Ordered partitions
You are encouraged to solve this task according to the task description, using any language you may know.
In this task we want to find the ordered partitions into fixed-size blocks. This task is related to [Combinations] in that it has to do with discrete mathematics and moreover a helper function to compute combinations is (probably) needed to solve this task.
partitions(arg1,arg2,...,argn)
should generate all distributions of the
elements in {1,...,sum{arg1,arg2,...,argn}}
into #{arg1,arg2,...,argn}
blocks of respective size arg1,arg2,...,argn
.
E.g. partitions(2,0,2)
would create:
{({1, 2}, {}, {3, 4}), ({1, 3}, {}, {2, 4}), ({1, 4}, {}, {2, 3}), ({2, 3}, {}, {1, 4}), ({2, 4}, {}, {1, 3}), ({3, 4}, {}, {1, 2})}
E.g. partitions(1,1,1)
would create:
{({1}, {2}, {3}), ({1}, {3}, {2}), ({2}, {1}, {3}), ({2}, {3}, {1}), ({3}, {1}, {2}), ({3}, {2}, {1})}
Note that the number of elements in the list is
(arg1+arg2+...+argn choose arg1) * (arg2+arg3+...+argn choose arg2) * ... * (argn choose argn)
(i.e. the multinomial coefficient). Also, partitions(1,1,1)
creates the permutations of {1,2,3}
and thus there would be 3! = 6
elements in the list.
Note: Do not use functions that are not in the standard library of the
programming language you use. Your file should be written so that it can
be executed on the command line and by default outputs the result of
partitions(2,0,2)
. If the programming language does not support polyvariadic
functions pass a list as an argument.
Notation
Remarks on the used notation for the task in order to understand it easierly.
{1,...,n}
denotes the set of consecutive numbers from 1
to n
, e.g.
{1,2,3}
if n = 3
. sum
is the function that takes as its sole argument
a set of natural numbers and computes the sum of the numbers, e.g.
sum{1,2,3} = 6
. arg1,arg2,...,argn
are the arguments - natural numbers -
that the sought function receives. The operator #
returns the number of
elements in a set, e.g. #{1,2,3} = 3
. ()
is a tuple, e.g. (1,2,3)
.
Haskell
<lang haskell> comb :: Int -> [a] -> a comb 0 _ = [[]] comb _ [] = [] comb k (x:xs) = [ x:cs | cs <- comb (k-1) xs ] ++ comb k xs
partitions :: [Int] -> [[[Int]]] partitions xs = p [1..sum xs] xs
where p _ [] = [[]] p xs (k:ks) = [ cs:rs | cs <- comb k xs, rs <- p (xs `minus` cs) ks ] minus xs ys = [ x | x <- xs, not $ x `elem` ys ]
main = do putStrLn $ show $ partitions [2,0,2] </lang>
An alternative where minus
is not needed anymore because comb
now not only
keeps the chosen elements but also the not chosen elements together in a tuple.
<lang haskell> comb :: Int -> [a] -> [([a],[a])] comb 0 xs = [([],xs)] comb _ [] = [] comb k (x:xs) = [ (x:cs,zs) | (cs,zs) <- comb (k-1) xs ] ++
[ (cs,x:zs) | (cs,zs) <- comb k xs ]
partitions :: [Int] -> [[[Int]]] partitions xs = p [1..sum xs] xs
where p _ [] = [[]] p xs (k:ks) = [ cs:rs | (cs,zs) <- comb k xs, rs <- p zs ks ]
main = do putStrLn $ show $ partitions [2,0,2] </lang>
Lua
A pretty verbose solution. Maybe somebody can replace with something terser/better.
<lang lua> --- Create a list {1,...,n}. local function range(n)
local res = {} for i=1,n do res[i] = i end return res
end
--- Return true if the element x is in t. local function isin(t, x)
for _,x_t in ipairs(t) do if x_t == x then return true end end return false
end
--- Return the sublist from index u to o (inclusive) from t. local function slice(t, u, o)
local res = {} for i=u,o do res[#res+1] = t[i] end return res
end
--- Compute the sum of the elements in t. -- Assume that t is a list of numbers. local function sum(t)
local s = 0 for _,x in ipairs(t) do s = s + x end return s
end
--- Generate all combinations of t of length k (optional, default is #t). local function combinations(m, r)
local function combgen(m, n) if n == 0 then coroutine.yield({}) end for i=1,#m do if n == 1 then coroutine.yield({m[i]}) else for m0 in coroutine.wrap(function() combgen(slice(m, i+1, #m), n-1) end) do coroutine.yield({m[i], unpack(m0)}) end end end end return coroutine.wrap(function() combgen(m, r) end)
end
--- Generate a list of partitions into fized-size blocks. local function partitions(...)
local function helper(s, ...) local args = {...} if #args == 0 then return {% templatetag openvariable %}{% templatetag closevariable %} end local res = {} for c in combinations(s, args[1]) do local s0 = {} for _,x in ipairs(s) do if not isin(c, x) then s0[#s0+1] = x end end for _,r in ipairs(helper(s0, unpack(slice(args, 2, #args)))) do res[#res+1] = {{unpack(c)}, unpack(r)} end end return res end return helper(range(sum({...})), ...)
end
-- Print the solution io.write "[" local parts = partitions(2,0,2) for i,tuple in ipairs(parts) do
io.write "(" for j,set in ipairs(tuple) do io.write "{" for k,element in ipairs(set) do io.write(element) if k ~= #set then io.write(", ") end end io.write "}" if j ~= #tuple then io.write(", ") end end io.write ")" if i ~= #parts then io.write(", ") end
end io.write "]" io.write "\n" </lang>
Python
<lang python> from itertools import combinations
def partitions(*args):
def p(s, *args): if not args: return [[]] res = [] for c in combinations(s, args[0]): s0 = [x for x in s if x not in c] for r in p(s0, *args[1:]): res.append([c] + r) return res s = range(sum(args)) return p(s, *args)
print partitions(2, 0, 2) </lang>
An equivalent but terser solution.
<lang python> from itertools import combinations as comb
def partitions(*args):
def minus(s1, s2): return [x for x in s1 if x not in s2] def p(s, *args): if not args: return [[]] return [[c] + r for c in comb(s, args[0]) for r in p(minus(s, c), *args[1:])] return p(range(1, sum(args) + 1), *args)
print partitions(2, 0, 2) </lang>