )          )        & sub\$(dummy !arg.)      |   )& noncontinuous\$(e r n i t)); `
Output:
```e n t
e n
e n i
e n i t
e i
e i t
e t
e r i
e r i t
e r t
e r n t
r i
r i t
r t
r n t
n t```

## C

Note: This specimen can only handle lists of length less than the number of bits in an int.

`#include <assert.h>#include <stdio.h> int main(int c, char **v){	unsigned int n = 1 << (c - 1), i = n, j, k;	assert(n); 	while (i--) {		if (!(i & (i + (i & -(int)i)))) // consecutive 1s			continue; 		for (j = n, k = 1; j >>= 1; k++)			if (i & j) printf("%s ", v[k]); 		putchar('\n');	} 	return 0;}`

Example use:

```\$ ./noncont 1 2 3 4
1 2 4
1 3 4
1 3
2 4
1 4
\$ ./noncont 1 2 3 4 5 6 7 8 9 0 | wc -l
968
```

Using "consecutive + gap + any subsequence" to produce disjointed sequences:

`#include <assert.h>#include <stdio.h>#include <stdlib.h> void binprint(unsigned int n, unsigned int m){	char c[sizeof(n) * 8 + 1];	int i = 0;	while (m >>= 1)	c[i++] = n & m ? '#' : '-';	c[i] = 0;	puts(c);} int main(int c, char **v){	unsigned int n, gap, left, right;	if (c < 2 || ! (n = 1 << atoi(v[1]))) n = 16; 	for (gap = 2; gap < n; gap <<= 1)		for (left = gap << 1; left < n; left |= left << 1)			for (right = 1; right < gap; right++)				binprint(left | right, n); 	return 0;}`

### Recursive method

Using recursion and a state transition table.

`#include <stdio.h> typedef unsigned char sint;enum states { s_blnk = 0, s_tran, s_cont, s_disj }; /* Recursively look at each item in list, taking both choices of   picking the item or not.  The state at each step depends on prvious   pickings, with the state transition table:	blank + no pick -> blank	blank + pick -> contiguous	transitional + no pick -> transitional	transitional + pick -> disjoint	contiguous + no pick -> transitional	contiguous + pick -> contiguous	disjoint + pick -> disjoint	disjoint + no pick -> disjoint   At first step, before looking at any item, state is blank.   Because state is known at each step and needs not be calculated,   it can be quite fast.*/unsigned char tbl[][2] = {	{ s_blnk, s_cont },	{ s_tran, s_disj },	{ s_tran, s_cont },	{ s_disj, s_disj },}; void pick(sint n, sint step, sint state, char **v, unsigned long bits){	int i, b;	if (step == n) {		if (state != s_disj) return;		for (i = 0, b = 1; i < n; i++, b <<= 1)			if ((b & bits)) printf("%s ", v[i]);		putchar('\n');		return;	} 	bits <<= 1;	pick(n, step + 1, tbl[state][0], v, bits); /* no pick */	pick(n, step + 1, tbl[state][1], v, bits | 1); /* pick */} int main(int c, char **v){	if (c - 1 >= sizeof(unsigned long) * 4)		printf("Too many items");	else		pick(c - 1, 0, s_blnk, v + 1, 0);	return 0;}`
running it:
```% ./a.out 1 2 3 4
1 3
1 4
2 4
1 2 4
1 3 4
% ./a.out 1 2 3 4 5 6 7 8 9 0 | wc -l
968```

## C++

` /* * Nigel Galloway, July 19th., 2017 - Yes well is this any better? */class N{  uint n,i,g,e,l;public:  N(uint n): n(n-1),i{},g{},e(1),l(n-1){}  bool hasNext(){    g=(1<<n)+e;for(i=l;i<n;++i) g+=1<<i;    if (l==2)             {l=--n; e=1; return true;}    if (e<((1<<(l-1))-1)) {++e;        return true;}                           e=1; --l;   return (l>0);  }  uint next() {return g;}}; `

Which may be used as follows:

` int main(){  N n(4);  while (n.hasNext()) std::cout << n.next() << "\t* " << std::bitset<4>(n.next()) << std::endl;} `
Output:
```9       * 1001
10      * 1010
11      * 1011
13      * 1101
5       * 0101
```

I can count the length of the sequence:

` int main(){  N n(31);  int z{};for (;n.hasNext();++z); std::cout << z << std::endl;} `
Output:
```2147483151
```

## Clojure

Here's a simple approach that uses the clojure.contrib.combinatorics library to generate subsequences, and then filters out the continuous subsequences using a naïve subseq test:

` (use '[clojure.contrib.combinatorics :only (subsets)]) (defn of-min-length [min-length]  (fn [s] (>= (count s) min-length))) (defn runs [c l]  (map (partial take l) (take-while not-empty (iterate rest c)))) (defn is-subseq? [c sub]  (some identity (map = (runs c (count sub)) (repeat sub)))) (defn non-continuous-subsequences [s]  (filter (complement (partial is-subseq? s)) (subsets s)))  (filter (of-min-length 2) (non-continuous-subsequences [:a :b :c :d])) `

## CoffeeScript

Use binary bitmasks to enumerate our sequences.

` is_contigous_binary = (n) ->  # return true if binary representation of n is  # of the form 1+0+  # examples:  #     0 true  #     1 true  #   100 true  #   110 true  #  1001 false  #  1010 false   # special case zero, or you'll get an infinite loop later  return true if n == 0   # first remove 0s from end  while n % 2 == 0    n = n / 2   # next, take advantage of the fact that a continuous  # run of 1s would be of the form 2^n - 1  is_power_of_two(n + 1) is_power_of_two = (m) ->  while m % 2 == 0    m = m / 2  m == 1 seq_from_bitmap = (arr, n) ->  # grabs elements from array according to a bitmap  # e.g. if n == 13 (1101), and arr = ['a', 'b', 'c', 'd'],  # then return ['a', 'c', 'd'] (flipping bits to 1011, so  # that least significant bit comes first)  i = 0  new_arr = []  while n > 0    if n % 2 == 1      new_arr.push arr[i]      n -= 1    n /= 2    i += 1  new_arr non_contig_subsequences = (arr) ->  # Return all subsqeuences from an array that have a "hole" in  # them.  The order of the subsequences is not specified here.   # This algorithm uses binary counting, so it is limited to  # small lists, but large lists would be unwieldy regardless.  bitmasks = [0...Math.pow(2, arr.length)]  (seq_from_bitmap arr, n for n in bitmasks when !is_contigous_binary n) arr = [1,2,3,4]console.log non_contig_subsequences arrfor n in [1..10]  arr = [1..n]  num_solutions = non_contig_subsequences(arr).length  console.log "for n=#{n} there are #{num_solutions} solutions" `
Output:
```> coffee non_contig_subseq.coffee
[ [ 1, 3 ],
[ 1, 4 ],
[ 2, 4 ],
[ 1, 2, 4 ],
[ 1, 3, 4 ] ]
for n=1 there are 0 solutions
for n=2 there are 0 solutions
for n=3 there are 1 solutions
for n=4 there are 5 solutions
for n=5 there are 16 solutions
for n=6 there are 42 solutions
for n=7 there are 99 solutions
for n=8 there are 219 solutions
for n=9 there are 466 solutions
for n=10 there are 968 solutions
```

