Algebraic aspects of increasing subsequences

We present a number of results relating partial Cauchy-Littlewood sums, int egrals over the compact classical groups, and increasing subsequences of pe rmutations. These include: integral formulae for the distribution of the lo ngest increasing subsequence of a random involution with constrained number of fixed points; new formulae for partial Cauchy-Littlewood sums, as well as new proofs of old formulae; relations of these expressions to orthogonal polynomials on the unit circle; and explicit bases for invariant spaces of the classical groups, together with appropriate generalizations of the str aightening algorithm.