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Séminaire Lotharingien de Combinatoire, 84B.3 (2020), 12 pp.

# Elia Bisi, Fabio Deelan Cunden, Shane Gibbons and Dan Romik

# Sorting Networks, Staircase Young Tableaux, and Last Passage Percolation

**Abstract.**
We present new combinatorial and probabilistic identities relating
three random processes: the oriented swap process on *n* particles,
the corner growth process, and the last passage percolation model. We
prove one of the probabilistic identities, relating a random vector of
last passage percolation times to its dual, using the duality between
the Robinson-Schensted-Knuth and Burge correspondences. A second
probabilistic identity, relating those two vectors to a vector of
"last swap times" in the oriented swap process, is conjectural. We
give a computer-assisted proof of this identity for *n* <= 6 after
first reformulating it as a purely combinatorial identity, and discuss
its relation to the Edelman-Greene correspondence.

Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.

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