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Cores, Shi arrangements, and Catalan numbers
Student-Run Discrete Math SeminarSpeaker: | Monica Vazirani, UC Davis |
Location: | 3106 MSB |
Start time: | Thu, Apr 9 2009, 1:10PM |
Catalan numbers are known to count many mathematical objects. (See Richard Stanley's Enumerative Combinatorics" or http://math.mit.edu/~rstan/ec/catalan.pdf and http://math.mit.edu/~rstan/ec/catadd.pdf for a list of over 150 different combinatorial interpretations.) Some of the more well-known include triangulations of an -gon or ways of closing up pairs of parentheses. In particular, the -th Catalan number counts dominant regions in the Shi arrangement (of type ) and partitions that are both -cores and -cores. This fits into a more general framework, considering the -Shi arrangement and partitions that are both -cores and -cores. In joint work with Susanna Fishel, we give a bijective proof of this result, (given necessary definitions along the way) using the techniques of J. Anderson.