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On $ q$-derangement numbers

Author: Michelle L. Wachs
Journal: Proc. Amer. Math. Soc. 106 (1989), 273-278
MSC: Primary 05A15; Secondary 05A30
MathSciNet review: 937015
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Abstract: We derive a $ q$-analogue of the classical formula for the number of derangements of an $ n$ element set. Our derivation is entirely analogous to the classical derivation, but relies on a descent set preserving bijection between the set of permutations with a given derangement part and the set of shuffles of two permutations.

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  • [A] Martin Aigner, Combinatorial theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 234, Springer-Verlag, Berlin-New York, 1979. MR 542445
  • [BW] A. Björner and M. Wachs, $ q$-Hook length formulas for forests, J. Combin. Th. Ser A (to appear).
  • [D$ _{1}$] J. Désarménien, Une autre interprétation du nombre de dérangements, Actes $ {8^e}$ Séminaire Lotharingien de Combinatoire, I.R.M.A. Strasbourg, 1984, pp. 11-16.
  • [D$ _{2}$] J. Désarménien, personal communication.
  • [G] I. Gessel, Counting permutations by descents, greater index, and cycle structure, unpublished work.
  • [GG] A. M. Garsia and I. Gessel, Permutation statistics and partitions, Adv. in Math. 31 (1979), no. 3, 288–305. MR 532836,
  • [GR] A. M. Garsia and J. Remmel, A combinatorial interpretation of 𝑞-derangement and 𝑞-Laguerre numbers, European J. Combin. 1 (1980), no. 1, 47–59. MR 576766,
  • [M] P. A. MacMahon, The Indices of Permutations and the Derivation Therefrom of Functions of a Single Variable Associated with the Permutations of any Assemblage of Objects, Amer. J. Math. 35 (1913), no. 3, 281–322. MR 1506186,
  • [DW] J. Désarménien and M. Wachs, Descentes des dérangements et mots circulaires, Actes 19 Séminaire Lotharinjien de Combinatoire, I.R.M.A. Strasbourg, 1988, 13-21.

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