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Sorting networks, staircase Young tableaux, and last passage percolation. (English) Zbl 1447.05218
Summary: 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 \(n \le 6\) after first reformulating it as a purely combinatorial identity, and discuss its relation to the Edelman-Greene correspondence.
MSC:
05E10 Combinatorial aspects of representation theory
60C05 Combinatorial probability
Software:
OrientedSwaps
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References:
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