# Combinatorial Properties of Fibonacci Arrays

• Manasi S. Kulkarni
• Kalpana Mahalingam
• Sivasankar Mohankumar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11436)

## Abstract

The non-trivial extension of Fibonacci words to Fibonacci arrays was proposed by Apostolico and Brimkov in order to study repetitions in arrays. In this paper we investigate several combinatorial as well as formal language theoretic properties of Fibonacci arrays. In particular, we show that the set of all Fibonacci arrays is a 2D primitive language (under certain conditions), count the number of borders in Fibonacci arrays, and show that the set of all Fibonacci arrays is a non-recognizable language. We also show that the set of all square Fibonacci arrays is a two dimensional code.

## Keywords

Fibonacci words Fibonacci arrays Recognizable picture language Two dimensional code Primitivity

## References

1. 1.
Alfred, S.P., Ingmar, L.: The Fabulous Fibonacci Numbers. Prometheus Books (2007)Google Scholar
2. 2.
Ali, D.: Twelve simple algorithms to compute Fibonacci numbers. CoRR. http://arxiv.org/abs/1803.07199 (2018)
3. 3.
Amir, A., Benson, G.: Two-dimensional periodicity in rectangular arrays. SIAM J. Comput. 27(1), 90–106 (1998)
4. 4.
Anselmo, M., Giammarresi, D., Madonia, M.: Two dimensional prefix codes of pictures. In: Béal, M.-P., Carton, O. (eds.) DLT 2013. LNCS, vol. 7907, pp. 46–57. Springer, Heidelberg (2013).
5. 5.
Anselmo, M., Giammarresi, D., Madonia, M.: Prefix picture codes: a decidable class of two-dimensional codes. Int. J. Found. Comput. Sci. 25(8), 1017–1031 (2014)
6. 6.
Anselmo, M., Madonia, M.: Two-dimensional comma-free and cylindric codes. Theor. Comput. Sci. 658, 4–17 (2017)
7. 7.
Apostolico, A., Brimkov, V.E.: Fibonacci arrays and their two-dimensional repetitions. Theor. Comput. Sci. 237(1–2), 263–273 (2000)
8. 8.
Berthé, V., Vuillon, L.: Tilings and rotations on the torus: a two-dimensional generalization of Sturmian sequences. Discret. Math. 223(1–3), 27–53 (2000)
9. 9.
Berthé, V., Vuillon, L.: Palindromes and two-dimensional Sturmian sequences. J. Autom. Lang. Comb. 6(2), 121–138 (2001)
10. 10.
Bozapalidis, S., Grammatikopoulou, A.: Picture codes. RAIRO-Theor. Inform. Appl. 40(4), 537–550 (2006)
11. 11.
Carlitz, L.: Fibonacci representations II. Fibonacci Q. 8, 113–134 (1970)
12. 12.
Chang, C., Wang, H.: Comparison of two-dimensional string matching algorithms. In: 2012 International Conference on Computer Science and Electronics Engineering, vol. 3, pp. 608–611. IEEE (2012)Google Scholar
13. 13.
Dallapiccola, R., Gopinath, A., Stellacci, F., Dal, N.L.: Quasi-periodic distribution of plasmon modes in two-dimensional Fibonacci arrays of metal nanoparticles. Opt. Express 16(8), 5544–5555 (2008)
14. 14.
Fu, K.S.: Syntactic Methods in Pattern Recognition, vol. 112. Elsevier, Amsterdam (1974)
15. 15.
Gamard, G., Richomme, G., Shallit, J., Smith, T.J.: Periodicity in rectangular arrays. Inf. Process. Lett. 118, 58–63 (2017)
16. 16.
Geizhals, S., Sokol, D.: Finding maximal 2-dimensional palindromes. In: The Proceedings of the 27th Annual Symposium on Combinatorial Pattern Matching, CPM 2016, vol. 54, no. 19, pp. 1–12. Dagstuhl (2016)Google Scholar
17. 17.
Giammarresi, D., Restivo, A.: Two-dimensional languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, pp. 215–267. Springer, Heidelberg (1997).
18. 18.
Hopcroft, J.E., Motwani, R., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation, vol. 24. Pearson (2006)Google Scholar
19. 19.
Kulkarni, M.S., Mahalingam, K.: Two-dimensional palindromes and their properties. In: Drewes, F., Martín-Vide, C., Truthe, B. (eds.) LATA 2017. LNCS, vol. 10168, pp. 155–167. Springer, Cham (2017).
20. 20.
Lothaire, M.: Algebraic Combinatorics on Words, vol. 90. Cambridge University Press, Cambridge (2002)
21. 21.
Luca, A.: A combinatorial property of the Fibonacci words. Inf. Process. Lett. 12(4), 193–195 (1981)
22. 22.
Margenstern, M.: An application of grossone to the study of a family of tilings of the hyperbolic plane. Appl. Math. Comput. 218(16), 8005–8018 (2012)
23. 23.
Margenstern, M.: Fibonacci words, hyperbolic tilings and grossone. Commun. Nonlinear Sci. Numer. Simul. 21(1–3), 3–11 (2015)
24. 24.
Mignosi, F., Restivo, A., Silva, P.V.: On Fine and Wilf’s theorem for bidimensional words. Theor. Comput. Sci. 292(1), 245–262 (2003)
25. 25.
Minsky, M., Papert, S.: Perceptrons. The MIT Press, Cambridge (1969)
26. 26.
Pal, D., Masami, I.: Primitive words and palindromes. In: Context-Free Languages and Primitive Words, pp. 423–435. World Scientific (2014)Google Scholar
27. 27.
Yu, S.S., Zhao, Y.K.: Properties of Fibonacci languages. Discret. Math. 224(1–3), 215–223 (2000)