Successions in integer partitions

A partition of an integer n is a representation n = a 1 + a 2 + ⋅⋅⋅ + a k , with integer parts 1≤ a 1 ≤ a 2 ≤…≤ a k . For any fixed positive integer  p , a p -succession in a partition is defined to be a pair of adjacent parts such that a i +1 − a i = p . We find generating functions for the number...

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Bibliographic Details
Published in:The Ramanujan journal 2009-04, Vol.18 (3), p.239-255
Main Authors: Knopfmacher, Arnold, Munagi, Augustine O.
Format: Article
Language:English
Subjects:
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Number Theory
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Summary:A partition of an integer n is a representation n = a 1 + a 2 + ⋅⋅⋅ + a k , with integer parts 1≤ a 1 ≤ a 2 ≤…≤ a k . For any fixed positive integer  p , a p -succession in a partition is defined to be a pair of adjacent parts such that a i +1 − a i = p . We find generating functions for the number of partitions of n with no p -successions, as well as for the total number of such successions taken over all partitions of  n . In the process, various interesting partition identities are derived. In addition, the Hardy-Ramanujan asymptotic formula for the number of partitions is used to obtain an asymptotic estimate for the average number of p -successions in the partitions of  n .
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-008-9140-2