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! y Factorial

Rank 0 -- operates on individual atoms of y, producing a result of the same shape -- WHY IS THIS IMPORTANT?



The Factorial of y.

Might be regarded as a special case of the Stope function (see link below), for start value identical to number of steps and step size _1:   n ^!._1 n .

   */ 5 4 3 2 1    NB. product (5th 'falling power' of 5)
120
   ! 5             NB. gets abbreviated
120
   ! i.6           NB. 0! = 1
1 1 2 6 24 120

More generally, it computes the Gamma function of (1+y). If we defined Ga(y) as Gamma of real argument (y) we get

   gamma=: ga=: !@<: : [:      NB. monadic Ga(y)= (y-1)!

   ga _1r4p1    NB. Ga(-π/4)
_5.42531
   ga 0         NB. Ga(0)
_
   ga -i.6
_ __ _ __ _ __
   ga 1 = ! 0   NB. Ga(1)= 1 = 0!
1
   ga 1p1       NB. Ga(π)
2.28804
   ga 2p1       NB. Ga(2π)
195.936
   !(<:2p1)     NB. (2π-1)!
195.936
 
   exp=: ^ : [:                NB. Exponential function, monad

   NB. calculating the (positive) intersections of exp(y)and ga(y) by
   NB. solving exp(y)- ga(y)= 0 numerically, using Newton-Raphson
   NB. with seed values 1 and 10, resp                          

   (exp-ga) VN^:_ (1)       NB. IP4=~ (0.52;1.69)
0.524922                   
   ga (exp-ga) VN^:_ (1)
1.69033
   (exp-ga) VN^:_ (10)      NB. IP5=~ (7.46;1743)
7.4636
   ga (exp-ga) VN^:_ (10)   NB. Gamma overtaking the Exponential
1743.42


x ! y Out Of

Rank 0 0 -- operates on individual atoms of x and y, producing a result of the same shape -- WHY IS THIS IMPORTANT?



Returns y-Combinations-x:  yCx  (read: x out of y  or: y pick x).

The number of ways of picking x balls (unordered) from a bag of y balls:

   10!10      NB. There is only 1 way of picking all 10 balls
1
   1!10       NB. There are 10 ways of choosing 1 ball from 10
10
   2!10       NB. 45 ways to pick 2 out of 10
45
   5!10
252

This example was found in K.E. Iverson, Computers and Mathematical Notation, E. Ambivalence; the table shows the first five rows of Pascal's Triangle:

   0 1 2 3 4 !/ 0 1 2 3 4       NB. 'Out Of' table "with meaningful zeros"
1 1 1 1 1
0 1 2 3 4
0 0 1 3 6
0 0 0 1 4
0 0 0 0 1

   |:!/~i.5                     NB. (transposed view)
1 0 0 0 0
1 1 0 0 0
1 2 1 0 0
1 3 3 1 0
1 4 6 4 1

Common uses

In Probability theory, also Functional Analysis (Infinite Series).


Related Primitives

Stope Function (^!._1 y) to count permutations.


See Also

John D. Cook, Binomial Coefficients discusses the domain of this operation, giving three definitions (from basic to general). There, x!y would be represented using a pixelated image, with x on the bottom and y on the top, both surrounded by extra large parenthesis. Something like this:

 {y \choose x}  or    \binom{y}{x}  in 'Math' (subset of LaTeX) notation.