Aunu Polynomials: Excedance Number in Aunu Involutions
DOI:
https://doi.org/10.56919/usci.2652.010Keywords:
Aunu-permutations, Involutions, permutation statisticsAbstract
An Involution is a Permutation which is equal to its own inverse. An involution is a permutation of a non-empty set which does not contain any permutation cycles of length greater than two (i.e., it consists exclusively of fixed points and transpositions). In this research, we obtain a generating function for the Aunu polynomials. The Aunu polynomials enumerate Aunu-permutation/Involutions according to their number of descents or their number of excedances. In this research, we generated Aunu polynomials of orders: , however, further investigation in the same pattern would lead to the generation of more polynomials of higher orders.
References
Alon, N., & Friedgut, E. (2000). On the number of permutations avoiding a given pattern. Journal of Combinatorics Theory Ser. A, 89, 133-140. DOI: https://doi.org/10.1006/jcta.1999.3002
Arratia, R. (1999). On the Stanley-Wilf conjecture for the number of permutations avoiding a given pattern. Electronic Journal of Combinatorics, 6(1), Note N1. DOI: https://doi.org/10.37236/1477
Bagno, E., Butman, A., & Garber, D. (2007). Statistics on the multi-colored permutation groups. Electronic Journal of Combinatorics, 14, #R24. DOI: https://doi.org/10.37236/942
Bóna, M. (1999). The solution of a conjecture of Stanley and Wilf for all layered patterns. Journal of Combinatorics Theory Ser. A, 85, 96-104. DOI: https://doi.org/10.1006/jcta.1998.2908
Brenti, F. (1994). q-Eulerian polynomials arising from Coxeter groups. Electronic Journal of Combinatorics, 15, 417-441. DOI: https://doi.org/10.1006/eujc.1994.1046
Brenti, F. (2000). A class of q-symmetric functions arising from plethysm. Journal of Combinatorics Theory Ser. A, 91, 137-170. DOI: https://doi.org/10.1006/jcta.2000.3092
Chen, W. Y. C., Tang, R. L., & Zhao, A. F. Y. (2009). Derangement polynomials and excedances of type B. Electronic Journal of Combinatorics, 16(2), #R15. DOI: https://doi.org/10.37236/81
Chow, T., & West, J. (1999). Permutations with forbidden subsequences and Chebyshev polynomials. Discrete Mathematics, 204, 119-128. DOI: https://doi.org/10.1016/S0012-365X(98)00384-7
Foata, D., & Schützenberger, M. (1970). Théorie géométrique des polynômes eulériens (Lecture Notes in Mathematics, Vol. 138). Springer-Verlag. DOI: https://doi.org/10.1007/BFb0060799
Ibrahim, A. A. (2008). Some transformation schemes involving the special (132)-avoiding permutation patterns and a binary coding: An algorithmic approach. Asian Journal of Algebra, 1(1), 10-14. DOI: https://doi.org/10.3923/aja.2008.10.14
Ibrahim, A. A., Sloane, N. J. S., et al. (2006). Integer Sequence A119626 [Online Encyclopedia of Integer Sequences]. http://www.research.att.com/~njas/sequence/?q=119626&sort=0&language=english&go=search
Isa, G. A., & Ibrahim, A. A. (2010). A new method of constructing a variety of finite group based on some succession scheme.
Klazar, M. (2000). The Füredi-Hajnal conjecture implies the Stanley-Wilf conjecture. In Proceedings Formal power series and algebraic combinatorics 2000 (pp. 250-255). Springer. DOI: https://doi.org/10.1007/978-3-662-04166-6_22
Krattenthaler, C. (2001). Permutations with restricted patterns and Dyck paths. Advances in Applied Mathematics, 27, 510-530. DOI: https://doi.org/10.1006/aama.2001.0747
Ksavrelof, G., & Zeng, J. (2003). Two involutions for signed excedance numbers. Sém. Lothar. Combin., 49, Art. B49e.
Mansour, T., & Mansour and Vainshtein, A. (2000). Restricted permutations, continued fractions, and Chebyshev polynomials. Electronic Journal of Combinatorics, 7, #R17. DOI: https://doi.org/10.37236/1495
Mansour, T., & Vainshtein, A. (2001). Restricted 132-avoiding permutations. Advances in Applied Mathematics, 26, 258-269. DOI: https://doi.org/10.1006/aama.2000.0719
Moustakas, V. P. (2018). The Eulerian distribution on the involutions of the hyperoctahedral group is unimodal. Electronic Journal of Combinatorics, 14, #R24.
Reifegerste, A. Reifegerste (2003). On the diagram of 132-avoiding permutations. European Journal of Combinatorics, 24, 759-776. DOI: https://doi.org/10.1016/S0195-6698(03)00065-9
Sedgewick, R., & Flajolet, P. (1996). An introduction to the analysis of algorithms. Addison-Wesley.
Zhao, A. F. Y. (2013). Excedance numbers for the permutations of type B. Electronic Journal of Combinatorics, 20(2), #P28. DOI: https://doi.org/10.37236/2375
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