Some basic power series expansions. This module is not re-exported by Math.Combinat.

Note: the "`convolveWithXXX`

" functions are much faster than the equivalent
`(XXX `convolve`)`

!

TODO: better names for these functions.

- unitSeries :: Num a => [a]
- convolve :: Num a => [a] -> [a] -> [a]
- convolveMany :: Num a => [[a]] -> [a]
- coinSeries :: [Int] -> [Integer]
- coinSeries' :: Num a => [(a, Int)] -> [a]
- convolveWithCoinSeries :: [Int] -> [Integer] -> [Integer]
- convolveWithCoinSeries' :: Num a => [(a, Int)] -> [a] -> [a]
- productPSeries :: [[Int]] -> [Integer]
- productPSeries' :: Num a => [[(a, Int)]] -> [a]
- convolveWithProductPSeries :: [[Int]] -> [Integer] -> [Integer]
- convolveWithProductPSeries' :: Num a => [[(a, Int)]] -> [a] -> [a]
- pseries :: [Int] -> [Integer]
- convolveWithPSeries :: [Int] -> [Integer] -> [Integer]
- pseries' :: Num a => [(a, Int)] -> [a]
- convolveWithPSeries' :: Num a => [(a, Int)] -> [a] -> [a]
- data Sign
- signValue :: Num a => Sign -> a
- signedPSeries :: [(Sign, Int)] -> [Integer]
- convolveWithSignedPSeries :: [(Sign, Int)] -> [Integer] -> [Integer]

# Documentation

unitSeries :: Num a => [a]Source

The series [1,0,0,0,0,...], which is the neutral element for the convolution.

convolve :: Num a => [a] -> [a] -> [a]Source

Convolution of series. The result is always an infinite list. Warning: This is slow!

convolveMany :: Num a => [[a]] -> [a]Source

Convolution of many series. Still slow!

# "Coin" series

coinSeries :: [Int] -> [Integer]Source

Power series expansion of

1 / ( (1-x^k_1) * (1-x^k_2) * ... * (1-x^k_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?

coinSeries' :: Num a => [(a, Int)] -> [a]Source

Generalization of the above to include coefficients: expansion of

1 / ( (1-a_1*x^k_1) * (1-a_2*x^k_2) * ... * (1-a_n*x^k_n) )

convolveWithCoinSeries :: [Int] -> [Integer] -> [Integer]Source

convolveWithCoinSeries' :: Num a => [(a, Int)] -> [a] -> [a]Source

# Reciprocals of products of polynomials

productPSeries :: [[Int]] -> [Integer]Source

Convolution of many `pseries`

, that is, the expansion of the reciprocal
of a product of polynomials

productPSeries' :: Num a => [[(a, Int)]] -> [a]Source

The same, with coefficients.

convolveWithProductPSeries :: [[Int]] -> [Integer] -> [Integer]Source

convolveWithProductPSeries' :: Num a => [[(a, Int)]] -> [a] -> [a]Source

This is the most general function in this module; all the others are special cases of this one.

# Reciprocals of polynomials

pseries :: [Int] -> [Integer]Source

The power series expansion of

1 / (1 - x^k_1 - x^k_2 - x^k_3 - ... - x^k_n)

convolveWithPSeries :: [Int] -> [Integer] -> [Integer]Source

Convolve with (the expansion of)

1 / (1 - x^k_1 - x^k_2 - x^k_3 - ... - x^k_n)

pseries' :: Num a => [(a, Int)] -> [a]Source

The expansion of

1 / (1 - a_1*x^k_1 - a_2*x^k_2 - a_3*x^k_3 - ... - a_n*x^k_n)

convolveWithPSeries' :: Num a => [(a, Int)] -> [a] -> [a]Source

Convolve with (the expansion of)

1 / (1 - a_1*x^k_1 - a_2*x^k_2 - a_3*x^k_3 - ... - a_n*x^k_n)

signedPSeries :: [(Sign, Int)] -> [Integer]Source

convolveWithSignedPSeries :: [(Sign, Int)] -> [Integer] -> [Integer]Source

Convolve with (the expansion of)

1 / (1 +- x^k_1 +- x^k_2 +- x^k_3 +- ... +- x^k_n)

Should be faster than using `convolveWithPSeries'`

.
Note: `Plus`

corresponds to the coefficient `-1`

in `pseries'`

(since
there is a minus sign in the definition there)!