gpPart: General partition polynomial

In kStatistics: Unbiased Estimators for Cumulant Products and Faa Di Bruno's Formula

General partition polynomial

Description

The function returns a general partition polynomial.

Usage

gpPart(n = 0)

Arguments

n

integer

Details

Faa di Bruno's formula gives the coefficients of the exponential formal power series composition f[g()] obtained from the composition of the exponential formal power series f with g. General partition polynomials in the variables y[1],...,y[n] are recovered from the Faa di Bruno's formula (output of the MFB function) in the case "composition of univariate f with univariate g" by setting f[i]=ai and g[i]=y[i], for i from 1 to n.

Value

string

the expression of the polynomial

Warning

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

Note

This function calls the MFB function in the kStatistics package.

Author(s)

Elvira Di Nardo [e-mail],
Giuseppe Guarino [e-mail]

References

C.A. Charalambides (2002) Enumerative Combinatoris, Chapman & Haii/CRC.

E. Di Nardo, 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 https://arxiv.org/abs/1012.6008)

See Also

MFB

Examples

# Return the general partition polynomial G[a1,a2; y1,y2], that is a2(y1^2) + a1(y2)
gpPart(2)

# Return the general partition polynomial G[a1,a2,a3,a4,a5; y1,y2,y3,y4,y5], that is 
# a5(y1^5) + 10a4(y1^3)(y2) + 15a3(y1)(y2^2) + 10a3(y1^2)(y3) + 10a2(y2)(y3) + 5a2(y1)(y4) 
# + a1(y5)
gpPart(5)

kStatistics documentation built on June 8, 2022, 5:05 p.m.