Generalized permutation patterns and a classification of the Mahonian statistics.

1]), and the generalized permutation group is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Our first goal is to generalize such properties to the case of generalized permutations.

A generalized permutation is defined to be a biword:

The integer n is called the length of the generalized permutation [alpha].

A generalized involution will be a generalized permutation a satisfying the further condition that each pair [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] occurs as many times as the pair [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We associate a generalized permutation with a pair (P, Q) of dual semistandard tableaux as follows.

In other words, [alpha] is a Baxter permutation if it avoids the

generalized permutation patterns 2 - 41 - 3 and 3 - 14 - 2 (see [3] for an introduction on

generalized permutation patterns).

Steingrimsson:

Generalized permutation patterns--a short survey, Permutation Patterns, St Andrews 2007, S.

Steingrimsson,

Generalized permutation patterns and a classification of the Mahonian statistics, Sem.

To our knowledge the above theorem is the first result of this type for

generalized permutation patterns of length larger than two.

On log surfaces Osamu Fujino and Hiromu Tanaka Zeta functions of

generalized permutations with application to their factorization formulas Shin-ya Koyama and Sachiko NAKAJIMA Above two, communicated by Shigefumi Mori, M.