Curtis Greene

Noncommutative Schur functions and their applications

We develop a theory of Schur functions in noncommuting variables, assuming commutation relations that are satisfied in many well-known associative algebras. As an application of our theory, we prove Schur-positivity and obtain generalized Littlewood-Richardson and Murnaghan-Nakayama rules for a large class of symmetric functions, including stable Schubert and Grothendieck polynomials.

A probabilistic proof of a formula for the number of Young tableaux of a given shape

Publisher Summary This chapter presents a probabilistic proof of a formula for the number of Young tableaux of a given shape. A Young tableau of shape λ is an arrangement of the integers 1, 2,…, n in the cells of the Ferrers diagram of λ such that all rows and columns form increasing sequences. The chapter also presents the problem of the occurrence of hook lengths, which do not seem to be involved naturally in any direct combinatorial correspondence. In any standard tableau, the integer n must appear at a corner, that is, a cell that is at the end of some row and at the end of a column.

Longest chains in the lattice of integer partitions ordered by majorization

A method is given for finding a chain of maximum length between two partitions λ ⩽ μ in the lattice of integer partitions, ordered by majorization. The main result is that chains in which covers of a certain kind (called ‘H-steps’) precede covers of another kind (called ‘V-steps’) all have the same length, and this length is maximal.

A Rational-Function Identity Related to the Murnaghan–Nakayama Formula for the Characters of S n

The Murnaghan–Nakayama formula for the characters of Sn is derived from Youngs seminormal representation, by a direct combinatorial argument. The main idea is a rational function identity which when stated in a more general form involves Möbius functions of posets whose Hasse diagrams have a planar embedding. These ideas are also used to give an elementary exposition of the main properties of Youngs seminormal representations.

Proof of a conjecture on immanants of the Jacobi-Trudi matrix

Abstract We prove a conjecture on characters of Sn which implies another conjecture (both due to Goulden and Jackson) that all immanants of the Jacobi-Trudi matrix H(ν,μ)= (h(νi-i)-(μj-j))i,j=1,…,n have nonnegative coefficients.

Posets of shuffles

Abstract We study properties of a lattice order defined on shuffles of subwords of a pair of fixed words.

The Möbius Function of a Partially Ordered Set

The history of the Mobius function has many threads, involving aspects of number theory, algebra, geometry, topology, and combinatorics. The subject received considerable focus from Rota’s by now classic paper in which the Mobius function of a partially ordered set emerged in clear view as an important object of study. On the one hand, it can be viewed as an enumerative tool, defined implicitly by the relations n n

A Class of Lattices with Möbius Function ź 1, 0

f(x) = sumlimits_{yx} {g(y)} {text{ and }}g(x) = sumlimits_{yx} {mu (y,x)f(y)}