Bijective counting of Kreweras walks and loopless triangulations

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• 作者：Olivier Bernardi
• 关键词：Planar walk ; Kreweras walk ; Planar map ; Triangulation ; Cubic map ; Bijection ; Counting
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2007
• 出版时间：July 2007
• 年：2007
• 卷：114
• 期：5
• 页码：931-956
• 全文大小：886 K

文摘

We consider lattice walks in the plane starting at the origin, remaining in the first quadrant i,j0 and made of West, South and North-East steps. In 1965, Germain Kreweras discovered a remarkably simple formula giving the number of these walks (with prescribed length and endpoint). Kreweras' proof was very involved and several alternative derivations have been proposed since then. But the elegant simplicity of the counting formula remained unexplained. We give the first purely combinatorial explanation of this formula. Our approach is based on a bijection between Kreweras walks and triangulations with a distinguished spanning tree. We obtain simultaneously a bijective way of counting loopless triangulations.