Add partial derangements (#78)

Author: Sedictious

README.md

@@ -18,6 +18,7 @@ This library provides the following functions:
  - `combinations(a,n)`: returns all combinations of `n` elements of indexable object `a`;
  - `combinations(a)`: returns combinations of all order by chaining calls to `combinations(a,n)`;
  - `derangement(n)`/`subfactorial(n)`: returns the number of permutations of n with no fixed points; always returns a `BigInt`;
+ - `partialderangement(n, k)`: returns the number of permutations of n with exactly k fixed points; always returns a `BigInt`;
  - `doublefactorial(n)`: returns the double factorial n!!; always returns a `BigInt`;
  - `fibonaccinum(n)`: the n-th Fibonacci number; always returns a `BigInt`;
  - `hyperfactorial(n)`: the n-th hyperfactorial, i.e. prod([i^i for i = 2:n]; always returns a `BigInt`;

src/factorials.jl

@@ -2,6 +2,7 @@
 
 export
     derangement,
+    partialderangement,
     factorial,
     subfactorial,
     doublefactorial,
@@ -52,6 +53,23 @@ function doublefactorial(n::Integer)
     return z[]
 end
 
+"""
+    partialderangement(n, k)
+    
+Compute the number of permutations of `n` with exactly k fixed points.
+"""
+function partialderangement(n::Integer, k::Integer)
+    if n < 0
+        throw(DomainError(n))
+    end
+    if k < 0 || k > n
+        throw(DomainError(k))
+    end
+    a = BigInt(n)
+    b = BigInt(k)
+    return subfactorial(a - b) * binomial(a, b)
+end
+
 # Hyperfactorial
 hyperfactorial(n::Integer) = prod(i->i^i, BigInt(2):n)
 

test/factorials.jl

@@ -11,6 +11,11 @@
 @test derangement(4) == subfactorial(4) == 9
 @test derangement(24) == parse(BigInt,"228250211305338670494289")
 
+# partialderangement
+@test partialderangement(7, 3) == 315
+@test_throws DomainError partialderangement(8, 9)
+@test_throws DomainError partialderangement(-8, 0)
+
 # doublefactorial
 @test doublefactorial(70) == parse(BigInt,"355044260642859198243475901411974413130137600000000")
 @test_throws DomainError doublefactorial(-1)