A general estimation for partitions with large difference: For even case

AbstractLet xN,i(n) denote the number of partitions of n with difference at least N and minimal component at least i, and yM,j(n) the number of partitions of n into parts which are ±j(modM). If N is even and i is co-prime with N+2i+1, we prove thatxN,i(n)⩾yN+2i+1,i(n) for any positive integer n. This result partially generalizes the Alder–Andrews conjecture. Moreover, we also prove thatyν,1(n)⩾yν,i(n) for any n if i