Continuously increasing subsequences of random multiset permutations

For a word π and an integer i, we define L(π) to be the length of the longest subsequence of the form i(i+1)⋯j for some i, and we let L1(π) be the length of the longest such subsequence beginning with 1. In this paper, we estimate the expected values of L1(π) and L(π) when π is chosen uniformly at random from all words which use each of the first n positive integers exactly m times. We show that E[L1(π)]∼m if n is sufficiently large in terms of m as m tends towards infinity, confirming a conjecture of Diaconis, Graham, He, and Spiro. We also show that E[L(π)] is asymptotic to the inverse gamma function Γ−1(n) if n is sufficiently large in terms of m as m tends towards infinity.