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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.
P Permutations of _{n}n objects.
P(n_{1}, ... ,n Permutations of _{k})n_{1} + ... + n objects such that there are _{k}n objects of type _{i}i.
V Variations of _{n}^{k}k objects selected from n objects without repetitions.
Variations of V_{n}^{k}k objects selected from n objects with repetitions.
C Combinations of _{n}^{k}k objects selected from n objects without repetitions.
Combinations of C_{n}^{k}k objects selected from n objects with repetitions.
D Subfactorial. Permutations of _{n}n objects such that none stay in place.
D Subfactorial. Permutations of _{n,r}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...th term of the series whose sum is 1/ 1 - 1/1! + 1/2! - ... +(-1)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.
P Permutations of _{n}n objects.
P(n_{1}, ... ,n Permutations of _{k})n_{1} + ... + n objects such that there are _{k}n objects of type _{i}i.
V Variations of _{n}^{k}k objects selected from n objects without repetitions.
Variations of V_{n}^{k}k objects selected from n objects with repetitions.
C Combinations of _{n}^{k}k objects selected from n objects without repetitions.
Also called the binomial coefficient. Combinations of C_{n}^{k}k objects selected from n objects with repetitions.
D Subfactorial. Orderings of _{n}n objects such that none stay in place.
D Subfactorial. Orderings of _{n,r}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 ^{n}(1/n!).
This is the n 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(n_{1}, ... ,n for _{k})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.
P Permutations of _{n}n objects.
P(n_{1}, ... ,n Permutations of _{k})n_{1} + ... + n objects such that there are _{k}n objects of type _{i}i.
V Variations of _{n}^{k}k objects selected from n objects without repetitions.
Variations of V_{n}^{k}k objects selected from n objects with repetitions.
C Combinations of _{n}^{k}k objects selected from n objects without repetitions.
Also called the binomial coefficient. Combinations of C_{n}^{k}k objects selected from n objects with repetitions.
D Subfactorial. Orderings of _{n}n objects such that none stay in place.
D Subfactorial. Orderings of _{n,r}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(n_{1}, ... ,n for _{k})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.
P Permutations of _{n}n objects.
P(n_{1}, ... ,n Permutations of _{k})n_{1} + ... + n objects such that there are _{k}n objects of type _{i}i.
V Variations of _{n}^{k}k objects selected from n objects without repetitions.
Variations of V_{n}^{k}k objects selected from n objects with repetitions.
C Combinations of _{n}^{k}k objects selected from n objects without repetitions.
Also called the binomial coefficient. Combinations of C_{n}^{k}k objects selected from n objects with repetitions.
D Subfactorial. Orderings of _{n}n objects such that none stay in place.
D Subfactorial. Orderings of _{n,r}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(n_{1}, ... ,n for _{k})k=2,3,4.
To be replaced by a single fuction when Equate provides variable number of arguments. Back to Equations |