Fahr, Philipp; Ringel, Claus MichaelUniBi
Abstract / Bemerkung
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 "hooks", or, equivalently, with additive functions for valued translation quivers. As for the Pascal triangle, we see that the numbers in these Fibonacci partition triangles arc 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's moves. (C) 2012 Elsevier Inc. All rights reserved.
Representations of quivers; Additive functions on translation quivers; 3-regular tree; Pascal triangle; Left hammocks; Valued translation quivers; Delannoy paths; Fibonacci modules; Kronecker quiver; Partition formulas; Fibonacci numbers
Advances in Mathematics
Fahr P, Ringel CM. The Fibonacci partition triangles. Advances in Mathematics. 2012;230(4-6):2513-2535.
Fahr, P., & Ringel, C. M. (2012). The Fibonacci partition triangles. Advances in Mathematics, 230(4-6), 2513-2535. doi:10.1016/j.aim.2012.04.010
Fahr, P., and Ringel, C. M. (2012). The Fibonacci partition triangles. Advances in Mathematics 230, 2513-2535.
Fahr, P., & Ringel, C.M., 2012. The Fibonacci partition triangles. Advances in Mathematics, 230(4-6), p 2513-2535.
P. Fahr and C.M. Ringel, “The Fibonacci partition triangles”, Advances in Mathematics, vol. 230, 2012, pp. 2513-2535.
Fahr, P., Ringel, C.M.: The Fibonacci partition triangles. Advances in Mathematics. 230, 2513-2535 (2012).
Fahr, Philipp, and Ringel, Claus Michael. “The Fibonacci partition triangles”. Advances in Mathematics 230.4-6 (2012): 2513-2535.