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

Let xN,i(n)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)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+1N+2i+1, we prove thatxN,i(n)⩾yN+2i+1,i(n)xN,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)yν,1(n)⩾yν,i(n) for any n   if i<ν/2i<ν/2 is co-prime with ν.