Triangular Record Highs

Computing expected triangular record highs in a random permutation

This medium-difficulty probability question tests your ability to combine linearity of expectation with careful case analysis over a permutation. It is the kind of problem quant firms use to assess whether you can decompose a complex structural property into independent indicator random variables and reason about their probabilities cleanly.

A triangular record high is a position that is simultaneously a left-to-right maximum (a record high) and locally dominant with respect to at least one immediate neighbor. The key insight is to use indicator variables for each position, compute the probability that it becomes a triangular record high, and sum these probabilities. The probability depends on the position's rank relative to its neighbors and its historical predecessors, which requires careful conditioning on the relative order of small local subsets of the permutation.

• Indicator random variables and linearity of expectation
• Record statistics and order statistics in random permutations
• Conditional probability on relative orderings of small sets
• Harmonic numbers and their role in permutation analysis