scholarly journals Walks, Partitions, and Normal Ordering

10.37236/5181 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Askar Dzhumadil'daev ◽  
Damir Yeliussizov
Keyword(s):  
Weyl Algebra ◽  
Differential Operators ◽  
Stirling Numbers ◽  
Set Partitions ◽  
Normal Ordering ◽  
Graph Decompositions

We describe the relation between graph decompositions into walks and the normal ordering of differential operators in the $n$-th Weyl algebra. Under several specifications, we study new types of restricted set partitions, and a generalization of Stirling numbers, which we call the $\lambda$-Stirling numbers.

scholarly journals On $xD$-Generalizations of Stirling Numbers and Lah Numbers via Graphs and Rooks

10.37236/6699 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Sen-Peng Eu ◽  
Tung-Shan Fu ◽  
Yu-Chang Liang ◽  
Tsai-Lien Wong
Keyword(s):  
Weyl Algebra ◽  
Stirling Numbers ◽  
Stirling Number ◽  
Combinatorial Structures ◽  
Normal Ordering ◽  
Rook Placements ◽  
Threshold Graphs ◽  
Combinatorial Interpretations ◽  
Ferrers Boards

This paper studies the generalizations of the Stirling numbers of both kinds and the Lah numbers in association with the normal ordering problem in the Weyl algebra $W=\langle x,D|Dx-xD=1\rangle$. Any word $\omega\in W$ with $m$ $x$'s and $n$ $D$'s can be expressed in the normally ordered form $\omega=x^{m-n}\sum_{k\ge 0} {{\omega}\brace {k}} x^{k}D^{k}$, where ${{\omega}\brace {k}}$ is known as the Stirling number of the second kind for the word $\omega$. This study considers the expansions of restricted words $\omega$ in $W$ over the sequences $\{(xD)^{k}\}_{k\ge 0}$ and $\{xD^{k}x^{k-1}\}_{k\ge 0}$. Interestingly, the coefficients in individual expansions turn out to be generalizations of the Stirling numbers of the first kind and the Lah numbers. The coefficients will be determined through enumerations of some combinatorial structures linked to the words $\omega$, involving decreasing forest decompositions of quasi-threshold graphs and non-attacking rook placements on Ferrers boards. Extended to $q$-analogues, weighted refinements of the combinatorial interpretations are also investigated for words in the $q$-deformed Weyl algebra.

scholarly journals The Group of Automorphisms of the Algebra of one-Sided Inverses of a Polynomial Algebra. II

2015 ◽  
Vol 58 (3) ◽  
pp. 543-580
Author(s):  
V. V. Bavula
Keyword(s):  
Weyl Algebra ◽  
Differential Operators ◽  
Polynomial Algebra ◽  
Group Of Automorphisms ◽  
Algebra Ii ◽  
Noetherian Property

AbstractThe algebra of one-sided inverses of a polynomial algebra Pn in n variables is obtained from Pn by adding commuting left (but not two-sided) inverses of the canonical generators of the algebra Pn. The algebra is isomorphic to the algebra of scalar integro-differential operators provided that char(K) = 0. Ignoring the non-Noetherian property, the algebra belongs to a family of algebras like the nth Weyl algebra An and the polynomial algebra P2n. Explicit generators are found for the group Gn of automorphisms of the algebra and for the group of units of (both groups are huge). An analogue of the Jacobian homomorphism AutK-alg (Pn) → K* is introduced for the group Gn (notice that the algebra is non-commutative and neither left nor right Noetherian). The polynomial Jacobian homomorphism is unique. Its analogue is also unique for n > 2 but for n = 1, 2 there are exactly two of them. The proof is based on the following theorem that is proved in the paper:

scholarly journals Modified approach to generalized Stirling numbers via differential operators

2010 ◽  
Vol 23 (1) ◽  
pp. 115-120 ◽  
Author(s):  
B.S. El-Desouky ◽  
Nenad P. Cakić ◽  
Toufik Mansour
Keyword(s):  
Differential Operators ◽  
Stirling Numbers ◽  
Generalized Stirling Numbers

scholarly journals Set partitions with successions and separations

2005 ◽  
Vol 2005 (3) ◽  
pp. 451-463 ◽  
Author(s):  
Augustine O. Munagi
Keyword(s):  
Stirling Numbers ◽  
Set Partitions ◽  
Consecutive Integers

