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The collection of reflecting hyperplanes of a finite Coxeter group is calleda reflection arrangement and it appears in many subareas of combinatorics andrepresentation theory. We focus on the problem of counting regions ofreflection arrangements and their deformations. Inspired by the recent work ofBernardi, we show that the notion of moves and sketches can be used to providea uniform and explicit bijection between regions of (the Catalan deformationof) a reflection arrangement and certain non-nesting partitions. We then usethe exponential formula to describe a statistic on these partitions such thatdistribution is given by the coefficients of the characteristic polynomial.Finally, we consider a sub-arrangement of type C arrangement called thethreshold arrangement and its Catalan and Shi deformations.

Sketches, moves and partitions: counting regions of deformations of reflection arrangements