The oriented swap process and last passage percolation

We present new probabilistic and combinatorial 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. The conjectural identity provides precise finite-$n$ and asymptotic predictions on the distribution of the absorbing time of the oriented swap process, thus conditionally solving an open problem posed by Angel, Holroyd and Romik.Comment: 36 pages, 6 figures. Full version of the FPSAC 2020 extended abstract arXiv:2003.0333

Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes

We put recent results by Chen, Deng, Du, Stanley and Yan on crossings and nestings of matchings and set partitions in the larger context of the enumeration of fillings of Ferrers shape on which one imposes restrictions on their increasing and decreasing chains. While Chen et al. work with Robinson-Schensted-like insertion/deletion algorithms, we use the growth diagram construction of Fomin to obtain our results. We extend the results by Chen et al., which, in the language of fillings, are results about $0$-$1$-fillings, to arbitrary fillings. Finally, we point out that, very likely, these results are part of a bigger picture which also includes recent results of Jonsson on $0$-$1$-fillings of stack polyominoes, and of results of Backelin, West and Xin and of Bousquet-M\'elou and Steingr\'\i msson on the enumeration of permutations and involutions with restricted patterns. In particular, we show that our growth diagram bijections do in fact provide alternative proofs of the results by Backelin, West and Xin and by Bousquet-M\'elou and Steingr\'\i msson.Comment: AmS-LaTeX; 27 pages; many corrections and improvements of short-comings; thanks to comments by Mireille Bousquet-Melou and Jakob Jonsson, the final section is now much more profound and has additional result