## Common Lisp

`(defun all-subsequences (list)  (labels ((subsequences (tail &optional (acc '()) (result '()))             "Return a list of the subsequence designators of the              subsequences of tail. Each subsequence designator is a              list of tails of tail, the subsequence being the first              element of each tail."             (if (endp tail)               (list* (reverse acc) result)               (subsequences (rest tail) (list* tail acc)                             (append (subsequences (rest tail) acc) result))))           (continuous-p (subsequence-d)             "True if the designated subsequence is continuous."             (loop for i in subsequence-d                   for j on (first subsequence-d)                   always (eq i j)))           (designated-sequence (subsequence-d)             "Destructively transforms a subsequence designator into              the designated subsequence."             (map-into subsequence-d 'first subsequence-d)))    (let ((nc-subsequences (delete-if #'continuous-p (subsequences list))))      (map-into nc-subsequences #'designated-sequence nc-subsequences))))`
Translation of: Scheme
`(defun all-subsequences2 (list)  (labels ((recurse (s list)             (if (endp list)                 (if (>= s 3)                     '(())                     '())                 (let ((x (car list))                       (xs (cdr list)))                   (if (evenp s)                       (append (mapcar (lambda (ys) (cons x ys))                                       (recurse (+ s 1) xs))                               (recurse s xs))                       (append (mapcar (lambda (ys) (cons x ys))                                       (recurse s xs))                               (recurse (+ s 1) xs)))))))    (recurse 0 list)))`

## D

### Recursive Version

Translation of: Python
`T[][] ncsub(T)(in T[] seq, in uint s=0) pure nothrow @safe {    if (seq.length) {        typeof(return) aux;        foreach (ys; ncsub(seq[1 .. \$], s + !(s % 2)))            aux ~= seq[0] ~ ys;        return aux ~ ncsub(seq[1 .. \$], s + s % 2);    } else        return new typeof(return)(s >= 3, 0);} void main() @safe {    import std.stdio;     [1, 2, 3].ncsub.writeln;    [1, 2, 3, 4].ncsub.writeln;    foreach (const nc; [1, 2, 3, 4, 5].ncsub)        nc.writeln;}`
Output:
```[[1, 3]]
[[1, 2, 4], [1, 3, 4], [1, 3], [1, 4], [2, 4]]
[1, 2, 3, 5]
[1, 2, 4, 5]
[1, 2, 4]
[1, 2, 5]
[1, 3, 4, 5]
[1, 3, 4]
[1, 3, 5]
[1, 3]
[1, 4, 5]
[1, 4]
[1, 5]
[2, 3, 5]
[2, 4, 5]
[2, 4]
[2, 5]
[3, 5]```

### Faster Lazy Version

This version doesn't copy the sub-arrays.

`struct Ncsub(T) {    T[] seq;     int opApply(int delegate(ref T[]) dg) const {        immutable n = seq.length;        int result;        auto S = new T[n];         OUTER: foreach (immutable i; 1 .. 1 << n) {            uint lenS;            bool nc = false;            foreach (immutable j; 0 .. n + 1) {                immutable k = i >> j;                if (k == 0) {                    if (nc) {                        auto auxS = S[0 .. lenS];                        result = dg(auxS);                        if (result)                            break OUTER;                    }                    break;                } else if (k % 2) {                    S[lenS] = seq[j];                    lenS++;                } else if (lenS)                    nc = true;            }        }         return result;    }} void main() {    import std.array, std.range;     //assert(24.iota.array.Ncsub!int.walkLength == 16_776_915);    auto r = 24.iota.array;    uint counter = 0;    foreach (s; Ncsub!int(r))        counter++;    assert(counter == 16_776_915);}`

### Generator Version

This version doesn't copy the sub-arrays, and it's a little slower than the opApply-based version.

`import std.stdio, std.array, std.range, std.concurrency; Generator!(T[]) ncsub(T)(in T[] seq) {    return new typeof(return)({        immutable n = seq.length;        auto S = new T[n];         foreach (immutable i; 1 .. 1 << n) {            uint lenS = 0;            bool nc = false;            foreach (immutable j; 0 .. n + 1) {                immutable k = i >> j;                if (k == 0) {                    if (nc)                        yield(S[0 .. lenS]);                    break;                } else if (k % 2) {                    S[lenS] = seq[j];                    lenS++;                } else if (lenS)                    nc = true;            }        }    });} void main() {    assert(24.iota.array.ncsub.walkLength == 16_776_915);     [1, 2, 3].ncsub.writeln;    [1, 2, 3, 4].ncsub.writeln;    foreach (const nc; [1, 2, 3, 4, 5].ncsub)        nc.writeln;}`

## Elixir

Translation of: Erlang
`defmodule RC do  defp masks(n) do    maxmask = trunc(:math.pow(2, n)) - 1    Enum.map(3..maxmask, &Integer.to_string(&1, 2))    |> Enum.filter_map(&contains_noncont(&1), &String.rjust(&1, n, ?0)) # padding  end   defp contains_noncont(n) do    Regex.match?(~r/10+1/, n)  end   defp apply_mask_to_list(mask, list) do    Enum.zip(to_char_list(mask), list)    |> Enum.filter_map(fn {include, _} -> include > ?0 end, fn {_, value} -> value end)  end   def ncs(list) do    Enum.map(masks(length(list)), fn mask -> apply_mask_to_list(mask, list) end)  endend IO.inspect RC.ncs([1,2,3])IO.inspect RC.ncs([1,2,3,4])IO.inspect RC.ncs('abcd')`
Output:
```[[1, 3]]
[[2, 4], [1, 4], [1, 3], [1, 3, 4], [1, 2, 4]]
```

## Erlang

Erlang's not optimized for strings or math, so this is pretty inefficient. Nonetheless, it works by generating the set of all possible "bitmasks" (represented as strings), filters for those with non-continuous subsequences, and maps from that set over the list. One immediate point for optimization that would complicate the code a bit would be to compile the regular expression, the problem being where you'd put it.

`-module(rosetta).-export([ncs/1]). masks(N) ->    MaxMask = trunc(math:pow(2, N)),    Total = lists:map(fun(X) -> integer_to_list(X, 2) end,                lists:seq(3, MaxMask)),    Filtered = lists:filter(fun(X) -> contains_noncont(X) end, Total),    lists:map(fun(X) -> string:right(X, N, \$0) end, Filtered). % padding contains_noncont(N) ->    case re:run(N, "10+1") of        {match, _} -> true;        nomatch -> false    end. apply_mask_to_list(Mask, List) ->    Zipped = lists:zip(Mask, List),    Filtered = lists:filter(fun({Include, _}) -> Include > 48 end, Zipped),    lists:map(fun({_, Value}) -> Value end, Filtered). ncs(List) ->    lists:map(fun(Mask) -> apply_mask_to_list(Mask, List) end,                masks(length(List))).`
Output:
```Eshell V5.10.1  (abort with ^G)
1> c(rosetta).
{ok,rosetta}
2> rosetta:ncs([1,2,3,4]).
[[2,4],[1,4],[1,3],[1,3,4],[1,2,4]]
```

## F#

### Generate only the non-continuous subsequences

` (*  A function to generate only the non-continuous subsequences.  Nigel Galloway July 20th., 2017*)let N n =  let     fn n = Seq.map (fun g->(2<<<n)+g)  let rec fg n = seq{if n>0 then yield! seq{1..((1<<<n)-1)}|>fn n; yield! fg (n-1)|>fn n}  Seq.collect fg ({1..(n-2)}) `

This may be used as follows:

` let Ng ng = N ng |> Seq.iter(fun n->printf "%2d -> " n; {0..(ng-1)}|>Seq.iter (fun g->if (n&&&(1<<<g))>0 then printf "%d " (g+1));printfn "")Ng 4 `
Output:
``` 5 -> 1 3
9 -> 1 4
10 -> 2 4
11 -> 1 2 4
13 -> 1 3 4
```

Counting the number of non-continuous subsequences is interesting:

