The Fibonacci partition triangles

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• 作者：Philipp Fahr^a ; ^{philfahr@gmx.net} ; Claus Michael Ringel^b ; ^c ; ^{ringel@math.uni-bielefeld.de}
• 关键词：Fibonacci numbers ; Partition formulas ; Representations of quivers ; Kronecker quiver ; Fibonacci modules ; 3-regular tree ; Pascal triangle ; Additive functions on translation quivers ; Valued translation quivers ; Left hammocks ; Delannoy paths
• 刊名：Advances in Mathematics
• 出版年：2012
• 期刊代码：62_00018708
• 类别：mt
• 出版时间：July-August, 2012
• 卷：230
• 期：4-6
• 页码：2513-2535
• 文件大小：3049 K

摘要

In two previous papers we have presented partition formulas for the Fibonacci numbers motivated by the appearance of the Fibonacci numbers in the representation theory of the 3-Kronecker quiver and its universal cover, the 3-regular tree. Here we show that the basic information can be rearranged in two triangles. They are quite similar to the Pascal triangle of the binomial coefficients, but in contrast to the additivity rule for the Pascal triangle, we now deal with additivity along 鈥渉ooks鈥? or, equivalently, with additive functions for valued translation quivers. As for the Pascal triangle, we see that the numbers in these Fibonacci partition triangles are given by evaluating polynomials. We show that the two triangles can be obtained from each other by looking at differences of numbers, it is sufficient to take differences along arrows and knight鈥檚 moves.