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Sets of double and triple weights of trees [article]

Elena Rubei
2011 arXiv   pre-print
Let T be a weighted tree with n leaves. Let D_i,j be the distance between the leaves i and j. Let D_i,j,k= (D_i,j + D_j,k +D_i,k)/2. We will call such numbers "triple weights" of the tree.  ...  By using the same ideas,we find also necessary and sufficient conditions for a set of real numbers indexed by 3-subsets of an n-set to be the set of the triple weights of a tree with n leaves.  ...  We call such numbers "triple weights" of the tree. More generally define the k-weights of the tree D i 1 ,....  ... 
arXiv:0712.3026v3 fatcat:qsxbzinqfrbjnnvzz7dcl3qa3u

Revisiting the Top-Down Computation of BDD of Spanning Trees of a Graph and Its Tutte Polynomial

Farley Soares OLIVEIRA, Hidefumi HIRAISHI, Hiroshi IMAI
2019 IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences  
with respect to its (proper) pathwidth, pw (ppw), and obtain a bound of O * (Bell min{pw +1,ppw} ), where Bell n denotes the n-th Bell number, defined as the number of partitions of a set of n elements  ...  We further investigate the case of complete graphs in terms of Bell numbers and related combinatorics, obtaining a time complexity bound of Bell n−O (n/ log n) .  ...  Acknowledgments This work was supported by JSPS KAKENHI Grant Numbers 15H01677, 16K12392, 17K12639. We thank the two anonymous reviewers for their constructive comments and suggestions.  ... 
doi:10.1587/transfun.e102.a.1022 fatcat:qc6hgdmksfgqtewa23guijffti

Combinatorial approach to generalized Bell and Stirling numbers and boson normal ordering problem

M. A. Méndez, P. Blasiak, K. A. Penson
2005 Journal of Mathematical Physics  
We treat a general form of a boson string which is shown to be associated with generalizations of Stirling and Bell numbers.  ...  We consider the numbers arising in the problem of normal ordering of expressions in canonical boson creation and annihilation operators.  ...  One of the authors ͑P.B.͒ wishes to thank the Polish Ministry of Scientific Research and Information Technology for support under Grant No. 1P03B 051 26.  ... 
doi:10.1063/1.1990120 fatcat:3iobncl7w5e65hhsdb3bghwxxu

Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning [article]

Zhang Jiang, Amir Kalev, Wojciech Mruczkiewicz, Hartmut Neven
2020 arXiv   pre-print
We show that using the ternary-tree mapping one can determine the elements of all k-fermion RDMs, to precision ϵ, by repeating a single quantum circuit for ≲ (2n+1)^k ϵ^-2 times.  ...  The mapping has a simple structure and is optimal in the sense that it is impossible to construct Pauli operators in any fermion-to-qubit mapping acting nontrivially on less than log_3(2n) qubits on average  ...  Special thanks to Ryan Babbush for describing the problem and the beautiful results he and collaborators found during a surfing trip.  ... 
arXiv:1910.10746v2 fatcat:yhds6pl5jre5zkm2s6vkmat6we

Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning

Zhang Jiang, Amir Kalev, Wojciech Mruczkiewicz, Hartmut Neven
2020 Quantum  
We introduce a fermion-to-qubit mapping defined on ternary trees, where any single Majorana operator on an n-mode fermionic system is mapped to a multi-qubit Pauli operator acting nontrivially on ⌈log3⁡  ...  We show that one can determine individual elements of all k-fermion RDMs in parallel, to precision ϵ, by repeating a single quantum circuit for ≲(2n+1)kϵ−2 times.  ...  Special thanks to Ryan Babbush for describing the problem and the beautiful results he and collaborators found during a surfing trip.  ... 
doi:10.22331/q-2020-06-04-276 fatcat:4szy53mgeva3jgsq4oeom3xt6y

Schröder Coloring and Applications [article]

Daniel Birmajer, Juan B. Gil, Juan D. Gil, Michael D. Weiner
2019 arXiv   pre-print
On the other hand, we derive partial Bell polynomial identities for the little and large Schr\"oder numbers, which allow us to obtain explicit enumeration formulas.  ...  integer slope, ordered rooted trees, and simple rooted outerplanar maps.  ...  (tree) Number of ordered trees with no vertex of outdegree 1 and having n+1 leaves. (poly) Number of dissections of a convex (n + 2)-gon by nonintersecting diagonals.  ... 
arXiv:1908.08103v1 fatcat:6riyospcwzfplasqu4k6h6veru

The inversion enumerator for labeled trees

C. L. Mallows, John Riordan
1968 Bulletin of the American Mathematical Society  
•It is also worth noting that (1) and the recurrence for Bell polynomials imply Jn+l(%) = ]C Cn-l,*(l + *+•••+ X k )J k +i(x)J n -. k (x). REFERENCE 1. G.  ...  permutation is the number of transpositions needed to restore the standard order), and with J r n (l)=w n~2 , which suggests labeled trees.  ... 
doi:10.1090/s0002-9904-1968-11888-9 fatcat:xq4r4caaezfbbggwmiaarjwfkm

