Restricted single or double signed patterns

Abstract

Let $E_n^r=\{[\tau]_a=(\tau_1^{(a_1)},...,\tau_n^{(a_n)})| \tau\in S_n,\ 1\leq a_i\leq r\}$ be the set of all signed permutations on the symbols 1,2,...,n with signs 1,2,...,r. We prove, for every 2-letter signed pattern $[\tau]_a$, that the number of $[\tau]_a$-avoiding signed permutations in $E_n^r$ is given by the formula $\sum\limits_{j=0}^n j!(r-1)^j{n\choose j}^2$. Also we prove that there are only one Wilf class for r=1, four Wilf classes for r=2, and six Wilf classes for $r\geq 3$.