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An involution on bicubic maps and β(0, 1)-trees

Claesson, Anders and Kitaev, Sergey and de Mier, Anna (2015) An involution on bicubic maps and β(0, 1)-trees. Australasian Journal of Combinatorics, 61 (1). pp. 1-18. ISSN 1034-4942

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Abstract

Bicubic maps are in bijection with β(0, 1)-trees. We introduce two new ways of decomposing β(0, 1)-trees. Using this we dene an endofunction on β(0, 1)-trees, and thus also on bicubic maps. We show that this endofunction is in fact an involution. As a consequence we are able to prove some surprising results regarding the joint equidistribution of certain pairs of statistics on trees and maps. Finally, we conjecture the number of fixed points of the involution.