History of Combinatorial Functions
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modified by Peter Koves on  2004-05-11 08:59:16

A set of equations to compute basic combinatorial functions. The following functions are defined (the normal notation for functions with repetitions uses overbar, however HTML does not have that capability. Hence underbar is used instead):

n!                   The factorial of n.
Pn                  Permutations of n objects.
P(n1, ... ,nk)   Permutations of n1 + ... + nk objects such that there are ni objects of type i.
Vnk                Variations of k objects selected from n objects without repetitions.
Vnk                Variations of k objects selected from n objects with repetitions.
Cnk                Combinations of k objects selected from n objects without repetitions.
Cnk                Combinations of k objects selected from n objects with repetitions.
Dn                 Subfactorial. Permutations of n objects such that none stay in place.
Dn,r               Subfactorial. Permutations of n objects such that exactly r stay in place.

The following are the actual functions defined:

_prod(n,k,p) = again(n>=k,p,n-1,k,np)
          Computes the product n(n-1)...k

_eser(n,i,r,s) = again(n>=i,r,n,i+1,r+s/!(i),-s)
          Computes 1 - 1/1! + 1/2! - ... +(-1)n(1/n!).
          This is the nth term of the series whose sum is 1/e; hence the name.

!(n) = _prod(n,1,1)

P(n) = !(n)

V(n,k) = _prod(n,n-k+1,1)

To be continued...

modified by Peter Koves on  2004-05-11 09:17:30

Author: Peter Koves


A set of equations to compute basic combinatorial functions. The following functions are defined (the normal notation for functions with repetitions uses overbar, however HTML does not have that capability. Hence underbar is used instead):

n!                   The factorial of n.
Pn                  Permutations of n objects.
P(n1, ... ,nk)  Permutations of n1 + ... + nk objects such that there are ni objects of type i.
Vnk                Variations of k objects selected from n objects without repetitions.
Vnk                Variations of k objects selected from n objects with repetitions.
Cnk                Combinations of k objects selected from n objects without repetitions.
                      Also called the binomial coefficient.
Cnk                Combinations of k objects selected from n objects with repetitions.
Dn                 Subfactorial. Orderings of n objects such that none stay in place.
Dn,r               Subfactorial. Orderings of n objects such that exactly r stay in place.


The following are the actual functions defined:

_prod(n,k,p) = again(n>=k,p,n-1,k,np)
          Computes the product n(n-1)...k

_eser(n,i,r,s) = again(n>=i,r,n,i+1,r+s/!(i),-s)
          Computes
1 - 1/1! + 1/2! - ... +(-1)n(1/n!).
          This is the n
th term of the series whose sum is 1/e; hence the name.

!(n) = _prod(n,1,1)

P(n) = !(n)

V(n,k) = _prod(n,n-k+1,1)

Vr(n,k) = n^k

C(n,k) = V(n,k)/P(k)

Cr(n,k) = C(n+k-1,k)

D(n) = !(n)_eser(n,1,1,-1)

Dr(n,r) = C(n,r)D(n-r)

Pr2(n1,n2) = !(n1+n2)/(!(n1)!(n2))
Pr3(n1,n2,n3) = !(n1+n2+n3)/(!(n1)!(n2)!(n3))
Pr4(n1,n2,n3,n4) = !(n1+n2+n3+n4)/(!(n1)!(n2)!(n3)!(n4))
          Implementations of P(n1, ... ,nk) for k=2,3,4.
          To be replaced by a single fuction when Equate provides variable number of arguments.

