Fast computation of Andrews’ smallest part statistic and conjectured congruences

Let spt ( n ) denote Andrews’ smallest part statistic. Andrews discovered congruences for spt ( n ) mod 5 , 7 and 13 which are reminiscent of Ramanujan’s classical partition congruences for moduli 5, 7, and 11. We create an algorithm exploiting a recursive pattern in Andrews’ smallest part statistic...

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Bibliographic Details
Published in:Discrete Applied Mathematics 2011-08, Vol.159 (13), p.1377-1380
Main Authors: Garrett, K.C., McEachern, C., Frederick, T., Hall-Holt, O.
Format: Article
Language:English
Subjects:
Algorithmics. Computability. Computer arithmetics
Algorithms
Applied sciences
Combinatorics
Combinatorics. Ordered structures
Computer science
control theory
systems
Congruences
Exact sciences and technology
Integer partitions
Mathematical models
Mathematics
Modular
Ramanujan congruences
Recursive
Residues
Sciences and techniques of general use
Statistics
Theoretical computing
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Summary:Let spt ( n ) denote Andrews’ smallest part statistic. Andrews discovered congruences for spt ( n ) mod 5 , 7 and 13 which are reminiscent of Ramanujan’s classical partition congruences for moduli 5, 7, and 11. We create an algorithm exploiting a recursive pattern in Andrews’ smallest part statistic, spt ( n ) , to generate modular residues of spt values in quadratic time and linear working memory. We use this algorithm to acquire the first million values of spt ( n ) . On the basis of the data, we make conjectures about the existence of hundreds of thousands of new congruences including a simple modulus 11 congruence that was found and proved independently by Garvan.
ISSN:0166-218X
1872-6771
DOI:10.1016/j.dam.2011.04.022