6-regular partitions: new combinatorial properties, congruences, and linear inequalities

We consider the number of the 6-regular partitions of n , b 6 ( n ), and give infinite families of congruences modulo 3 (in arithmetic progression) for b 6 ( n ). We also consider the number of the partitions of n into distinct parts not congruent to ± 2 modulo 6, Q 2 ( n ), and investigate connections between b 6 ( n ) and Q 2 ( n ) providing new combinatorial interpretations for these partition functions. In this context, we discover new infinite families of linear inequalities involving Euler’s partition function p ( n ). Infinite families of linear inequalities involving the 6-regular partition function b 6 ( n ) and the distinct partition function Q 2 ( n ) are proposed as open problems.

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