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

Anders Claesson, Sergey Kitaev, Anna de Mier

Research output: Contribution to journalArticle

1 Citation (Scopus)

### 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.
Original language English 1-18 18 Australasian Journal of Combinatorics 61 1 Published - 2015

Involution
Equidistribution
Bijection
Fixed point
Statistics

### Keywords

• bicubic maps
• statistics
• mathematical trees
• mathematical modeling

### Cite this

Claesson, Anders ; Kitaev, Sergey ; de Mier, Anna. / An involution on bicubic maps and β(0, 1)-trees. In: Australasian Journal of Combinatorics. 2015 ; Vol. 61, No. 1. pp. 1-18.
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An involution on bicubic maps and β(0, 1)-trees. / Claesson, Anders; Kitaev, Sergey; de Mier, Anna.

In: Australasian Journal of Combinatorics, Vol. 61, No. 1, 2015, p. 1-18.

Research output: Contribution to journalArticle

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AU - Claesson, Anders

AU - Kitaev, Sergey

AU - de Mier, Anna

PY - 2015

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N2 - 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.

AB - 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.

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KW - statistics

KW - mathematical trees

KW - mathematical modeling

UR - http://ajc.maths.uq.edu.au/?page=get_volumes&volume=61

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