Generating restricted classes of involutions, Bell and Stirling permutations

Maddalena Poneti, Vincent Vajnovszki
2010 European journal of combinatorics (Print)  
It generates permutations at each recursive step and slight modifications of it produce generating algorithms for Bell permutations and involutions.  ...  We obtain, as particular cases, generating algorithms for permutations counted by the Stirling numbers of the first and second kind, even permutations, fixed-point-free involutions and derangements.  ...  We denote by Tail(π ) the set of tails of π ∈ S n and by B n,k the set of Bell permutations with k cycles, and B n,k is counted by the Stirling number of the second kind [17, pp. 33 ].  ... 
doi:10.1016/j.ejc.2009.03.028 fatcat:tyh5ovhcaja7lgnov2ryl2tsri

Page 5652 of Mathematical Reviews Vol. , Issue 88k [page]

1988 Mathematical Reviews  
The authors introduce modified Bell numbers and derive a recur- rence for them, which is useful for calculating high precision values of the Bell numbers.  ...  A multivari- ate analogue is also derived and several examples cited. Frank K.  ... 

Bijections and the Riordan group

Louis W. Shapiro
2003 Theoretical Computer Science  
In many cases this leads to both a combinatorial interpretation and to ECO rewriting rules.  ...  One of the cornerstone ideas in mathematics is to take a problem and to look at it in a bigger space. In this paper we examine combinatorial sequences in the context of the Riordan group.  ...  have k descendants and they will have the labels c 1 ; c 2 ; : : : ; c k . we then count the number of nodes at each level in the tree where the root is at level 0.  ... 
doi:10.1016/s0304-3975(03)00227-5 fatcat:pymkcufk5bbsvb4g5ksxlpsyk4

Error-correcting entanglement swapping using a practical logical photon encoding [article]

Paul Hilaire, Edwin Barnes, Sophia E. Economou, Frédéric Grosshans
2021 arXiv   pre-print
Our approach uses a tree graph state logical encoding, which can be produced deterministically with a few quantum emitters, and achieves near-deterministic logical photonic Bell state measurements while  ...  Therefore, photon losses and the 50\% success probablity upper bound of Bell state measurements pose a critical limitation to photonic quantum technologies.  ...  (c) Performance of the static (top) and dynamic (bottom) protocols as a function of the number of photons in the tree, for η = 95% and E = 10 −5 .  ... 
arXiv:2101.11082v4 fatcat:hz7g2je2a5brrpup2yueuxvlwi

Page 624 of Mathematical Reviews Vol. , Issue 87b [page]

1987 Mathematical Reviews  
Also, the determination of all trees T for which the Ramsey number r(T, Kim) = m+n-2 follows from this characterization. Ralph Faudree (Memphis, Tenn.) Chung, F. R. K. (1-BELL); 87b:05071 Erdés, P.  ...  K. (1-BELL); Graham, R. L. (1-BELL) On complete bipartite subgraphs contained in spanning tree complements. Studies in pure mathematics, 83-90, Birkhauser, Basel-Boston, Mass., 1983.  ... 

Synchronization Properties of Trees in the Kuramoto Model

Anthony H. Dekker, Richard Taylor
2013 SIAM Journal on Applied Dynamical Systems  
For several classes of tree, and for both uniform and Gaussian vertex frequency distributions, we provide tight closed form bounds and empirical expressions for the expected value of the critical coupling  ...  We also provide several bounds on the expected value of the critical coupling for all trees.  ...  However, as Figure 9 shows, the binary tree is intermediate between the chain and the dumb-bell. 3.5. Tadpoles.  ... 
doi:10.1137/120899728 fatcat:brqlced3a5c2hmih3x7icdei64

Page 2796 of Mathematical Reviews Vol. , Issue 97E [page]

1997 Mathematical Reviews  
97e:1 1031 11 power series, evaluate the nth derivative of a composite function, calculate Bernoulli, Euler, Bell and other numbers, and evaluate Bell and Gauss polynomials, the cycle index, etc.  ...  Let F(a) denote this tree grown to depth k, and let .7,*(a) denote the pruned tree resulting from the removal of all nodes n = 0 mod 3.  ... 

Ranked Schröder Trees [article]

Olivier Bodini and Antoine Genitrini and Mehdi Naima
2019 arXiv   pre-print
Through the paper, we present these links, exhibit some parameters in typical large trees and conclude the studies with efficient uniform samplers.  ...  In particular, it does not answer the question: how many different phylogenetic stories lead to the creation of n species and what is the average time to get there?  ...  The one-to-one correspondence between weakly increasing Schröder trees and ordered Bell numbers gives the combinatorial proof of the distribution for (g n,k ).  ... 
arXiv:1808.08376v5 fatcat:szhbdu3urvbepaf37iiy6j7mru
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