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Dokos et al. recently conjectured that the distribution polynomial fn(q) on the set of permutations of size n avoiding the pattern 321 for the number of inversions is given by:
with f0(q) = 1, which was later proven in the affirmative, see . In this note, we provide a new proof of this conjecture, based on the scanning-elements algorithm described in , and present an identity obtained by equating two explicit formulas for the generating function .
- Inversion number
- Continued fractions
- Cheng, S.-E., S. Elizalde, A. Kasraoui, B. E. Sagan. Inversion and major index polynomials, Preprint, http://arxiv.org/pdf/1112.6014.pdf.
- Dokos, T., T. Dwyer, B. P. Johnson, B. E. Sagan, K. Selsor. Permutation patterns and statistics, Discrete Math., Vol. 312, 2012, 2760–2775.
- Firro, G., T. Mansour. Three-letter-pattern-avoiding permutations and functional equations, Electron. J. Combin., Vol. 13, 2006, #R51.
- Fürlinger, J., J. Hofbauer, q-Catalan numbers, J. Combin. Theory Ser. A, Vol. 40, 1985, No. 2, 248–264.
Cite this paper
Mansour, T. & Shattuck, M. (2014). On a recurrence related to 321-avoiding permutations. Notes on Number Theory and Discrete Mathematics, 20(2), 74-78.