Partitions of large multipartites with congruence conditions.
Spencer, D. (1973) Partitions of large multipartites with congruence conditions. Doctoral thesis, University of Surrey (United Kingdom)..

Text
10804518.pdf Available under License Creative Commons Attribution Noncommercial Share Alike. Download (5MB)  Preview 
Abstract
Our object is to obtain formulae for the number of partitions of the vector (n[l],...,n[j]) into parts whose components lie in certain congruence classes (modulo M), where M, n[l],...,n[j] are all positive integers. The asymptotic expression obtained for the generating function of multipartite partitions enables us to consider all large n[l],...,n[j] each of approximately the same order of magnitude. If M = p (a prime) partitions always exist. For composite values of M nonzero partitions always exist if the Vector (n[l],...,n[j])is "parallel" to a certain jdimensional "plane" which is determined by the residues (modulo M) of the problem. This can be interpreted geometrically when j=3. We find, even for large values of n[l],...,n[j], that the chance of a nonzero partition of (n[l],...,n[j]) occurring is very small and in the asymptotic formulae obtained this is caused by the strong thinning effect of a certain exponential sum which is zero except when each n(1 < 1 < j) lies in a certain residue class (mod M). Certain special cases of these formulae are considered for j=2 . In order to estimate the number of partitions in these formulae the evaluation of a certain integral I (z[l],...,z[j]) is required. This is considered in the last six sections where asymptotic expansions are obtained for I (z[l],...,z[j]) when every z is small and approximate expressions are given for I (z[l],...,z[j]) when every z is real. Finally, exact formulae are given when every z is real and rational.
Item Type:  Thesis (Doctoral)  

Divisions :  Theses  
Authors : 


Date :  1973  
Contributors : 


Additional Information :  Thesis (Ph.D.)University of Surrey (United Kingdom), 1973.  
Depositing User :  EPrints Services  
Date Deposited :  22 Jun 2018 14:26  
Last Modified :  06 Nov 2018 16:53  
URI:  http://epubs.surrey.ac.uk/id/eprint/848043 
Actions (login required)
View Item 
Downloads
Downloads per month over past year