On a Distribution Function Arising in Computational Biology

where Rk,r is the number of permutations of the integers {1, . . . , k} that contain an increasing subsequence of length at least r. Let Xy denote a positive integer valued random variable such that Prob (Xy ≥ r) = F (r; y). If R k,r denotes the complement of Rk,r, i.e. the number of permutations σ ∈ Sk all of whose increasing subsequences have length strictly less than r, then clearly R k,r = # {σ ∈ Sk : lk(σ) < r} = # {σ ∈ Sk : lk(σ) ≤ r − 1} := fk,r−1 where lk(σ) is the length of the longest increasing subsequence in σ ∈ Sk. Remarks 1. F (r; y) is a distribution function in r with parameter y. 2. Dropping the requirement of consistent ordering has the effect of replacing Rk,r by k! in (1). Thus the segments are Poisson distributed with parameter y. By convention, f0,r := 1 for all r and we note that fk,r = k! if k ≤ r. It is also convenient to introduce the parameter t ≥ 0, y = t. If we define Dr(t) = ∞