A cycle lemma for permutation inversions

  • Authors:

  • S. Brunetti;A. Del Lungo;F. Del Ristoro

  • Affiliations:

  • Dipartimento di Matematica e Informatica Teorica, Università di Siena, Via del Capitano 15, 53100 Siena, Italy;Dipartimento di Matematica e Informatica Teorica, Università di Siena, Via del Capitano 15, 53100 Siena, Italy;DSI, Università di Firenze, Via Lombroso 6/17, 50134, Firenze, Italy

  • Venue:
  • Discrete Mathematics
  • Year:
  • 2002

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Abstract

In this paper we study some properties of the inversion statistic. Some enumerative results concerning the permutations of the multiset {x1m1, x2m2} with respect to the inversion parameter are established and it is shown that these depend on gcd (m1, m2). Using a "cycle lemma", a combinatorial proof of the results is given. Moreover, some applications to the Gaussian binomial coefficient are illustrated.