Some inequalities on cranks of cubic partitions modulo 4 and 6

A partition is a cubic partition if its even parts come in two colors. The crank for cubic partitions which explains cubic partition congruences modulo powers of 3 combinatorially, was defined by Kim. Motivated by the works on inequalities of ranks and cranks for certain partitions, in this paper, w...

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Published in:Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A, Matemáticas Físicas y Naturales. Serie A, Matemáticas, 2023, Vol.117 (1), Article 31
Main Authors: Fan, Yan, Liu, Eric H., Xia, Ernest X. W.
Format: Article
Language:English
Subjects:
Applications of Mathematics
Congruences
Eccentrics
Inequalities
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Original Paper
Theoretical
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Summary:A partition is a cubic partition if its even parts come in two colors. The crank for cubic partitions which explains cubic partition congruences modulo powers of 3 combinatorially, was defined by Kim. Motivated by the works on inequalities of ranks and cranks for certain partitions, in this paper, we establish the generating functions for M ′ ( r , m , n ) which counts the number of cubic partitions of n whose crank is congruent to r modulo m . In addition, we determine the signs of the differences M ′ ( r , m , n ) - M ′ ( s , m , n ) with m ∈ { 2 , 3 , 4 , 6 } and 0 ≤ r < s ≤ m - 1 for large enough integers n by utilizing asymptotic formulas of eta quotients due to Chern and q -series techniques.
ISSN:1578-7303
1579-1505
DOI:10.1007/s13398-022-01366-7