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A combinatorial approach to partitions with parts in the gaps

Dennis Eichhorn (1998)

Acta Arithmetica

Many links exist between ordinary partitions and partitions with parts in the “gaps”. In this paper, we explore combinatorial explanations for some of these links, along with some natural generalizations. In particular, if we let p k , m ( j , n ) be the number of partitions of n into j parts where each part is ≡ k (mod m), 1 ≤ k ≤ m, and we let p * k , m ( j , n ) be the number of partitions of n into j parts where each part is ≡ k (mod m) with parts of size k in the gaps, then p * k , m ( j , n ) = p k , m ( j , n ) .

A formula for the number of solutions of a restricted linear congruence

K. Vishnu Namboothiri (2021)

Mathematica Bohemica

Consider the linear congruence equation x 1 + ... + x k ≡ b ( mod n s ) for b ∈ ℤ , n , s ∈ ℕ . Let ( a , b ) s denote the generalized gcd of a and b which is the largest l s with l ∈ ℕ dividing a and b simultaneously. Let d 1 , ... , d τ ( n ) be all positive divisors of n . For each d j ∣ n , define 𝒞 j , s ( n ) = { 1 ≤ x ≤ n s : ( x , n s ) s = d j s } . K. Bibak et al. (2016) gave a formula using Ramanujan sums for the number of solutions of the above congruence equation with some gcd restrictions on x i . We generalize their result with generalized gcd restrictions on x i and prove that for the above linear congruence, the number of solutions...

BG-ranks and 2-cores.

Chen, William Y.C., Ji, Kathy Q., Wilf, Herbert S. (2006)

The Electronic Journal of Combinatorics [electronic only]

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