2025/09/14 by Levy, Gilead
#05A17 #05C07 #11P81 #11P82 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2510.00007
We study the distributions of parts in random integer partitions subject to general arithmetic restrictions. In particular, we enumerate restricted graphical partitions of an even integer n and identify the conditions under which the fraction of graphical partitions, relative to all restricted partitions, is maximal. We prove that this maximal fraction is asymptotically O(n-1/2). Furthermore, for any set of arithmetic restrictions, we establish the existence of a minimal lower bound on the parts beyond which the influence of these restrictions on the fraction of graphical partitions becomes negligible; in this regime, the fraction depends primarily on the choice of this lower bound. We highlight a key example of partitions restricted to powers of 2, where the critical lower bound is found to be (1)/(2)nlog2+O(log n).