```> Seq.length (N 20);;
Real: 00:00:00.169, CPU: 00:00:00.169, GC gen0: 0, gen1: 0
val it : int = 1048365
> Seq.length (N 23);;
Real: 00:00:01.238, CPU: 00:00:01.239, GC gen0: 0, gen1: 0
val it : int = 8388331
> Seq.length (N 24);;
Real: 00:00:02.520, CPU: 00:00:02.523, GC gen0: 0, gen1: 0
val it : int = 16776915
> Seq.length (N 25);;
Real: 00:00:04.926, CPU: 00:00:04.930, GC gen0: 0, gen1: 0
val it : int = 33554106
```

### Generate all subsequences and filter out the continuous

` (*  A function to filter out continuous subsequences.  Nigel Galloway July 24th., 2017*)let Nonseq n=  let fn = function    |((n,0),true )->(n+1,1)    |((n,_),false)->(n,0)    |(n,_)        ->n  {5..(1<<<n)-1}|>Seq.choose(fun i->if fst({0..n-1}|>Seq.takeWhile(fun n->(1<<<(n-1))<i)|>Seq.fold(fun n g->fn (n,(i&&&(1<<<g)>0)))(0,0)) > 1 then Some(i) else None) `

Again counting the number of non-continuous subsequences

```> Seq.length (Nonseq 20);;
Real: 00:00:02.356, CPU: 00:00:02.389, GC gen0: 183, gen1: 0
val it : int = 1048365
> Seq.length (Nonseq 23);;
Real: 00:00:20.714, CPU: 00:00:20.950, GC gen0: 1571, gen1: 0
val it : int = 8388331
> Seq.length (Nonseq 24);;
Real: 00:00:43.129, CPU: 00:00:43.601, GC gen0: 3216, gen1: 0
val it : int = 16776915
> Seq.length (Nonseq 25);;
Real: 00:01:28.853, CPU: 00:01:29.869, GC gen0: 6577, gen1: 0
val it : int = 33554106
```

### Conclusion

Find a better filter or use the generator.

## Go

Generate the power set (power sequence, actually) with a recursive function, but keep track of the state of the subsequence on the way down. When you get to the bottom, if state == non-continuous, then include the subsequence. It's just filtering merged in with generation.

`package main import "fmt" const ( // state:    m   = iota // missing:  all elements missing so far    c          // continuous:  all elements included so far are continuous    cm         // one or more continuous followed by one or more missing    cmc        // non-continuous subsequence) func ncs(s []int) [][]int {    if len(s) < 3 {        return nil    }    return append(n2(nil, s[1:], m), n2([]int{s[0]}, s[1:], c)...)} var skip = []int{m, cm, cm, cmc}var incl = []int{c, c, cmc, cmc} func n2(ss, tail []int, seq int) [][]int {    if len(tail) == 0 {        if seq != cmc {            return nil        }        return [][]int{ss}    }    return append(n2(append([]int{}, ss...), tail[1:], skip[seq]),        n2(append(ss, tail[0]), tail[1:], incl[seq])...)} func main() {    ss := ncs([]int{1, 2, 3, 4})    fmt.Println(len(ss), "non-continuous subsequences:")    for _, s := range ss {        fmt.Println("  ", s)    }}`
Output:
```5 non-continuous subsequences:
[2 4]
[1 4]
[1 3]
[1 3 4]
[1 2 4]
```

`action p x = if p x then succ x else x fenceM p q s []     = guard (q s) >> return []fenceM p q s (x:xs) = do  (f,g) <- p   ys <- fenceM p q (g s) xs  return \$ f x ys ncsubseq = fenceM [((:), action even), (flip const, action odd)] (>= 3) 0`
Output:
```*Main> ncsubseq [1..3]
[[1,3]]
*Main> ncsubseq [1..4]
[[1,2,4],[1,3,4],[1,3],[1,4],[2,4]]
*Main> ncsubseq [1..5]
[[1,2,3,5],[1,2,4,5],[1,2,4],[1,2,5],[1,3,4,5],[1,3,4],[1,3,5],[1,3],[1,4,5],[1,4],[1,5],[2,3,5],[2,4,5],[2,4],[2,5],[3,5]]```

### Filtered templates

This implementation works by computing templates of all possible subsequences of the given length of sequence, discarding the continuous ones, then applying the remaining templates to the input list.

`continuous = null . dropWhile not . dropWhile id . dropWhile notncs xs = map (map fst . filter snd . zip xs) \$           filter (not . continuous) \$             mapM (const [True,False]) xs`

### Recursive

Recursive method with powerset as helper function.

`import Data.List poset = foldr (\x p -> p ++ map (x:) p) [[]] ncsubs [] = [[]]ncsubs (x:xs) = tail \$ nc [x] xs  where    nc [_] [] = [[]]    nc (_:x:xs) [] = nc [x] xs    nc  xs (y:ys) = (nc (xs++[y]) ys) ++ map (xs++) (tail \$ poset ys)`
Output:
``` *Main> ncsubs "aaa"
["aa"]
(0.00 secs, 0 bytes)
*Main> ncsubs [9..12]
[[10,12],[9,10,12],[9,12],[9,11],[9,11,12]]
(0.00 secs, 522544 bytes)
*Main> ncsubs []
[[]]
(0.00 secs, 0 bytes)
*Main> ncsubs [1]
[]
(0.00 secs, 0 bytes)
```

A disjointed subsequence is a consecutive subsequence followed by a gap, then by any nonempty subsequence to its right:

`import Data.List (subsequences, tails, delete) disjoint a = concatMap (cutAt a) [1..length a - 2] where	cutAt s n = [a ++ b |	b <- delete [] (subsequences right),				a <- init (tails left) ] where		(left, _:right) = splitAt n s main = print \$ length \$ disjoint [1..20]`

Build a lexicographic list of consecutive subsequences, and a list of all subsequences, then subtract one from the other:

`import Data.List (inits, tails) subseqs = foldr (\x s -> [x] : map (x:) s ++ s) [] consecs = concatMap (tail.inits) . tails minus [] [] = []minus (a:as) bb@(b:bs)	| a == b = minus as bs	| otherwise = a:minus as bb disjoint s = (subseqs s) `minus` (consecs s) main = mapM_ print \$ disjoint [1..4]`

## J

We select those combinations where the end of the first continuous subsequence appears before the start of the last continuous subsequence:

`allmasks=: 2 #:@[email protected]^ #firstend=:1 0 i.&1@E."1 ]laststart=: 0 1 {:@[email protected]"1 ]noncont=: <@#~ (#~ firstend < laststart)@allmasks`

Example use:

`   noncont 1+i.4┌───┬───┬───┬─────┬─────┐│2 4│1 4│1 3│1 3 4│1 2 4│└───┴───┴───┴─────┴─────┘   noncont 'aeiou'┌──┬──┬──┬───┬───┬──┬──┬───┬──┬───┬───┬────┬───┬───┬────┬────┐│iu│eu│eo│eou│eiu│au│ao│aou│ai│aiu│aio│aiou│aeu│aeo│aeou│aeiu│└──┴──┴──┴───┴───┴──┴──┴───┴──┴───┴───┴────┴───┴───┴────┴────┘   #noncont i.10968`

Alternatively, since there are relatively few continuous sequences, we could specifically exclude them:

`contmasks=: a: ;@,  1 <:/[email protected]&.>@[email protected]+ #noncont=: <@#~ (allmasks -. contmasks)`

(we get the same behavior from this implementation)

