413 research outputs found
Boson Operator Ordering Identities from Generalized Stirling and Eulerian Numbers
Ordering identities in the Weyl-Heisenberg algebra generated by single-mode
boson operators are investigated. A boson string composed of creation and
annihilation operators can be expanded as a linear combination of other such
strings, the simplest example being a normal ordering. The case when each
string contains only one annihilation operator is already combinatorially
nontrivial. Two kinds of expansion are derived: (i) that of a power of a string
in lower powers of another string , and (ii) that of a power
of in twisted versions of the same power of . The expansion
coefficients are shown to be, respectively, the generalized Stirling numbers of
Hsu and Shiue, and certain generalized Eulerian numbers. Many examples are
given. These combinatorial numbers are binomial transforms of each other, and
their theory is developed, emphasizing schemes for computing them: summation
formulas, Graham-Knuth-Patashnik (GKP) triangular recurrences, terminating
hypergeometric series, and closed-form expressions. The results on the first
type of expansion subsume a number of previous results on the normal ordering
of boson strings.Comment: 36 pages (preprint format
Binomial Eulerian polynomials for colored permutations
Binomial Eulerian polynomials first appeared in work of Postnikov, Reiner and
Williams on the face enumeration of generalized permutohedra. They are
-positive (in particular, palindromic and unimodal) polynomials which
can be interpreted as -polynomials of certain flag simplicial polytopes and
which admit interesting Schur -positive symmetric function
generalizations. This paper introduces analogues of these polynomials for
-colored permutations with similar properties and uncovers some new
instances of equivariant -positivity in geometric combinatorics.Comment: Final version; minor change
On a Class of Combinatorial Sums Involving Generalized Factorials
The object of this paper is to show that generalized Stirling numbers can be effectively used to evaluate a class of combinatorial sums involving generalized factorials
Mixed volumes of hypersimplices, root systems and shifted young tableaux
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010.Cataloged from PDF version of thesis.Includes bibliographical references (p. 40-41).This thesis consists of two parts. In the first part, we start by investigating the classical permutohedra as Minkowski sums of the hypersimplices. Their volumes can be expressed as polynomials whose coefficients - the mixed Eulerian numbers - are given by the mixed volumes of the hypersimplices. We build upon results of Postnikov and derive various recursive and combinatorial formulas for the mixed Eulerian numbers. We generalize these results to arbitrary root systems [fee], and obtain cyclic, recursive and combinatorial formulas for the volumes of the weight polytopes ([fee]-analogues of permutohedra) as well as the mixed [fee]-Eulerian numbers. These formulas involve Cartan matrices and weighted paths in Dynkin diagrams, and thus enable us to extend the theory of mixed Eulerian numbers to arbitrary matrices whose principal minors are invertible. The second part deals with the study of certain patterns in standard Young tableaux of shifted shapes. For the staircase shape, Postnikov found a bijection between vectors formed by the diagonal entries of these tableaux and lattice points of the (standard) associahedron. Using similar techniques, we generalize this result to arbitrary shifted shapes.by Dorian Croitoru.Ph.D
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