P: Number of partitions of Q with k or less parts.

In rpartitions: Code for integer partitioning

Description

This function was derived using the following theorem and proposition. The number of partitions of Q with k or less parts equals the number of partitions of Q with k or less as the largest part (see Bona 2006). This is a mathematical symmetry, i.e. congruency. Additionally, the number of partitions of Q with k or less parts equals the number of partitions of Q+k with k as the largest part when k>0, i.e. P(Q + k, k). We do not have a source for this proposition, but it can be shown when enumerating the entire feasible set or using the Sage computing enviornment

Usage

  P(D, Q, k, use_c, use_hash)

Arguments

D	lookup table for numbers of partitions of Q having k or less parts (or k or less as the largest part), i.e. P(Q, Q + k)
Q	total (i.e., sum across all k or n parts)
k	the number of parts and also the size of the largest part (congruency)
use_c	boolean, if TRUE the number of partitions is computed in c
use_hash	boolean, if TRUE then a hash table is used instead of R's native list to store the information

Value

a two element list, the first element is D the lookup table and the second element is the number of partitions for the specified Q and k value.

References

Bona, M. (2006). A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory. 2nd Ed. World Scientific Publishing Co. Singapore.

Examples

P(list(), 100, 10, FALSE, FALSE)

rpartitions documentation built on May 1, 2019, 8:03 p.m.