## Java

`public class NonContinuousSubsequences {     public static void main(String args[]) {        seqR("1234", "", 0, 0);    }     private static void seqR(String s, String c, int i, int added) {        if (i == s.length()) {            if (c.trim().length() > added)                System.out.println(c);        } else {            seqR(s, c + s.charAt(i), i + 1, added + 1);            seqR(s, c + ' ', i + 1, added);        }    }}`
```12 4
1 34
1 3
1  4
2 4```

## JavaScript

Uses powerset() function from here. Uses a JSON stringifier from http://www.json.org/js.html

Works with: SpiderMonkey
`function non_continuous_subsequences(ary) {    var non_continuous = new Array();    for (var i = 0; i < ary.length; i++) {        if (! is_array_continuous(ary[i])) {            non_continuous.push(ary[i]);        }    }    return non_continuous;} function is_array_continuous(ary) {    if (ary.length < 2)        return true;    for (var j = 1; j < ary.length; j++) {        if (ary[j] - ary[j-1] != 1) {            return false;        }    }    return true;} load('json2.js'); /* http://www.json.org/js.html */ print(JSON.stringify( non_continuous_subsequences( powerset([1,2,3,4]))));`
Output:
`[[1,3],[1,4],[2,4],[1,2,4],[1,3,4]]`

## jq

Works with: jq version 1.4

In order to handle arrays of more than a handful of elements, we define non_continuous_subsequences/0 as a generator; that is, it produces a stream of arrays, each of which is a non-continuous subsequence of the given sequence.

Since the non-continuous subsequences are dense in the set of all subsets, we will use the powerset approach, and accordingly begin by defining subsets/0 as a generator.

`# Generate a stream of subsets of the input arraydef subsets:  if length == 0 then []  else .[0] as \$first    | (.[1:] | subsets)     | ., ([\$first] + .)  end ; # Generate a stream of non-continuous indices in the range 0 <= i < .def non_continuous_indices:  [range(0;.)] | subsets  | select(length > 1 and length != 1 + .[length-1] - .[0]) ; def non_continuous_subsequences:  (length | non_continuous_indices) as \$ix  | [.[ \$ix[] ]] ;`

Example: To show that the above approach can be used for relatively large n, let us count the number of non-continuous subsequences of [0, 1, ..., 19].

`def count(f): reduce f as \$i (0; . + 1); count( [range(0;20)] | non_continuous_subsequences) `
Output:
```\$ jq -n -f powerset_generator.jq
1048365
```

## Julia

Works with: Julia version 0.6

This solution uses an iterator over non-contiguous sub-sequences, NCSubSeq. In the spirit of Julia's permutations and combinations built-ins, NCSubSeq provides an array of indices that can be used to create each subsequence from the full sequence. Sub-sequences are indexed by integers whose bit patterns indicate which members are included.

NCSubSeq works by filtering indices according to whether all 1s in these indices have bit pattern that are contiguous (using the iscontseq functions). This is an easy to implement approach. Greater efficiency might be achieved by exploiting the property that a sequence is contiguous if and only if its index is a difference of two powers of 2. This property is used to create the length(NCSubSeq(n)) function, which gives the number of non-contiguous sub-sequences of a sequence of length n.

NCSubSeq works transparently for sequence lengths up to WORD_SIZE-1 (typically 63). It can be extended to work for longer sequences by casting n to a larger integer, e.g. using Big(n). A more polished implementation would handle this extension behind the scenes.

Iterator and Functions

`iscontseq(n::Integer) = count_zeros(n) == leading_zeros(n) + trailing_zeros(n)iscontseq(n::BigInt)  = !ismatch(r"0", rstrip(bin(n), '0')) function makeint2seq(n::Integer)    const idex = collect(1:n)    function int2seq(m::Integer)        d = digits(m, 2, n)        idex[d .== 1]    end    return int2seqend struct NCSubSeq{T<:Integer}    n::Tend mutable struct NCSubState{T<:Integer}    m::T    m2s::Functionend Base.iteratorsize(::NCSubSeq) = Base.HasLength()Base.length(a::NCSubSeq) = 2 ^ a.n - a.n * (a.n + 1) ÷ 2 - 1 Base.start(a::NCSubSeq) = NCSubState(5, makeint2seq(a.n))Base.done(a::NCSubSeq, as::NCSubState) = 2 ^ a.n - 3 < as.mfunction Base.next(a::NCSubSeq, as::NCSubState)    s = as.m2s(as.m)    as.m += 1    while iscontseq(as.m)        as.m += 1    end    return (s, as)end n = 4println("Testing NCSubSeq for ", n, " items:\n ", join(NCSubSeq(n), " ")) s = "Rosetta"cs = split(s, "")m = 10n = length(NCSubSeq(length(s))) - mprintln("\nThe first and last ", m, " NC sub-sequences of \"", s, "\":")for (i, a) in enumerate(NCSubSeq(length(cs)))    i <= m || n < i || continue    println(@sprintf "%6d %s" i join(cs[a], ""))    i == m || continue    println("    .. ......")end using IterTools.chain println("\nThe first and last ", m, " NC sub-sequences of \"", s, "\"")for x in IterTools.chain(1:10, 20:10:40, big.(50:50:200))    @printf "%7d → %d\n" x length(NCSubSeq(x))end`
Output:
```Testing NCSubSeq for 4 items:
[1, 3] [1, 4] [2, 4] [1, 2, 4] [1, 3, 4]

The first and last 10 NC sub-sequences of "Rosetta":
1 Rs
2 Re
3 oe
4 Roe
5 Rse
6 Rt
7 ot
8 Rot
9 st
10 Rst
.. ......
90 otta
91 Rotta
92 stta
93 Rstta
94 ostta
95 Rostta
96 Retta
97 oetta
98 Roetta
99 Rsetta

The first and last 10 NC sub-sequences of "Rosetta"
1 → 0
2 → 0
3 → 1
4 → 5
5 → 16
6 → 42
7 → 99
8 → 219
9 → 466
10 → 968
20 → 1048365
30 → 1073741358
40 → 1099511626955
50 → 1125899906841348
100 → 1267650600228229401496703200325
150 → 1427247692705959881058285969449495136382735298
200 → 1606938044258990275541962092341162602522202993782792835281275```

## Kotlin

`// version 1.1.2 fun <T> ncs(a: Array<T>) {     fun generate(m: Int, k: Int, c: IntArray) {        if (k == m) {            if (c[m - 1] != c[0] + m - 1) {                for (i in 0 until m)  print("\${a[c[i]]} ")                                println()            }        }        else {             for (j in 0 until a.size) {                if (k == 0 || j > c[k - 1]) {                    c[k] = j                    generate(m, k + 1, c)                }            }        }    }     for (m in 2 until a.size) {        val c = IntArray(m)         generate(m, 0, c)    }    }   fun main(args: Array<String>) {    val a = arrayOf(1, 2, 3, 4)    ncs(a)    println()    val ca = arrayOf('a', 'b', 'c', 'd', 'e')    ncs(ca)}`
Output:
```1 3
1 4
2 4
1 2 4
1 3 4

a c
a d
a e
b d
b e
c e
a b d
a b e
a c d
a c e
a d e
b c e
b d e
a b c e
a b d e
a c d e
```

## Mathematica

We make all the subsets then filter out the continuous ones:

`GoodBad[i_List]:=Not[MatchQ[Differences[i],{1..}|{}]]n=5Select[Subsets[Range[n]],GoodBad]`

gives back:

` {{1,3},{1,4},{1,5},{2,4},{2,5},{3,5},{1,2,4},{1,2,5},{1,3,4},{1,3,5},{1,4,5},{2,3,5},{2,4,5},{1,2,3,5},{1,2,4,5},{1,3,4,5}}`

