knight {spray}

Generating function for a chess knight and king

Description

Generating function for a chess knight and king on an arbitrarily-dimensioned chessboard

Usage

knight(d=2)
king(d=2)

Arguments

d

Dimensionality of the board, defaulting to 2

Value

Returns the generating function of the piece in question.

Note

The pieces are forced to move; if they have the option of not moving, add 1 to the returned spray. The vignette contains a short discussion.

Author(s)

Robin K. S. Hankin

Examples

knight()  # default 2D chess board
king()    # ditto

knight()^2  # generating function for two knight's moves

## How many ways can a knight return to its starting square in 6 moves?
constant(knight()^6)

## How many in 6 or fewer?
constant((1+knight())^6)

## Where does a randomly-moving knight end up?
d <- xyz(2)
kt <- (1+knight())*d^2/9
persp(1:25,1:25,as.array(d*kt^6))


## what is the probability that a 4D king is a knight's move from
##   (0,0,0,0) after 6 moves?

sum(coeffs(((king(4)/80)^4)[knight(4)]))