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Enumerative combinatorics, graph theory, order theory, posets, matroids, designs and other discrete structures. It also includes algebraic, analytic and probabilistic combinatorics.

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Probability of words summing to $1$ in $S_n$ or $\mathrm{PGL}_2(n)$

$\DeclareMathOperator\PGL{PGL}$I believe the best result in this direction is due to M. Larsen & A. Shalev (2012); see this paper. I'll summarize their results here. This doesn't answer the questions …
Yotam Shomroni's user avatar