We present a number of results relating partial Cauchy-Littlewood sums, integrals over the compact classical groups, and increasing subsequences of permutations. These include: integral formulae for the distribution of the longest 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 straightening algorithm.
"Algebraic aspects of increasing subsequences." Duke Math. J. 109 (1) 1 - 65, 15 July 2001. https://doi.org/10.1215/S0012-7094-01-10911-3