Recursive Record Filtering and Longest Decreasing Subsequences

Abstract

We consider a recursive record-filtering procedure, which we informally call Disappear-Sort, acting as a sort of parallel to traditional patience sorting. Let $D_n$ denote the number of passes required to eliminate a sequence of length $n$ sampled as i.i.d.\ copies of a continuous random variable, where each pass retains the left-to-right records and applies the same rule recursively to the remaining entries. For the non-resampling procedure, we associate to a permutation $p_n\in S_n$ a natural poset and show that the recursive Disappear-Sort layers form an antichain decomposition of this poset. This provides an order-theoretic interpretation of the procedure and identifies the total number of passes with $L(p_n)$, the length of the longest decreasing subsequence of $p_n$. Equivalently, after reversing the comparison direction, the Disappear-Sort layers coincide with the piles arising in classical patience sorting. For a uniformly random permutation, the pass count therefore has the same distribution as the first-column length of the tableau shape produced by the Robinson--Schensted correspondence. We use this classical connection to express $\mathbb{E}[D_n]$ as a sum over partitions and standard Young tableaux. Established results on Plancherel-random Young diagrams then imply $\mathbb{E}[D_n]\sim 2\sqrt{n}$, with fluctuations on the $n^{1/6}$ scale governed by the Tracy--Widom distribution. We also consider a resampling variant in which the nonrecord entries are replaced after each pass by a fresh independent sample of the same size, and derive an exact recurrence for its expected number of passes involving the unsigned Stirling numbers of the first kind. We conclude with an $O(n\log n)$ implementation for computing the non-resampling pass count.