## Nim

Translation of: Python
`import sequtils proc ncsub[T](se: seq[T], s = 0): seq[seq[T]] =  result = @[]  if se.len > 0:    let      x = se[0..0]      xs = se[1 .. -1]      p2 = s mod 2      p1 = (s + 1) mod 2    for ys in ncsub(xs, s + p1):      result.add(x & ys)    result.add(ncsub(xs, s + p2))  elif s >= 3:    result.add(@[]) echo "ncsub(", toSeq 1.. 3, ") = ", ncsub(toSeq 1..3)echo "ncsub(", toSeq 1.. 4, ") = ", ncsub(toSeq 1..4)echo "ncsub(", toSeq 1.. 5, ") = ", ncsub(toSeq 1..5)`
Output:
```ncsub(@[1, 2, 3]) = @[@[1, 3]]
ncsub(@[1, 2, 3, 4]) = @[@[1, 2, 4], @[1, 3, 4], @[1, 3], @[1, 4], @[2, 4]]
ncsub(@[1, 2, 3, 4, 5]) = @[@[1, 2, 3, 5], @[1, 2, 4, 5], @[1, 2, 4], @[1, 2, 5], @[1, 3, 4, 5], @[1, 3, 4], @[1, 3, 5], @[1, 3], @[1, 4, 5], @[1, 4], @[1, 5], @[2, 3, 5], @[2, 4, 5], @[2, 4], @[2, 5], @[3, 5]]```

## OCaml

`let rec fence s = function    [] ->      if s >= 3 then        [[]]      else        []   | x :: xs ->      if s mod 2 = 0 then        List.map          (fun ys -> x :: ys)          (fence (s + 1) xs)        @          fence s xs      else        List.map          (fun ys -> x :: ys)          (fence s xs)        @          fence (s + 1) xs let ncsubseq = fence 0`
Output:
```# ncsubseq [1;2;3];;
- : int list list = [[1; 3]]
# ncsubseq [1;2;3;4];;
- : int list list = [[1; 2; 4]; [1; 3; 4]; [1; 3]; [1; 4]; [2; 4]]
# ncsubseq [1;2;3;4;5];;
- : int list list =
[[1; 2; 3; 5]; [1; 2; 4; 5]; [1; 2; 4]; [1; 2; 5]; [1; 3; 4; 5]; [1; 3; 4];
[1; 3; 5]; [1; 3]; [1; 4; 5]; [1; 4]; [1; 5]; [2; 3; 5]; [2; 4; 5];
[2; 4]; [2; 5]; [3; 5]]```

## Oz

A nice application of finite set constraints. We just describe what we want and the constraint system will deliver it:

`declare  fun {NCSubseq SeqList}     Seq = {FS.value.make SeqList}     proc {Script Result}        %% the result is a subset of Seq        {FS.subset Result Seq}         %% at least one element of Seq is missing        local Gap in           {FS.include Gap Seq}           {FS.exclude Gap Result}           %% and this element is between the smallest           %% and the largest elements of the subsequence           Gap >: {FS.int.min Result}           Gap <: {FS.int.max Result}        end         %% enumerate all such sets        {FS.distribute naive [Result]}     end  in     {Map {SearchAll Script} FS.reflect.lowerBoundList}  endin  {Inspect {NCSubseq [1 2 3 4]}}`

## PARI/GP

Just a simple script, but it's I/O bound so efficiency isn't a concern. (Almost all subsequences are non-contiguous so looping over all possibilities isn't that bad. For length 20 about 99.98% of subsequences are non-contiguous.)

`noncontig(n)=n>>=valuation(n,2);n++;n>>=valuation(n,2);n>1;nonContigSubseq(v)={  for(i=5,2^#v-1,    if(noncontig(i),      print(vecextract(v,i))    )  )};nonContigSubseq([1,2,3])nonContigSubseq(["a","b","c","d","e"])`
Output:
```[1, 3]

["a", "c"]
["a", "d"]
["b", "d"]
["a", "b", "d"]
["a", "c", "d"]
["a", "e"]
["b", "e"]
["a", "b", "e"]
["c", "e"]
["a", "c", "e"]
["b", "c", "e"]
["a", "b", "c", "e"]
["a", "d", "e"]
["b", "d", "e"]
["a", "b", "d", "e"]
["a", "c", "d", "e"]```

## Perl

`my (\$max, @current);sub non_continuous {        my (\$idx, \$has_gap, \$found) = @_;         for (\$idx .. \$max) {                push @current, \$_;                # print "@current\n" if \$has_gap; # uncomment for huge output                \$found ++ if \$has_gap;                \$found += non_continuous(\$_ + 1, \$has_gap)   if \$_ < \$max;                pop @current;                \$has_gap = @current;   # don't set gap flag if it's empty still        }        \$found;} \$max = 20;      # 1048365 sequences, 10 seconds-ishprint "found ", non_continuous(1), " sequences\n";`

## Perl 6

Works with: rakudo version 2015-09-24
`sub non_continuous_subsequences ( *@list ) {    @list.combinations.grep: { 1 != all( .[ 0 ^.. .end] Z- .[0 ..^ .end] ) }} say non_continuous_subsequences( 1..3 )».gist;say non_continuous_subsequences( 1..4 )».gist;say non_continuous_subsequences(   ^4 ).map: {[<a b c d>[.list]].gist};`
Output:
```((1 3))
((1 3) (1 4) (2 4) (1 2 4) (1 3 4))
([a c] [a d] [b d] [a b d] [a c d])```

## Phix

Straightforward recursive implementation, the only minor trick is that a gap does not mean non-contiguous until you actually take something later.
Counts non-contiguous subsequences of sequences 1..20 in just over 0.5 seconds

`bool countonly = falseinteger count = 0 procedure ncs(sequence rest, integer ri=0, sequence taken={}, bool contig=false, bool gap=false)    if ri>=length(rest) then        if contig then            if countonly then                count += 1            else                ?taken            end if        end if    else        ri += 1        ncs(rest,ri,taken&rest[ri],gap,gap)        ncs(rest,ri,taken,contig,length(taken)!=0)    end ifend procedure ncs({1,2,3})?"==="ncs({1,2,3,4})?"==="countonly = trueatom t0 = time()sequence s = {}for i=1 to 20 do    count = 0    ncs(tagset(i))    s = append(s,count)end for?time()-t0?s`
Output:
```{1,3}
"==="
{1,2,4}
{1,3,4}
{1,3}
{1,4}
{2,4}
"==="
0.515
{0,0,1,5,16,42,99,219,466,968,1981,4017,8100,16278,32647,65399,130918,261972,524097,1048365}
```

## PicoLisp

Translation of: Scheme
`(de ncsubseq (Lst)   (let S 0      (recur (S Lst)         (ifn Lst            (and (>= S 3) '(NIL))            (let (X (car Lst)  XS (cdr Lst))               (ifn (bit? 1 S)  # even                  (conc                     (mapcar '((YS) (cons X YS))                        (recurse (inc S) XS) )                     (recurse S XS) )                  (conc                     (mapcar '((YS) (cons X YS))                        (recurse S XS) )                     (recurse (inc S) XS) ) ) ) ) ) ) )`

## Pop11

We modify classical recursive generation of subsets, using variables to keep track if subsequence is continuous.

