Recursive Record Filtering and Longest Decreasing Subsequences

Abstract

We consider a recursive record-filtering procedure, which we informally call Disappear-Sort. Let $D_n$ denote the random variable giving the required number of passes in Disappear-Sort to eliminate a sequence of length $n$ sampled as i.i.d. copies of a continuous random variable $X$, where each pass retains the left-to-right records and discards all remaining entries. We show that this procedure admits two natural probabilistic interpretations. For the resampling variant we prove that $d_n=\mathbb{E}[D_n]$ satisfies an exact recurrence involving the unsigned Stirling numbers of the first kind. For the non-resampling variant, we associate to a permutation $p_n\in S_n$ a natural poset and prove that the recursive Disappear-Sort layers form an antichain decomposition of this poset. We deduce that the total number of passes equals $L(p_n)$, where $L(p_n)$ is the length of the longest decreasing subsequence of $p_n$. We then show that for a uniform random permutation of size $n$, the expectation $\mathbb{E}[D_n]$ of this second variant coincides with the expected first-column length of a Plancherel-random Young diagram. Using the Robinson--Schensted correspondence, we obtain an exact formula for this expectation in terms of partitions and standard Young tableaux, and classical Plancherel asymptotics then yield $\mathbb{E}[D_n]\sim 2\sqrt{n}$, with fluctuations on the $n^{1/6}$ scale governed by the Tracy--Widom law derived by Baik, Deift and Johansson. We conclude with an $O(n\log n)$ implementation.