2,948 research outputs found
A Simple Bijection for the Regions of the Shi Arrangement of Hyperplanes
The Shi arrangement is the arrangement of affine hyperplanes
in of the form or , for . It dissects into regions, as was first proved
by Shi. We give a simple bijective proof of this result. Our bijection
generalizes easily to any subarrangement of containing the
hyperplanes and to the extended Shi arrangements
Statistics on parallelogram polyominoes and a q,t-analogue of the Narayana numbers
We study the statistics area, bounce and dinv on the set of parallelogram
polyominoes having a rectangular m times n bounding box. We show that the
bi-statistics (area, bounce) and (area, dinv) give rise to the same
q,t-analogue of Narayana numbers which was introduced by two of the authors in
[arXiv:1208.0024]. We prove the main conjectures of that paper: the
q,t-Narayana polynomials are symmetric in both q and t, and m and n. This is
accomplished by providing a symmetric functions interpretation of the
q,t-Narayana polynomials which relates them to the famous diagonal harmonics
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