Version 27 of Permutations

Updated 2008-04-16 06:40:17 by arjen

SS 16Apr2004: I wrote these two functions some time ago because someone posted in comp.lang.python a related question, i.e. just for fun, but probably this can be of real use. The functions are useful to compute the permutations of a set, actually you in theory only need [permutations] - that returns a list of permutations of the input set (also a Tcl list), but it's very memory-consuming to generate the permutations and then use foreach against it, so there is another function called [foreach-permutation] that runs all the permutations of the set and run a script against every one. The code is Copyright(C) 2004 Salvatore Sanfilippo, and is under the Tcl 8.4 license.


 % !source
 source antilib.tcl 
 % permutations {a b c}
 {a b c} {a c b} {b a c} {b c a} {c a b} {c b a}
 % foreach-permutation p {a b c} {puts $p}
 a b c
 a c b
 b a c
 b c a
 c a b
 c b a


 # Return a list with all the permutations of elements in list 'items'.
 # Example: permutations {a b} > {{a b} {b a}}
 proc permutations items {
     set l [llength $items]
     if {[llength $items] < 2} {
        return $items
     } else {
        for {set j 0} {$j < $l} {incr j} {
            foreach subcomb [permutations [lreplace $items $j $j]] {
                lappend res [concat [lindex $items $j] $subcomb]
        return $res

 # Like foreach but call 'body' for every permutation of the elements
 # in the list 'items', setting the variable 'var' to the permutation.
 # Example: foreach-permutation x {a b} {puts $x}
 # Will output:
 # a b          
 # b a
 proc foreach-permutation {var items body} {
     set l [llength $items]
     if {$l < 2} {
        uplevel [list set $var [lrange $items 0 0]]
        uplevel $body
     } else {
        for {set j 0} {$j < $l} {incr j} {
            foreach-permutation subcomb [lreplace $items $j $j] {
                uplevel [list set $var [concat [lrange $items $j $j] $subcomb]]
                uplevel $body

RS: Here's my take, a tiny hot summer day balcony fun project:

 proc permute {list {prefix ""}} {
    if ![llength $list] {return [list $prefix]}
    set res {}
    foreach e $list {
        lappend res [permute [l- $list $e] [concat $prefix $e]]
    join $res
 proc l- {list e} {
    set pos [lsearch $list $e]
    lreplace $list $pos $pos

Lars H: RS's permute, tidied up:

 proc permute {list {prefix ""}} {
    if {![llength $list]} then {return [list $prefix]}
    set res [list]
    set n 0
    foreach e $list {
        eval [list lappend res]\
          [permute [lreplace $list $n $n] [linsert $prefix end $e]]
        incr n
    return $res

Vince says that once upon a time he came across a very clever algorithm which had a state-based approach to generating permutations, which worked something like this:

    while {[getNextPermutation privateState permVar]} {
        puts $permVar

i.e. the procedure maintains some state 'privateState' which allows it to iterate through the permutations one by one, and therefore avoids the need to pass in a $script as in the foreach-permutation example above. Unfortunately, I can't remember that algorithm right now...

RS: nice challenge! I don't claim to have the optimal variant (in fact, this is terribly slow - see timings below), but the following code snippets work for me. If you normalize the set to be permuted to integers 0 < i < 10, you can write each permutation as a decimal number by just slapping the digits together. These decimals are ordered increasingly, with deltas that seem always to be multiples of 9.

 1 2 3 4 -> 1234
 1 2 4 3 -> 1243

First, a function to check whether an integer is a "perm" number, i.e. contains all the digits 1..n, where n is its number of digits:

 proc isperm x {
   foreach i [iota [string length $x]] {
      if ![contains [incr i] $x] {return 0}
   return 1

A little integer range generator - [iota 4] -> {0 1 2 3}

 proc iota n {
   set res {}
   for {set i 0} {$i<$n} {incr i} {lappend res $i}
   set res

A tight wrapper for substring containment:

 proc contains {substr string} {expr {[string first $substr $string]>=0}}

So given one permutation number, we add 9 as often as needed to reach the next perm number

 proc nextperm x {
   set length [string length $x]
   while 1 {
      if [isperm [incr x 9]] {return $x}
      if {[string length $x]>$length} return 

When at end (no further permutation possible), an empty string is returned.