`define ncsubseq(l);    lvars acc = [], gap_started = false, is_continuous = true;    define do_it(l1, l2);        dlocal gap_started;        lvars el, save_is_continuous = is_continuous;        if l2 = [] then            if not(is_continuous) then                cons(l1, acc) -> acc;            endif;        else            front(l2) -> el;            back(l2) -> l2;            not(gap_started) and is_continuous -> is_continuous;            do_it(cons(el, l1), l2);            save_is_continuous -> is_continuous;            not(l1 = []) or gap_started -> gap_started;            do_it(l1, l2);        endif;    enddefine;    do_it([], rev(l));    acc;enddefine; ncsubseq([1 2 3 4 5]) =>`
Output:
```[[1 3] [1 4] [2 4] [1 2 4] [1 3 4] [1 5] [2 5] [1 2 5] [3 5] [1 3 5]
[2 3 5] [1 2 3 5] [1 4 5] [2 4 5] [1 2 4 5] [1 3 4 5]]```

## PowerShell

`Function SubSequence ( [Array] \$S, [Boolean] \$all=\$false ){   \$sc = \$S.count   if( \$sc -gt ( 2 - [Int32] \$all ) ) {      [void] \$sc--      0..\$sc | ForEach-Object {         \$gap = \$_         "\$( \$S[ \$_ ] )"         if( \$gap -lt \$sc )         {            SubSequence ( ( \$gap + 1 )..\$sc | Where-Object { \$_ -ne \$gap } ) ( ( \$gap -ne 0 ) -or \$all ) | ForEach-Object {               [String]::Join( ',', ( ( [String]\$_ ).Split(',') | ForEach-Object {                  \$lt = \$true               } {                  if( \$lt -and ( \$_ -gt \$gap ) )                  {                     \$S[ \$gap ]                     \$lt = \$false                  }                  \$S[ \$_ ]               } {                  if( \$lt )                  {                     \$S[ \$gap ]                  }               }               ) )            }         }      }      #[String]::Join( ',', \$S)   } else {       \$S | ForEach-Object { [String] \$_ }    }} Function NonContinuous-SubSequence ( [Array] \$S ){   \$sc = \$S.count   if( \$sc -eq 3 )   {      [String]::Join( ',', \$S[ ( 0,2 ) ] )   } elseif ( \$sc -gt 3 ) {      [void] \$sc--      \$gaps = @()      \$gaps += ( ( NonContinuous-SubSequence ( 1..\$sc ) ) | ForEach-Object {         \$gap1 = ",\$_,"         "0,{0}" -f ( [String]::Join( ',', ( 1..\$sc | Where-Object { \$gap1 -notmatch "\$_," } ) ) )      } )      \$gaps += 1..( \$sc - 1 )      2..( \$sc - 1 ) | ForEach-Object {         \$gap2 = \$_ - 1         \$gaps += ( ( SubSequence ( \$_..\$sc ) ) | ForEach-Object {            "\$gap2,\$_"         } )      }      #Write-Host "S \$S gaps \$gaps"      \$gaps | ForEach-Object {         \$gap3 = ",\$_,"         "\$( 0..\$sc | Where-Object { \$gap3 -notmatch ",\$_," } | ForEach-Object {            \$S[\$_]         } )" -replace ' ', ','      }   } else {       \$null   }} ( NonContinuous-SubSequence 'a','b','c','d','e' ) | Select-Object length, @{Name='value';Expression={ \$_ } } | Sort-Object length, value | ForEach-Object { \$_.value }`

## Prolog

Works with SWI-Prolog.
We explain to Prolog how to build a non continuous subsequence of a list L, then we ask Prolog to fetch all the subsequences.

` % fetch all the subsequencesncsubs(L, LNCSL) :-	setof(NCSL, one_ncsubs(L, NCSL), LNCSL). % how to build one subsequenceone_ncsubs(L, NCSL) :-	extract_elem(L, NCSL);	(   sublist(L, L1),	    one_ncsubs(L1, NCSL)). % extract one element of the list% this element is neither the first nor the last.extract_elem(L, NCSL) :-	length(L, Len),	Len1 is Len - 2,	between(1, Len1, I),	nth0(I, L, Elem),	select(Elem, L, NCS1),	(   NCSL = NCS1; extract_elem(NCS1, NCSL)). % extract the first or the last element of the listsublist(L, SL) :-	(L = [_|SL];	reverse(L, [_|SL1]),	reverse(SL1, SL)). `

Example :

`?- ncsubs([a,e,i,o,u], L).L = [[a,e,i,u],[a,e,o],[a,e,o,u],[a,e,u],[a,i],[a,i,o],[a,i,o,u],[a,i,u],[a,o],[a,o,u],[a,u],[e,i,u],[e,o],[e,o,u],[e,u],[i,u]]`

## Python

Translation of: Scheme
`def ncsub(seq, s=0):    if seq:        x = seq[:1]        xs = seq[1:]        p2 = s % 2        p1 = not p2        return [x + ys for ys in ncsub(xs, s + p1)] + ncsub(xs, s + p2)    else:        return [[]] if s >= 3 else []`
Output:
```>>> ncsub(range(1, 4))
[[1, 3]]
>>> ncsub(range(1, 5))
[[1, 2, 4], [1, 3, 4], [1, 3], [1, 4], [2, 4]]
>>> ncsub(range(1, 6))
[[1, 2, 3, 5], [1, 2, 4, 5], [1, 2, 4], [1, 2, 5], [1, 3, 4, 5], [1, 3, 4],
[1, 3, 5], [1, 3], [1, 4, 5], [1, 4], [1, 5], [2, 3, 5], [2, 4, 5], [2, 4],
[2, 5], [3, 5]]```

A faster Python + Psyco JIT version:

`from sys import argvimport psyco def C(n, k):    result = 1    for d in xrange(1, k+1):        result *= n        n -= 1        result /= d    return result # http://oeis.org/A002662nsubs = lambda n: sum(C(n, k) for k in xrange(3, n+1)) def ncsub(seq):    n = len(seq)    result = [None] * nsubs(n)    pos = 0     for i in xrange(1, 2 ** n):        S  = []        nc = False        for j in xrange(n + 1):            k = i >> j            if k == 0:                if nc:                    result[pos] = S                    pos += 1                break            elif k % 2:                S.append(seq[j])            elif S:                nc = True    return result from sys import argvimport psycopsyco.full()n = 10 if len(argv) < 2 else int(argv[1])print len( ncsub(range(1, n)) )`

## R

The idea behind this is to loop over the possible lengths of subsequence, finding all subsequences then discarding those which are continuous.

`ncsub <- function(x){   n <- length(x)   a <- seq_len(n)   seqlist <- list()   for(i in 2:(n-1))   {      seqs <- combn(a, i)                                                          # Get all subseqs      ok <- apply(seqs, 2, function(x) any(diff(x)!=1))                            # Find noncts ones      newseqs <- unlist(apply(seqs[,ok], 2, function(x) list(x)), recursive=FALSE) # Convert matrix to list of its columns      seqlist <- c(seqlist, newseqs)                                               # Append to existing list    }   lapply(seqlist, function(index) x[index])}# Example usagencsub(1:4)ncsub(letters[1:5])`

## Racket

Take a simple subsets definition:

` (define (subsets l)  (if (null? l) '(())      (append (for/list ([l2 (subsets (cdr l))]) (cons (car l) l2))              (subsets (cdr l))))) `

since the subsets are returned in their original order, it is also a sub-sequences function.

Now add to it a "state" counter which count one for each chunk of items included or excluded. It's always even when we're in an excluded chunk (including the beginning) and odd when we're including items -- increment it whenever we switch from one kind of chunk to the other. This means that we should only include subsequences where the state is 3 (included->excluded->included) or more. Note that this results in code that is similar to the "Generalized monadic filter" entry, except a little simpler.

` #lang racket(define (non-continuous-subseqs l)  (let loop ([l l] [x 0])    (if (null? l) (if (>= x 3) '(()) '())        (append (for/list ([l2 (loop (cdr l) (if (even? x) (add1 x) x))])                  (cons (car l) l2))                (loop (cdr l) (if (odd? x) (add1 x) x))))))(non-continuous-subseqs '(1 2 3 4));; => '((1 2 4) (1 3 4) (1 3) (1 4) (2 4)) `

## REXX

This REXX version also works with non-numeric (alphabetic) items   (as well as numbers).

