TAILIEUCHUNG - Báo cáo toán học: "Double-dimer pairings and skew Young diagrams"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Double-dimer pairings and skew Young diagrams. | Double-dimer pairings and skew Young diagrams Richard W. Kenyon Brown University Providence RI 02912 USA David B. Wilson Microsoft Research Redmond WA 98052 USA Submitted Jul 12 2010 Accepted Jun 1 2011 Published Jun 14 2011 2010 Mathematics Subject Classification 05A19 60C05 82B20 05C05 05C50 Abstract We study the number of tilings of skew Young diagrams by ribbon tiles shaped like Dyck paths in which the tiles are vertically decreasing . We use these quantities to compute pairing probabilities in the double-dimer model Given a planar bipartite graph G with special vertices called nodes on the outer face the doubledimer model is formed by the superposition of a uniformly random dimer configuration perfect matching of G together with a random dimer configuration of the graph formed from G by deleting the nodes. The double-dimer configuration consists of loops doubled edges and chains that start and end at the boundary nodes. We are interested in how the chains connect the nodes. An interesting special case is when the graph is e Z X N and the nodes are at evenly spaced locations on the boundary R as the grid spacing e 0. 1 Introduction Among the combinatorial objects counted by Catalan numbers Cn 2n n n 1 are balanced parentheses expressions BPEs Dyck paths and noncrossing pairings see Sta99 exercise . A word of length 2n in the symbols and is said to be a balanced parentheses expression if it can be reduced to the empty word by successively removing subwords . Equivalently there are n s and n s and the number of s minus the number of s in any initial segment is nonnegative see Sta99 ex. r . A Dyck path of length 2n is a map h 0 1 2 . 2n . with h 0 h 2n 0 and h i 1 h i 1. There is a bijection between Dyck paths and BPEs defined by letting h i be the number of s minus the number of s in the prefix of length i see Sta99 ex. i . Dyck paths h are also in bijective correspondence with noncrossing pairings n that is pairings of points 1 2 . 2n arranged

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