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