`/*REXX program lists all the  non─continuous subsequences  (NCS),  given a sequence.    */parse arg list                                   /*obtain the arguments from the  C. L. */if list='' | list==','  then list=1 2 3 4 5      /*Not specified?  Then use the default.*/say 'list=' space(list);        say              /*display the list to the terminal.    */w=words(list)                                    /*W:  is the number of items in list.  */\$=left(123456789, w)                             /*build a string of decimal digits.    */tail=right(\$, max(0, w-2))                       /*construct a fast tail for comparisons*/#=0                                              /* [↓]      L:   length of  Jth  item. */    do j=13  to left(\$,1) || tail;  L=length(j)  /*step through list (using smart start)*/    if verify(j, \$)\==0  then iterate            /*Not one of the chosen  (sequences) ? */    f=left(j,1)                                  /*use the fist decimal digit of  J.    */    NCS=0                                        /*there isn't a non─continuous subseq. */            do k=2  to L;      _=substr(j, k, 1) /*extract a single decimal digit of  J.*/            if _ <=  f    then iterate j         /*if next digit ≤, then skip this digit*/            if _ \== f+1  then NCS=1             /*it's OK as of now  (that is, so far).*/            f=_                                  /*now have a  new  next decimal digit. */            end   /*k*/     if \NCS  then iterate                        /*not OK?  Then skip this number (item)*/    #=#+1                                        /*Eureka!  We found a number (or item).*/    @=;     do m=1  for L                        /*build a sequence string to display.  */            @[email protected]  word(list, substr(j, m, 1))     /*pick off a number (item) to display. */            end   /*m*/     say 'a non─continuous subsequence: '    @    /*show the non─continuous subsequence. */    end         /*j*/sayif #==0  then #='no'                             /*make it look more gooder Angleshy.   */say  #  "non─continuous subsequence"s(#)     'were found.'exit                                             /*stick a fork in it,  we're all done. *//*──────────────────────────────────────────────────────────────────────────────────────*/s:  if arg(1)==1  then return '';     return word(arg(2) 's', 1)           /*pluralizer.*/`

output   when using the input:   1 2 3 4

```list= 1 2 3 4

a non-continuous subsequence:  1 3
a non-continuous subsequence:  1 4
a non-continuous subsequence:  2 4
a non-continuous subsequence:  1 2 4
a non-continuous subsequence:  1 3 4

5 non-continuous subsequences were found.
```

output   when using the following input:   a e I o u

```list= a e I o u

a non-continuous subsequence:  a I
a non-continuous subsequence:  a o
a non-continuous subsequence:  a u
a non-continuous subsequence:  e o
a non-continuous subsequence:  e u
a non-continuous subsequence:  I u
a non-continuous subsequence:  a e o
a non-continuous subsequence:  a e u
a non-continuous subsequence:  a I o
a non-continuous subsequence:  a I u
a non-continuous subsequence:  a o u
a non-continuous subsequence:  e I u
a non-continuous subsequence:  e o u
a non-continuous subsequence:  a e I u
a non-continuous subsequence:  a e o u
a non-continuous subsequence:  a I o u

16 non-continuous subsequences were found.
```

output   when using the following [channel Islands (Great Britain)] as input:   Alderney Guernsey Herm Jersey Sark

```list= Alderney Guernsey Herm Jersey Sark

a non-continuous subsequence:  Alderney Herm
a non-continuous subsequence:  Alderney Jersey
a non-continuous subsequence:  Alderney Sark
a non-continuous subsequence:  Guernsey Jersey
a non-continuous subsequence:  Guernsey Sark
a non-continuous subsequence:  Herm Sark
a non-continuous subsequence:  Alderney Guernsey Jersey
a non-continuous subsequence:  Alderney Guernsey Sark
a non-continuous subsequence:  Alderney Herm Jersey
a non-continuous subsequence:  Alderney Herm Sark
a non-continuous subsequence:  Alderney Jersey Sark
a non-continuous subsequence:  Guernsey Herm Sark
a non-continuous subsequence:  Guernsey Jersey Sark
a non-continuous subsequence:  Alderney Guernsey Herm Sark
a non-continuous subsequence:  Alderney Guernsey Jersey Sark
a non-continuous subsequence:  Alderney Herm Jersey Sark

16 non-continuous subsequences were found.
```

output   when using the following [six noble gases] as input:   helium neon argon krypton xenon radon

```list= helium neon argon krypton xenon radon

a non-continuous subsequence:  helium argon
a non-continuous subsequence:  helium krypton
a non-continuous subsequence:  helium xenon
a non-continuous subsequence:  neon krypton
a non-continuous subsequence:  neon xenon
a non-continuous subsequence:  argon xenon
a non-continuous subsequence:  helium neon krypton
a non-continuous subsequence:  helium neon xenon
a non-continuous subsequence:  helium neon radon
a non-continuous subsequence:  helium argon krypton
a non-continuous subsequence:  helium argon xenon
a non-continuous subsequence:  helium argon radon
a non-continuous subsequence:  helium krypton xenon
a non-continuous subsequence:  helium krypton radon
a non-continuous subsequence:  helium xenon radon
a non-continuous subsequence:  neon argon xenon
a non-continuous subsequence:  neon argon radon
a non-continuous subsequence:  neon krypton xenon
a non-continuous subsequence:  neon krypton radon
a non-continuous subsequence:  neon xenon radon
a non-continuous subsequence:  argon krypton radon
a non-continuous subsequence:  argon xenon radon
a non-continuous subsequence:  helium neon argon xenon
a non-continuous subsequence:  helium neon argon radon
a non-continuous subsequence:  helium neon krypton xenon
a non-continuous subsequence:  helium neon krypton radon
a non-continuous subsequence:  helium neon xenon radon
a non-continuous subsequence:  helium argon krypton xenon
a non-continuous subsequence:  helium argon krypton radon
a non-continuous subsequence:  helium argon xenon radon
a non-continuous subsequence:  helium krypton xenon radon
a non-continuous subsequence:  neon argon krypton radon
a non-continuous subsequence:  neon argon xenon radon
a non-continuous subsequence:  neon krypton xenon radon
a non-continuous subsequence:  helium neon argon krypton radon
a non-continuous subsequence:  helium neon argon xenon radon
a non-continuous subsequence:  helium neon krypton xenon radon
a non-continuous subsequence:  helium argon krypton xenon radon

42 non-continuous subsequences were found.
```

## Ring

` # Project : Non-continuous subsequences# Date    : 2018/02/03# Author : Gal Zsolt [~ CalmoSoft ~]# Email   : <[email protected]> load "stdlib.ring"list = [1,2,3,4]items = newlist(pow(2,len(list))-1,len(list))see "For [1, 2, 3, 4] non-continuous subsequences are:" + nlpowerset(list,4)showarray(items,4)see nl list = [1,2,3,4,5]items = newlist(pow(2,len(list))-1,len(list))see "For [1, 2, 3, 4, 5] non-continuous subsequences are:" + nlpowerset(list,5)showarray(items,5) func showarray(items,ind)        for n = 1 to len(items)             flag = 0             for m = 1 to ind - 1                  if items[n][m] = 0 or items[n][m+1] = 0                     exit                 ok                    if (items[n][m] + 1) != items[n][m+1]                     flag = 1                     exit                 ok            next            if flag = 1               see "["                str = ""               for x = 1 to len(items[n])                    if items[n][x] != 0                         str = str + items[n][x] + " "                    ok               next                 str = left(str, len(str) - 1)                 see str + "]" + nl            ok        next func powerset(list,ind)        num = 0        num2 = 0        items = newlist(pow(2,len(list))-1,ind)        for i = 2 to (2 << len(list)) - 1 step 2             num2 = 0             num = num + 1             for j = 1 to len(list)                   if i & (1 << j)                      num2 = num2 + 1                      if list[j] != 0                        items[num][num2] = list[j]                     ok                  ok             next        next        return items `

Output:

```For [1, 2, 3, 4] non-continuous subsequences are:
[1 3]
[1 4]
[2 4]
[1 2 4]
[1 3 4]

For [1, 2, 3, 4, 5] non-continuous subsequences are:
[1 3]
[1 4]
[2 4]
[1 2 4]
[1 3 4]
[1 5]
[2 5]
[1 2 5]
[3 5]
[1 3 5]
[2 3 5]
[1 2 3 5]
[1 4 5]
[2 4 5]
[1 2 4 5]
[1 3 4 5]
```

## Ruby

Translation of: Tcl

Uses code from Power Set.

