2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78499We 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.LaTeX+amsmath+eepic; 52 pages. Expanded introduction, new references, other minor changesCombinatoricsProbabilityRepresentation Theorytext