Back to Equations

modified by Russ Webb on  2004-05-11 10:02:19

Author: Peter Koves
Download: Yeq1_Comb.PDB


A set of equations to compute basic combinatorial functions. The following functions are defined (the normal notation for functions with repetitions uses overbar, however HTML does not have that capability. Hence underbar is used instead):

n!                   The factorial of n.
Pn                  Permutations of n objects.
P(n1, ... ,nk)  Permutations of n1 + ... + nk objects such that there are ni objects of type i.
Vnk                Variations of k objects selected from n objects without repetitions.
Vnk                Variations of k objects selected from n objects with repetitions.
Cnk                Combinations of k objects selected from n objects without repetitions.
                      Also called the binomial coefficient.
Cnk                Combinations of k objects selected from n objects with repetitions.
Dn                 Subfactorial. Orderings of n objects such that none stay in place.
Dn,r               Subfactorial. Orderings of n objects such that exactly r stay in place.


The following are the actual functions defined:

_prod(n,k,p) = again(n>=k,p,n-1,k,np)
          Computes the product n(n-1)...k

_eser(n,i,r,s) = again(n>=i,r,n,i+1,r+s/!(i),-s)
          Computes 1 - 1/1! + 1/2! - ... +(-1)n(1/n!).
          This is the nth term of the series whose sum is 1/e; hence the name.

!(n) = _prod(n,1,1)

P(n) = !(n)

V(n,k) = _prod(n,n-k+1,1)

Vr(n,k) = n^k

C(n,k) = V(n,k)/P(k)

Cr(n,k) = C(n+k-1,k)

D(n) = !(n)_eser(n,1,1,-1)

Dr(n,r) = C(n,r)D(n-r)

Pr2(n1,n2) = !(n1+n2)/(!(n1)!(n2))
Pr3(n1,n2,n3) = !(n1+n2+n3)/(!(n1)!(n2)!(n3))
Pr4(n1,n2,n3,n4) = !(n1+n2+n3+n4)/(!(n1)!(n2)!(n3)!(n4))
          Implementations of P(n1, ... ,nk) for k=2,3,4.
          To be replaced by a single fuction when Equate provides variable number of arguments.

Back to Equations

modified by Peter Koves on  2004-05-11 15:28:06

Author: Peter Koves
Download: Yeq1_Comb.PDB


A set of equations to compute basic combinatorial functions. The following functions are defined (the normal notation for functions with repetitions uses overbar, however HTML does not have that capability. Hence underbar is used instead):

n!                   The factorial of n.
Pn                  Permutations of n objects.
P(n1, ... ,nk)  Permutations of n1 + ... + nk objects such that there are ni objects of type i.
Vnk                Variations of k objects selected from n objects without repetitions.
Vnk                Variations of k objects selected from n objects with repetitions.
Cnk                Combinations of k objects selected from n objects without repetitions.
                      Also called the binomial coefficient.
Cnk                Combinations of k objects selected from n objects with repetitions.
Dn                 Subfactorial. Orderings of n objects such that none stay in place.
Dn,r               Subfactorial. Orderings of n objects such that exactly r stay in place.


The following are the actual functions defined:

_prod(n,k,p) = again(n>=k,p,n-1,k,np)
          Computes the product n(n-1)...k

_eser(n,i,r,s) = again(n>=i,r,n,i+1,r+s/!(i),-s)
          Computes 1 - 1/1! + 1/2! - ... +(-1)n(1/n!).
          The limiting sum of this series is 1/e; hence the name.

!(n) = _prod(n,1,1)

P(n) = !(n)

V(n,k) = _prod(n,n-k+1,1)

Vr(n,k) = n^k

C(n,k) = V(n,k)/P(k)

Cr(n,k) = C(n+k-1,k)

D(n) = !(n)_eser(n,1,1,-1)

Dr(n,r) = C(n,r)D(n-r)

Pr2(n1,n2) = !(n1+n2)/(!(n1)!(n2))
Pr3(n1,n2,n3) = !(n1+n2+n3)/(!(n1)!(n2)!(n3))
Pr4(n1,n2,n3,n4) = !(n1+n2+n3+n4)/(!(n1)!(n2)!(n3)!(n4))
          Implementations of P(n1, ... ,nk) for k=2,3,4.
          To be replaced by a single fuction when Equate provides variable number of arguments.

Back to Equations