The Theory of Partitions

Abstract

This book develops the theory of partitions. Simply put, the partitions of a number are the ways of writing that number as sums of positive integers. For example, the five partitions of 4 are 4, 3+1, 2+2, 2+1+1, and 1+1+1+1. Surprisingly, such a simple matter requires some deep mathematics for its study. This book considers the many theoretical aspects of this subject, which have in turn recently found applications to statistical mechanics, computer science and other branches of mathematics. With minimal prerequisites, this book is suitable for students as well as researchers in combinatorics, analysis, and number theory.

Gauge sector statistics of intersecting D‐brane models

“…particle representation mult. Table 3 Realisation of quarks and leptons for hypercharges (2) and (3) of (65), which can only be realised for the first choice of gauge groups in (63). In figure 18(a) we show the frequency distributions of the total rank of gauge groups in the hidden sector.…”

“…This is nothing else but the number of unordered partitions of L. Since we are not interested in an exact solution, but rather an approximative result, suitable for a statistical analysis and further generalisation to the more ambitious task of solving the tadpole equation, let us attack this by means of the saddle point approximation [2,102]. As a first step to solve (13), let's consider…”

Generation of alternating groups by pairs of conjugates

“…The number of integers n that can be the degree of a primitive group contained properly in Alt (n) is vanishingly small . 1Ylore precisely (1 in almost all partitions of n.…”

Restricting SBH Ambiguity via Restriction Enzymes

“…The number of integer partitions of n is asymptotically: as n → ∞, by Hardy and Ramanujan [1]. Thus the information content of a restriction digest is lg(a(n)) ≈ (lg e)( 2n/3) bits.…”

“…This preference for short cutters can be understood through the theory of integer partitions [1]. We observe that each complete digest (including multiplicities) returns a partition of the sequence length n. Since, on average, an r-cutter cuts every 4 r bases, the expected number of parts resulting from such a digest is n/4 r .…”