126,477 research outputs found
Algebraic aspects of increasing subsequences
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.Comment: LaTeX+amsmath+eepic; 52 pages. Expanded introduction, new references,
other minor change
Partitioning the hypercube into smaller hypercubes
Denote by Q_d the d-dimensional hypercube. Addressing a recent question we
estimate the number of ways the vertex set of Q_d can be partitioned into
vertex disjoint smaller cubes. Among other results, we prove that the
asymptotic order of this function is not much larger than the number of perfect
matchings of Q_d. We also describe several new (and old) questions.Comment: Proofs slightly shortene
Algebraic dependence of commuting elements in algebras
The aim of this paper to draw attention to several aspects of the algebraic
dependence in algebras. The article starts with discussions of the algebraic
dependence problem in commutative algebras. Then the Burchnall-Chaundy
construction for proving algebraic dependence and obtaining the corresponding
algebraic curves for commuting differential operators in the Heisenberg algebra
is reviewed. Next some old and new results on algebraic dependence of commuting
q-difference operators and elements in q-deformed Heisenberg algebras are
reviewed. The main ideas and essence of two proofs of this are reviewed and
compared. One is the algorithmic dimension growth existence proof. The other is
the recent proof extending the Burchnall-Chaundy approach from differential
operators and the Heisenberg algebra to the q-deformed Heisenberg algebra,
showing that the Burchnall-Chaundy eliminant construction indeed provides
annihilating curves for commuting elements in the q-deformed Heisenberg
algebras for q not a root of unity.Comment: LaTeX, 14 pages, no figure
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