kStatistics (version 2.0)

pPart: Partition polynomial

Description

The function generates the partition polynomial of degree n, whose coefficients are the number of partitions of n with k parts for k from 1 to n.

Usage

pPart(n = 0)

Arguments

n

integer

Value

string

expression of the partition polynomial of degree n

Warning

The value of the first parameter is the same as the MFB function in the univariate with univariate case composition.

Details

Faa di Bruno's formula gives the coefficients of the exponential formal power series obtained from the composition f(g()) of exponential formal power series. The partition polynomial F[n] of degree n is obtained using the Faa di Bruno's formula obtained from the MFB function in the case "composition of univariate f with univariate g" with f[i]=1/i!, g[i]^k=(i!)^k*k!*y for each i and k from 1 to n. Note the symbolic substitution g[i]^k=(i!)^k*k!*y, where the power of g[i] appears in its definition. This function is an example of application of Faa di Bruno's formula by using the symbolic calculus with two indexes.

References

E. Di Nardo E., G. Guarino, D. Senato (2011) A new algorithm for computing the multivariate Faa di Bruno's formula. Appl. Math. Comp. 217, 6286--6295. (download from http://arxiv.org/abs/1012.6008)

See Also

MFB

Examples

# NOT RUN {
# Return the partition polynomial F[5]
pPart(5)

# Return the partition polynomial F[11] and its evaluation when y=7  
#
s<-pPart(11)          # run the command
s<-paste0("1",s)      # add the coefficient to the first term (fixed command)
s<-gsub(" y","1y",s)  # replace the variable y without coefficient (fixed command)
s<-gsub("y", "*7",s)  # assignment y = 7
eval(parse(text=s))   # evaluation of the expression (fixed command)
# }