About A005802 sequence equivalent definitions

2017-12-13 17:42:01

OEIS sequence A005802 has several equivalent definitions, but there are two of them that I don't understand:

-first "the dimension of SL(3)-invariants in $V^n$ tensor $(V^*)^n$, where $V$ is the standard 3-dimensional representation of SL(3) and $V^*$ is its dual."

I don't understand this, because if $v_1 \otimes \dots \otimes v_n \otimes w_1 \otimes \dots \otimes w_n$ is SL(3)-invariant, then in particular each $v_i$ is an eigenvector of all $M \in SL(3)$, but then $v_i=0$ and the dimension is zero.

-second "the number of doubly-alternating permutations of length 2n with no four-term increasing subsequence (i.e., 1234-avoiding doubly-alternating permutations). The doubly-alternating permutations (counted by sequence A007999) are those permutations $w$ such that both $w$ and $w^{-1}$ have descent set {2, 4, 6, ...}." But if $w$ has descent set {2,4,6,...}, it means $w(3)