Riordan paths and derangements

Chen, William Y. C.; Deng, Eva Y. P.; Yang, Laura L. M.

doi:10.1016/j.disc.2007.05.001

Cited by 5 publications

(5 citation statements)

References 10 publications

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“…Now we compute C ∞ (t) by considering the so called "Riordan paths" [32,33] (Motzkin paths [34] with no horizontal steps at the bottom line) weighted through q and r. (see Fig. 3).…”

Section: Autocorrelator At T = ∞mentioning

confidence: 99%

Dissipative spin chain as a non-Hermitian Kitaev ladder

Shibata

Katsura

2019

Phys. Rev. B

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We derive exact results for the Lindblad equation for a quantum spin chain (one-dimensional quantum compass model) with dephasing noise. The system possesses doubly degenerate nonequilibrium steady states due to the presence of a conserved charge commuting with the Hamiltonian and Lindblad operators. We show that the system can be mapped to a non-Hermitian Kitaev model on a two-leg ladder, which is solvable by representing the spins in terms of Majorana fermions. This allows us to study the Liouvillian gap, the inverse of relaxation time, in detail. We find that the Liouvillian gap increases monotonically when the dissipation strength γ is small, while it decreases monotonically for large γ, implying a kind of phase transition in the first decay mode. The Liouvillian gap and the transition point are obtained in closed form in the case where the spin chain is critical. We also obtain the explicit expression for the autocorrelator of the edge spin. The result implies the suppression of decoherence when the spin chain is in the topological regime.

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“…Now we compute C ∞ (t) by considering the so called "Riordan paths" [32,33] (Motzkin paths [34] with no horizontal steps at the bottom line) weighted through q and r. (see Fig. 3).…”

Section: Autocorrelator At T = ∞mentioning

confidence: 99%

Dissipative spin chain as a non-Hermitian Kitaev ladder

Shibata

Katsura

2019

Phys. Rev. B

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“…The explicit formula for the kth Riordan numbers is as in Eq. (5) (see [10] for a simple combinatorial proof), and asymptotically r k = Θ * (3 k ).…”

Section: The Order Of the Ring Componentmentioning

confidence: 99%

“…The DB-matching of size k with specified χ and z (the label of the leftmost point on U) will be denoted by DB(k, χ, z). 10 The dual trees of DB-matchings have the following structure (we denote the vertices of D(M ) identically to the corresponding faces of the map of M ): There is a path B 0 D 1 D 2 . .…”

Section: Small Components For Even K (Pairs)mentioning

confidence: 99%

Disjoint Compatibility Graph of Non-Crossing Matchings of Points in Convex Position

Aichholzer

Asinowski

Miltzow

2015

Electron. J. Combin.

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Let X 2k be a set of 2k labeled points in convex position in the plane. We consider geometric non-intersecting straight-line perfect matchings of X 2k . Two such matchings, M and M ′ , are disjoint compatible if they do not have common edges, and no edge of M crosses an edge of M ′ . Denote by DCM k the graph whose vertices correspond to such matchings, and two vertices are adjacent if and only if the corresponding matchings are disjoint compatible. We show that for each k ≥ 9, the connected components of DCM k form exactly three isomorphism classes -namely, there is a certain number of isomorphic small components, a certain number of isomorphic medium components, and one big component. The number and the structure of small and medium components is determined precisely.

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“…2 . The paths with such constraints and j = 0 are known in mathematical literature as Riordan paths (and their multiplicity as Riordan numbers 24 25 26 . For j ≥ 0 these are Riordan arrays 27 , whose generating functions are known to be 27 …”

Section: Resultsmentioning

confidence: 99%

From entanglement witness to generalized Catalan numbers

Cohen

Hansen

Itzhaki

2016

Sci Rep

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Being extremely important resources in quantum information and computation, it is vital to efficiently detect and properly characterize entangled states. We analyze in this work the problem of entanglement detection for arbitrary spin systems. It is demonstrated how a single measurement of the squared total spin can probabilistically discern separable from entangled many-particle states. For achieving this goal, we construct a tripartite analogy between the degeneracy of entanglement witness eigenstates, tensor products of SO(3) representations and classical lattice walks with special constraints. Within this framework, degeneracies are naturally given by generalized Catalan numbers and determine the fraction of states that are decidedly entangled and also known to be somewhat protected against decoherence. In addition, we introduce the concept of a “sterile entanglement witness”, which for large enough systems detects entanglement without affecting much the system’s state. We discuss when our proposed entanglement witness can be regarded as a sterile one.

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Riordan paths and derangements

Cited by 5 publications

References 10 publications

Dissipative spin chain as a non-Hermitian Kitaev ladder

Dissipative spin chain as a non-Hermitian Kitaev ladder

Disjoint Compatibility Graph of Non-Crossing Matchings of Points in Convex Position

From entanglement witness to generalized Catalan numbers

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