KBK 2005-02-17 : In response to a request from RS, here's a pair of procedures that return the lexicographically first permutation of a set of elements, and the lexicographically next permutation given the current permutation. The general principle is:

  • Empty lists and singletons have no "next permutation".
  • If you have a permutation {p q r s t}, the preferred choice for the "next permutation" is p, followed by the "next permutation" of {q r s t}.
  • If {q r s t} doesn't have a "next permutation", then the next possibility is to choose the element after p in sequence (without loss of generality, let it be q), and return it followed by the lexicographically first permutation of {p r s t}.
  • If p was the greatest element in the set {p q r s t}, and the above two tests fail, then we have the last permutation.
 # Procedure that forms the lexicographically first permutation of a list of
 # elements.

 proc firstperm { list } {
     lsort $list

 # Procedure that accepts a permutation of a set of elements and returns
 # the next permutatation in lexicographic sequence.  The optional
 # "partial" arg is a list of elements that is prepended to the return
 # value.

 proc nextperm { perm { partial {}} } {

     # If a permutation is of a single element, there's no
     # "next permutation."

     if { [llength $perm] <= 1 } {
         return {}

     # Try to hold the first element fixed, and make the "next permutation"
     # of the remaining elements.

     set first [lindex $perm 0]
     set p2 $partial
     lappend p2 $first
     set next [nextperm [lrange $perm 1 end] $p2]
     if {[llength $next] > 0} {
         return $next

     # If the remaining elements were in descending sequence (that is,
     # were the last permutation of those elements), choose the
     # lexicographically next "first element". Fail if the "first element"
     # of the permutation was the lexicographically first.

     set elements [lsort $perm]
     set idx [lsearch -exact $elements $first]
     incr idx
     if { $idx >= [llength $elements] } {
         return {}

     # Place the new first element at the head of the permutation, and
     # follow with the remaining choices in ascending order.

     set ret $partial
     lappend ret [lindex $elements $idx]
     foreach e [lreplace $elements $idx $idx] {
         lappend ret $e
     return $ret


 # Demonstration - permute four elements

 for { set p [firstperm {alfa bravo charlie delta}] } \
     { [llength $p] > 0 } \
     { set p [nextperm $p] } \
         puts $p

RS: Ah, Kevin beat me to it... Here's my solution, based on the observation that reorganisation starts at the last pair of ascending neighbors - from that position, the minimal element greater than the smaller neighbor is moved to front, and the rest sorted:

 proc nextperm perm {
    #-- determine last ascending neighbors
    set last ""
    for {set i 0} {$i<[llength $perm]-1} {incr i} {
        if {[lindex $perm $i]<[lindex $perm [expr {$i+1}]]} {
            set last $i
    if {$last ne ""} {
        set pivot [lindex $perm $last]
        #-- find smallest successor greater than pivot
        set successors [lrange $perm $last end]
        set minSucc ""
        foreach i $successors {
            if {$i>$pivot && ($minSucc eq "" || $i<$minSucc)} {
                set minSucc $i
        concat [lrange $perm 0 [expr {$last-1}]] [list $minSucc] \
            [lsort [l- $successors $minSucc]]

This generally useful function removes an element from a list by value:

 proc l- {list element} {
    set pos [lsearch -exact $list $element]
    lreplace $list $pos $pos

The code passes both numeric and non-numeric tests, producing the ordered sequence of permutations like Kevin's test:

 for {set set [lsort {Tom Dick Harry Bob}]} {$set ne ""} {} {
    puts $set; set set [nextperm $set]
 for {set set [lsort {1 2 3 4}]} {$set ne ""} {} {
    puts $set; set set [nextperm $set]

So let's compare the two versions! (I renamed them by author)

 proc try {cmd set} {
    for {set perm [lsort $set]} {[llength $perm]} {} {set perm [$cmd $perm]} 
 % time {try nextpermRS {1 2 3 4}} 100
 21001 microseconds per iteration
 % time {try nextpermKBK {1 2 3 4}} 100
 18305 microseconds per iteration

After replacing two info exists tests with comparison against initial "", I however get

 % time {try nextpermRS {1 2 3 4}} 100
 14206 microseconds per iteration

Vince: Wow, that's great! It would be good to place one of these procedures in tcllib as a permutation iterator...

