Abstract
The purpose of this paper is to present some enumerative results concerning the permutations of the multiset \(\left\{ {\chi \frac{{m1}}{1},\chi \frac{{m2}}{2}, \ldots ,\chi \frac{{mr}}{r}} \right\}\) having inversion number congruent to k modulo n, with k < n. We show that, for some m, the enumeration of this family of permutations is strictly connected to gcd(m l, m 2, ..., m r ). Using a “cyclic lemma”, a combinatorial proof of the results is given.
This work is partially supported by MURST project: Modelli di calcolo innovativi: metodi sintattici e combinatori.
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Brunetti, S., Lungo, A.D., Ristoro, F.D. (2000). A General Cyclic Lemma for Multiset Permutation Inversions. In: Krob, D., Mikhalev, A.A., Mikhalev, A.V. (eds) Formal Power Series and Algebraic Combinatorics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04166-6_12
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DOI: https://doi.org/10.1007/978-3-662-04166-6_12
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