`class Array  def func_power_set    inject([[]]) { |ps,item|    # for each item in the Array      ps +                      # take the powerset up to now and add      ps.map { |e| e + [item] } # it again, with the item appended to each element    }  end   def non_continuous_subsequences    func_power_set.reject {|seq| continuous?(seq)}  end   def continuous?(seq)    seq.each_cons(2) {|a, b| return false if a.succ != b}    true  endend p (1..3).to_a.non_continuous_subsequencesp (1..4).to_a.non_continuous_subsequencesp (1..5).to_a.non_continuous_subsequencesp ("a".."d").to_a.non_continuous_subsequences`
Output:
```[[1, 3]]
[[1, 3], [1, 4], [2, 4], [1, 2, 4], [1, 3, 4]]
[[1, 3], [1, 4], [2, 4], [1, 2, 4], [1, 3, 4], [1, 5], [2, 5], [1, 2, 5], [3, 5], [1, 3, 5],
[2, 3, 5], [1, 2, 3, 5], [1, 4, 5], [2, 4, 5], [1, 2, 4, 5], [1, 3, 4, 5]]
[["a", "c"], ["a", "d"], ["b", "d"], ["a", "b", "d"], ["a", "c", "d"]]
```

It is not the value of the array element and when judging continuation in the position, it changes as follows.

`class Array  def continuous?(seq)    seq.each_cons(2) {|a, b| return false if index(a)+1 != index(b)}    true  endend p %w(a e i o u).non_continuous_subsequences`
Output:
`[["a", "i"], ["a", "o"], ["e", "o"], ["a", "e", "o"], ["a", "i", "o"], ["a", "u"], ["e", "u"], ["a", "e", "u"], ["i", "u"], ["a", "i", "u"], ["e", "i", "u"], ["a", "e", "i", "u"], ["a", "o", "u"], ["e", "o", "u"], ["a", "e", "o", "u"], ["a", "i", "o", "u"]]`

## Scheme

`(define (ncsubseq lst)  (let recurse ((s 0)                (lst lst))    (if (null? lst)        (if (>= s 3)            '(())            '())        (let ((x (car lst))              (xs (cdr lst)))          (if (even? s)              (append               (map (lambda (ys) (cons x ys))                    (recurse (+ s 1) xs))               (recurse s xs))              (append               (map (lambda (ys) (cons x ys))                    (recurse s xs))               (recurse (+ s 1) xs)))))))`
Output:
```> (ncsubseq '(1 2 3))
((1 3))
> (ncsubseq '(1 2 3 4))
((1 2 4) (1 3 4) (1 3) (1 4) (2 4))
> (ncsubseq '(1 2 3 4 5))
((1 2 3 5) (1 2 4 5) (1 2 4) (1 2 5) (1 3 4 5) (1 3 4) (1 3 5) (1 3) (1 4 5) (1 4) (1 5) (2 3 5) (2 4 5) (2 4) (2 5) (3 5))```

## Seed7

`\$ include "seed7_05.s7i"; const func array bitset: ncsub (in bitset: seq, in integer: s) is func  result    var array bitset: subseq is 0 times {};  local    var bitset: x is {};    var bitset: xs is {};    var bitset: ys is {};  begin    if seq <> {} then      x := {min(seq)};      xs := seq - x;      for ys range ncsub(xs, s + 1 - s rem 2) do        subseq &:= x | ys;      end for;      subseq &:= ncsub(xs, s + s rem 2);    elsif s >= 3 then      subseq &:= {};    end if;  end func; const proc: main is func  local    var bitset: seq is {};  begin    for seq range ncsub({1, 2, 3, 4}, 0) do      writeln(seq);    end for;  end func;`
Output:
```{1, 2, 4}
{1, 3, 4}
{1, 3}
{1, 4}
{2, 4}
```

## Sidef

Translation of: Perl
`func non_continuous(min, max, subseq=[], has_gap=false) {     static current = [];     range(min, max).each { |i|        current.push(i);        has_gap && subseq.append([current...]);        i < max && non_continuous(i.inc, max, subseq, has_gap);        current.pop;        has_gap = current.len;    }     subseq;} say non_continuous(1, 3);say non_continuous(1, 4);say non_continuous("a", "d");`
Output:
```[[1, 3]]
[[1, 2, 4], [1, 3], [1, 3, 4], [1, 4], [2, 4]]
[["a", "b", "d"], ["a", "c"], ["a", "c", "d"], ["a", "d"], ["b", "d"]]
```

## Standard ML

`fun fence s [] =      if s >= 3 then        [[]]      else        []   | fence s (x :: xs) =      if s mod 2 = 0 then        map          (fn ys => x :: ys)          (fence (s + 1) xs)        @          fence s xs      else        map          (fn ys => x :: ys)          (fence s xs)        @          fence (s + 1) xs fun ncsubseq xs = fence 0 xs`
Output:
```- ncsubseq [1,2,3];
val it = [[1,3]] : int list list
- ncsubseq [1,2,3,4];
val it = [[1,2,4],[1,3,4],[1,3],[1,4],[2,4]] : int list list
- ncsubseq [1,2,3,4,5];
val it =
[[1,2,3,5],[1,2,4,5],[1,2,4],[1,2,5],[1,3,4,5],[1,3,4],[1,3,5],[1,3],
[1,4,5],[1,4],[1,5],[2,3,5],...] : int list list```

## Tcl

This Tcl implementation uses the subsets function from Power Set, which is acceptable as that conserves the ordering, as well as a problem-specific test function is_not_continuous and a generic list filter lfilter:

` proc subsets l {     set res [list [list]]     foreach e \$l {         foreach subset \$res {lappend res [lappend subset \$e]}     }     return \$res } proc is_not_continuous seq {     set last [lindex \$seq 0]     foreach e [lrange \$seq 1 end] {         if {\$e-1 != \$last} {return 1}         set last \$e     }     return 0 } proc lfilter {f list} {     set res {}     foreach i \$list {if [\$f \$i] {lappend res \$i}}     return \$res } % lfilter is_not_continuous [subsets {1 2 3 4}]{1 3} {1 4} {2 4} {1 2 4} {1 3 4}`

## Ursala

To do it the lazy programmer way, apply the powerset library function to the list, which will generate all continuous and non-continuous subsequences of it, and then delete the subsequences that are also substrings (hence continuous) using a judicious combination of the built in substring predicate (K3), negation (Z), and distributing filter (K17) operator suffixes. This function will work on lists of any type. To meet the requirement for structural equivalence, the list items are first uniquely numbered (num), and the numbers are removed afterwards (rSS).

`#import std noncontinuous = num; ^rlK3ZK17rSS/~& powerset #show+ examples = noncontinuous 'abcde'`
Output:
```abce
abd
abde
abe
ac
acd
acde
ace
ae
bce
bd
bde
be
ce```

## zkl

Translation of: JavaScript
`fcn non_continuous_subsequences(ary){   pwerSet(ary).filter(fcn(list){(not isContinuous(list)) })}fcn isContinuous(ary){   if(ary.len()<2) return(True);   foreach n in (ary.len()-1){ if(1+ary[n]!=ary[n+1]) return(False); }   return(True);}non_continuous_subsequences(T(1,2,3,4)).println();`
`fcn pwerSet(list){  (0).pump(list.len(),List,List,Utils.Helpers.pickNFrom.fp1(list),     T(T,Void.Write,Void.Write) ) .append(list)}`
`fcn brokenSubsequences(str){   pwerSet(str.split("")).apply("concat")   .filter('wrap(substr){ (not str.holds(substr)) })}brokenSubsequences("1234").println();`
Output:
```L(L(1,3),L(1,4),L(2,4),L(1,2,4),L(1,3,4))
L("13","14","24","124","134")
```