RS: The morning after this "shootout", I wanted to give the initial integer-based solution a try too. I changed it to accept and return a list:

 proc nextperm0 perm  {
   set length [llength $perm]
   set x [join $perm ""]
   while {[string length $x]==$length} {
      if [isperm [incr x 9]] {return [split $x ""]}

But the timing was devastating, so better don't use this if time matters:

 % time {try nextperm0 {1 2 3 4}} 100
 508775 microseconds per iteration

35 times slower than nextpermRS ... But back to Vince's original request for a permutation iterator, it's now easily done (and doesn't even require a separate state variable):

 proc getNextPerm _perm {
    upvar 1 $_perm perm
    set perm [nextpermRS $perm]
    expr {[llength $perm]>0}

#-- Test:

 % set perm {1 2 3}
 1 2 3
 % while {[getNextPerm perm]} {puts $perm}
 1 3 2
 2 1 3
 2 3 1
 3 1 2
 3 2 1

KBK 2005-02-18 As always, Donald Knuth seems to have the last word on the subject. The very first algorithm in (Fascicle 2b of ''The Art of Computer Programming' volume 4) generates all permutations, using a "next permutation' method that appears quite nice indeed. Converted to Tcl, it looks like:

 # Procedure that accepts a permutation of a set of elements and returns
 # the next permutatation in lexicographic sequence. 

 proc nextperm { perm } {

     # Find the smallest subscript j such that we have already visited
     # all permutations beginning with the first j elements.

     set j [expr { [llength $perm] - 1 }]
     set ajp1 [lindex $perm $j]
     while { $j > 0 } {
         incr j -1
         set aj [lindex $perm $j]
         if { [string compare $ajp1 $aj] > 0 } {
             set foundj {}
         set ajp1 $aj
     if { ![info exists foundj] } return

     # Find the smallest element greater than the j'th among the elements
     # following aj. Let its index be l, and interchange aj and al.

     set l [expr { [llength $perm] - 1 }]
     while { $aj >= [set al [lindex $perm $l]] } {
         incr l -1
     lset perm $j $al
     lset perm $l $aj

     # Reverse a_j+1 ... an

     set k [expr {$j + 1}]
     set l [expr { [llength $perm] - 1 }]
     while { $k < $l } {
         set al [lindex $perm $l]
         lset perm $l [lindex $perm $k]
         lset perm $k $al
         incr k
         incr l -1

     return $perm


RS considers factoring out the swapping of two list elements like this:

 proc lswap {_list i j} {
    upvar $_list list
    if {$i != $j} {
       set tmp      [lindex $list $i]
       lset list $i [lindex $list $j]
       lset list $j $tmp
    } else {set list}

See also (but no nextperm algorithm there)

AM Here is a small experiment with Group theory and permutations

KPV Recently, I needed the related function of Selection: all k size subsets of n elements. Here's a recursive function that returns the complete list.

 proc Select {k l} {
    if {$k == 0} {return {}}
    if {$k == [llength $l]} {return [list $l]}

    set all {}
    incr k -1
    for {set i 0} {$i < [llength $l]-$k} {incr i} {
        set first [lindex $l $i]
        if {$k == 0} {
            lappend all $first
        } else {
            foreach s [Select $k [lrange $l [expr {$i+1}] end]] {
                set ans [concat $first $s]
                lappend all $ans
    return $all

Are the above procedures truly generating "all permutations"? Shouldn't subsets (results that don't contain "all" the numbers in the original) be generated as well? For example:

123 132 231 213 321 312 12 13 21 23 31 32 1 2 3

Or is that defined as something else?

escargo 15 Apr 2008 - admits to both definitions.

AM (16 april 2008) - on the other hand does not: it clearly speaks of permutations as a one-to-one map from a set to itself. This excludes mappings of a set to a proper subset, like M: {1, 2, 3} -> {1, 2}

Category Mathematics