Some permutations with forbidden subsequences and their inversion number

A permutation avoids the subpattern i3 has no subsequence having all the same pairwise comparisons as , and we write ∈ S( ). We examine the classes of permutations, S(321); S(321; 37 142) and S(4231; 4132), enumerated, respectively by the famous Catalan, Motzkin and Schr; oder number sequences. We determine their generating functions according to their length, number of active sites and inversion number. We also <nd the average inversion number for each class. Finally, we describe some bijections between these classes of permutations and some classes of parallelogram polyominoes, from which we deduce some relations between the parameters of Motzkin and Schr; oder permutations. c © 2001 Elsevier Science B.V. All rights reserved.