Geodesic
Sketches, moves and partitions: counting regions of Catalan deformations of reflection arrangements
Priyavrat Deshpande1 ; Krishna Menon2
1Chennai Mathematical Institute
2KTH Royal Institute of Technology
The electronic journal of combinatorics, Tome 32 (2025) no. 4
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The collection of reflecting hyperplanes of a finite Coxeter group is called a reflection arrangement and it appears in many subareas of combinatorics and representation theory. We focus on the problem of counting regions of reflection arrangements and their deformations. Inspired by the recent work of Bernardi, we show that the notion of moves and sketches can be used to provide a uniform and explicit bijection between regions of (the Catalan deformation of) a reflection arrangement and certain non-nesting partitions. We then use the exponential formula to describe a statistic on these partitions such that distribution is given by the coefficients of the characteristic polynomial. Finally, we consider a sub-arrangement of type C arrangement called the threshold arrangement and its Catalan and Shi deformations.
DOI : 10.37236/13001
Classification : 52C35, 05A15
Mots-clés : Catalan deformations, reflection arrangements

Priyavrat Deshpande  1   ; Krishna Menon  2

1 Chennai Mathematical Institute
2 KTH Royal Institute of Technology 
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Priyavrat Deshpande; Krishna Menon. Sketches, moves and partitions: counting regions of Catalan deformations of reflection arrangements. The electronic journal of combinatorics, Tome 32 (2025) no. 4. doi: 10.37236/13001

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