Enumerating rc-Invariant Permutations with No Long Decreasing Subsequences

We use the Robinson-Schensted-Knuth correspondence and Schutzenberger's evacuation of standard tableaux to enumerate permutations and involutions which are invariant under the reverse-complement map and which have no decreasing subsequences of length k. These enumerations are in terms of numbers of permutations with no decreasing subsequences of length approximately k/2; we use known results concerning these quantities to give explicit formulas when k <= 6.