Successions in integer partitions |
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| Authors: | Arnold Knopfmacher Augustine O Munagi |
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| Institution: | (1) The John Knopfmacher Centre for Applicable Analysis and Number Theory, University of the Witwatersrand, P.O. Wits, 2050 Johannesburg, South Africa |
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| Abstract: | 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.
This material is based upon work supported by the National Research Foundation under grant number 2053740. |
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| Keywords: | Partition p-succession k-part succession Factorial moment Identity |
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