Partitions of the set{1,2,…,n}are classified as having successions if a block contains consecutive integers, and separated otherwise. This paper constructs enumeration formulas for such set partitions and some variations using Stirling numbers of the second kind.

scholarly journals On ζn-Weyl algebra Wr(ζn, Z)

1989 ◽  
Vol 113 ◽  
pp. 153-159 ◽  
Author(s):  
Hisasi Morikawa
Keyword(s):  
Associative Algebra ◽  
Weyl Algebra ◽  
Differential Operators

Weyl algebra is an associative algebra generated by two elements â and a over R such that the generating relation is given byâa — aâ = 1,which is isomorphic to the algebra of differential operators

scholarly journals Enumerating set partitions by the number of positions between adjacent occurrences of a letter

2010 ◽  
Vol 4 (2) ◽  
pp. 284-308 ◽  
Author(s):  
Toufik Mansour ◽  
Mark Shattuck ◽  
Stephan Wagner
Keyword(s):  
Generating Functions ◽  
Stirling Numbers ◽  
Asymptotic Estimates ◽  
Explicit Formulas ◽  
Set Partitions ◽  
Minimal Elements

A partition ? of the set [n] = {1, 2,...,n} is a collection {B1,...,Bk} of nonempty disjoint subsets of [n] (called blocks) whose union equals [n]. Suppose that the subsets Bi are listed in increasing order of their minimal elements and ? = ?1, ?2...?n denotes the canonical sequential form of a partition of [n] in which iEB?i for each i. In this paper, we study the generating functions corresponding to statistics on the set of partitions of [n] with k blocks which record the total number of positions of ? between adjacent occurrences of a letter. Among our results are explicit formulas for the total value of the statistics over all the partitions in question, for which we provide both algebraic and combinatorial proofs. In addition, we supply asymptotic estimates of these formulas, the proofs of which entail approximating the size of certain sums involving the Stirling numbers. Finally, we obtain comparable results for statistics on partitions which record the total number of positions of ? of the same letter lying between two letters which are strictly larger.

DIXMIER'S PROBLEM 6 FOR SOMEWHAT COMMUTATIVE ALGEBRAS AND DIXMIER'S PROBLEM 3 FOR THE RING OF DIFFERENTIAL OPERATORS ON A SMOOTH IRREDUCIBLE AFFINE CURVE

2005 ◽  
Vol 04 (05) ◽  
pp. 577-586 ◽  
Author(s):  
V. V. BAVULA
Keyword(s):  
Weyl Algebra ◽  
Differential Operators ◽  
Short Proof ◽  
Prime Factor ◽  
Affirmative Answer ◽  
Inner Derivation ◽  
Enveloping Algebra ◽  
Commutative Algebras ◽  
Rings Of Differential Operators ◽  
Kirillov Dimension

In [6], J. Dixmier posed six problems for the Weyl algebra A1 over a field K of characteristic zero. Problems 3, 5,and 6 were solved respectively by Joseph and Stein [7]; the author [1]; and Joseph [7]. Problems 1, 2, and 4 are still open. For an arbitrary algebra A, Dixmier's problem 6 is essentially aquestion: whether an inner derivation of the algebra A of the type ad f(a), a ∈ A, f(t) ∈ K[t], deg t(f(t)) > 1, has a nonzero eigenvalue. We prove that the answer is negative for many classes of algebras (e.g., rings of differential operators [Formula: see text] on smooth irreducible algebraic varieties, all prime factor algebras of the universal enveloping algebra [Formula: see text] of a completely solvable algebraic Lie algebra [Formula: see text]). This gives an affirmative answer (with a short proof) to an analogue of Dixmier's Problem 6 for certain algebras of small Gelfand–Kirillov dimension, e.g. the ring of differential operators [Formula: see text] on a smooth irreducible affine curve X, Usl(2), etc. (see [3] for details). In this paper an affirmative answer is given to an analogue of Dixmier's Problem 3 but for the ring [Formula: see text].

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