Representing elements of the Weyl algebra by labeled trees

Asakly, Walaa; Mansour, Toufik; Schork, Matthias

doi:10.1063/1.4792655

Cited by 5 publications

(9 citation statements)

References 18 publications

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“…Given a Dyck word w, let W w be the set of (labels of) unit squares that lie below the staircase path of w, and completely above the line x = y. For example, if w = (xD) n then W w = ∅, and if w = xxDxxDxDDD then W w = {(1, 2), (2, 3), (2,4), (3,4), (3,5), (4, 5)}. Define a graph H w on vertex set {1, .…”

Section: A Closely Related Combinatorial Interpretationmentioning

confidence: 99%

“…. , y i }, (2,4), (3,5)}, and the edge set of H w is {{1, 2}} ∪ {{2, 3}, {2, 4}, {3, 4}} ∪ {{3, 4}, {3, 5}, {4, 5}}, i.e., it is composed of cliques on the vertex sets {1, 2}, {2, 3, 4}, and {3, 4, 5} (see Figure 2, where the coordinates of the peaks of the path are underlined).…”

Section: A Closely Related Combinatorial Interpretationmentioning

confidence: 99%

“…The connection to rook polynomials was revisited by Blasiak et al in [5], and in [27] the same authors explored connections to Feynman diagrams. In [3], Asakly et al showed how to read off the normal order of a word from a certain labeled tree, and in [20] Ma et al explored connections between normal ordering and context-free grammars.…”

Section: Introductionmentioning

confidence: 99%

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Combinatorially interpreting generalized Stirling numbers

Engbers

Galvin

Hilyard

2015

European Journal of Combinatorics

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Let $w$ be a word in alphabet $\{x,D\}$ with $m$ $x$'s and $n$ $D$'s. Interpreting "$x$" as multiplication by $x$, and "$D$" as differentiation with respect to $x$, the identity $wf(x) = x^{m-n}\sum_k S_w(k) x^k D^k f(x)$, valid for any smooth function $f(x)$, defines a sequence $(S_w(k))_k$, the terms of which we refer to as the {\em Stirling numbers (of the second kind)} of $w$. The nomenclature comes from the fact that when $w=(xD)^n$, we have $S_w(k)={n \brace k}$, the ordinary Stirling number of the second kind. Explicit expressions for, and identities satisfied by, the $S_w(k)$ have been obtained by numerous authors, and combinatorial interpretations have been presented. Here we provide a new combinatorial interpretation that retains the spirit of the familiar interpretation of ${n \brace k}$ as a count of partitions. Specifically, we associate to each $w$ a quasi-threshold graph $G_w$, and we show that $S_w(k)$ enumerates partitions of the vertex set of $G_w$ into classes that do not span an edge of $G_w$. We also discuss some relatives of, and consequences of, our interpretation, including $q$-analogs and bijections between families of labelled forests and sets of restricted partitions.Comment: To appear in Eur. J. Combin., doi:10.1016/j.ejc.2014.07.00

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Section: A Closely Related Combinatorial Interpretationmentioning

confidence: 99%

Section: A Closely Related Combinatorial Interpretationmentioning

confidence: 99%

See 1 more Smart Citation

Combinatorially interpreting generalized Stirling numbers

Engbers

Galvin

Hilyard

2015

European Journal of Combinatorics

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“…I would encourage readers to look in this direction. In Asakly et al (2013) the relation of the Weyl algebra to labeled rooted trees is given but I do not see a clear relation with our work.…”

Section: Inductive Proofmentioning

confidence: 64%

“…Acknowledgments J. H. Przytycki was partially supported by the GWU REF Grant, and Simons Collaboration Grant-316446. The author would like to thank a referee for directing his attention to Asakly et al (2013) and Schork (2011, 2016).…”

mentioning

confidence: 99%

q-Polynomial Invariant of Rooted Trees

Przytycki

2016

Arnold Math J.

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Abstract. We describe in this note a new invariant of rooted trees. We argue that the invariant is interesting on it own, and that it has connections to knot theory and homological algebra. However, the real reason that we propose this invariant to readers is that we deal here with an elementary, interesting, new mathematics, and after reading this essay readers can take part in developing the topic, inventing new results and connections to other disciplines of mathematics, and likely, statistical mechanics, and combinatorial biology. We also provide a (free) translation of the paper in Polish.

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Hiking a generalized Dyck path: A tractable way of calculating multimode boson evolution operators

Brádler

2015

Computer Physics Communications

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A time evolution operator in the interaction picture is given by exponentiating an interaction Hamiltonian H. Important examples of Hamiltonians, often encountered in quantum optics, condensed matter and high energy physics, are of a general form H = r(A † − A), where A is a multimode boson operator and r is the coupling constant. If no simple factorization formula for the evolution operator exists, the calculation of the evolution operator is a notoriously difficult problem. In this case the only available option may be to Taylor expand the operator in r and act on a state of interest ψ. But this brute-force method quickly hits the complexity barrier since the number of evaluated expressions increases exponentially. We relate a combinatorial structure called Dyck paths to the action of a boson word (monomial) and a large class of monomial sums on a quantum state ψ. This allows us to cross the exponential gap and make the problem of a boson unitary operator evaluation computationally tractable by achieving polynomial-time complexity for an extensive family of physically interesting multimode Hamiltonians. We further test our method on a cubic boson Hamiltonian whose Taylor series is known to diverge for all nonzero values of the coupling constant and an analytic continuation via a Padé approximant must be performed.

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Representing elements of the Weyl algebra by labeled trees

Abstract: It is shown how arbitrary elements of the Weyl algebra can be represented by labeled plane trees using normal ordering. Several examples are treated and combinatorial aspects are discussed. Also, possible avenues for future research are described.

Cited by 5 publications

References 18 publications

Combinatorially interpreting generalized Stirling numbers

Combinatorially interpreting generalized Stirling numbers

q-Polynomial Invariant of Rooted Trees

Hiking a generalized Dyck path: A tractable way of calculating multimode boson evolution operators

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