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Algebraic aspects of increasing subsequences. (English) Zbl 1007.05096

Summary: 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.

MSC:

05E15 Combinatorial aspects of groups and algebras (MSC2010)
05A15 Exact enumeration problems, generating functions
05E05 Symmetric functions and generalizations
60C05 Combinatorial probability
05E35 Orthogonal polynomials (combinatorics) (MSC2000)
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