2025/09/14 by Gajdzica, Krystian, Heim, Bernhard, Neuhauser, Markus
#05A17 #11P82 (Primary) 05A20 (Secondary) #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2509.11246
In this paper we provide a classification on the sign distribution of ΔE,ℓ(n):= pE,ℓ (n)2 - pE,ℓ (n-1) pE,ℓ (n+1), where ∑n =0∞ pE,ℓ (n) qn := ∏n ∈ S (1 - qn )^-fℓ(n), (ℓ ∈ ℕ, f1≡ 1). We take the product over 1∈ S ⊂ ℕ and denote the complement by E, the set of exceptions. In the case of ℓ=1 and E the multiples of k, pE,1( n) represents the number of k-regular partitions. More generally, let fℓ satisfy a certain growth condition. We determine the signs of ΔE,ℓ (n) for ℓ large. The signs mainly depend on the occurrence of subsets of \2,3,4,5\ as a part of the exception set and the residue class of n modulo r, where r depends on E. For example, let 2,3 ∈ S and 4 an exception. Let n be large. Then for almost all ℓ we have ΔE,ℓ (n) gt;0 for n≡ 2 \pmod3. If we assume 3,4 ∈ S and 2 an exception. Let n be large. Then for almost all ℓ we have ΔE,ℓ (n) lt; 0 for n≡ 2 \pmod3. Note that this property is independent of the integers k∈ S,k>4.