Math.Combinat.Numbers

A few important number sequences.

See the "On-Line Encyclopedia of Integer Sequences", http://www.research.att.com/~njas/sequences/ .

Documentation

paritySign :: Integral a => a -> Integer

(-1)^k

factorial :: Integral a => a -> Integer

A000142.

doubleFactorial :: Integral a => a -> Integer

A006882.

binomial :: Integral a => a -> a -> Integer

A007318.

multinomial :: Integral a => [a] -> Integer

Catalan numbers

catalan :: Integral a => a -> Integer

Catalan numbers. OEIS:A000108.

catalanTriangle :: Integral a => a -> a -> Integer

Catalan's triangle. OEIS:A009766. Note:

 catalanTriangle n n == catalan n
 catalanTriangle n k == countStandardYoungTableaux (toPartition [n,k])

Stirling numbers

signedStirling1stArray :: Integral a => a -> Array Int Integer

Rows of (signed) Stirling numbers of the first kind. OEIS:A008275. Coefficients of the polinomial (x-1)*(x-2)*...*(x-n+1). This function uses the recursion formula.

signedStirling1st :: Integral a => a -> a -> Integer

(Signed) Stirling numbers of the first kind. OEIS:A008275. This function uses signedStirling1stArray, so it shouldn't be used to compute many Stirling numbers.

unsignedStirling1st :: Integral a => a -> a -> Integer

(Unsigned) Stirling numbers of the first kind. See signedStirling1st.

stirling2nd :: Integral a => a -> a -> Integer

Stirling numbers of the second kind. OEIS:A008277. This function uses an explicit formula.

Bernoulli numbers

bernoulli :: Integral a => a -> Rational

Bernoulli numbers. bernoulli 1 == -1%2 and bernoulli k == 0 for k>2 and odd. This function uses the formula involving Stirling numbers of the second kind. Numerators: A027641, denominators: A027642.

Power series

coinSeries :: [Int] -> [Integer]

Power series expansion of

 @1 / ( (1-x^a_1) * (1-x^a_2) * ... * (1-x^a_n) )@

Example:

(coinSeries [2,3,5]) !! k is the number of ways to pay k dollars with coins of two, three and five dollars.

TODO: better name?