Abstract:  Based on Bóna's condition for the balanced property of the number of of cycles of permutations, we give a general criterion for the balanced property in terms of the generating function of a statistic. We show that the q-derangement numbers and the q-Catalan numbers satisfy the balanced property.

  AMS Classification:  05A05, 05A15

  Keywords:  balanced property, major index, derangement, Catalan word

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