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Sets of double and triple weights of trees
[article]

2011
*
arXiv
*
pre-print

Let T be a weighted

arXiv:0712.3026v3
fatcat:qsxbzinqfrbjnnvzz7dcl3qa3u
*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 ,.... ...##
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Revisiting the Top-Down Computation of BDD of Spanning Trees of a Graph and Its Tutte Polynomial

2019
*
IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences
*

with respect to its (proper) pathwidth, pw (ppw),

doi:10.1587/transfun.e102.a.1022
fatcat:qc6hgdmksfgqtewa23guijffti
*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. ...##
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Combinatorial approach to generalized Bell and Stirling numbers and boson normal ordering problem

2005
*
Journal of Mathematical Physics
*

We treat a general form of a boson string which is shown to be associated with generalizations of Stirling

doi:10.1063/1.1990120
fatcat:3iobncl7w5e65hhsdb3bghwxxu
*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. ...##
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Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning
[article]

2020
*
arXiv
*
pre-print

We show that using the ternary-

arXiv:1910.10746v2
fatcat:yhds6pl5jre5zkm2s6vkmat6we
*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. ...##
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Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning

2020
*
Quantum
*

We introduce a fermion-to-qubit mapping defined on ternary

doi:10.22331/q-2020-06-04-276
fatcat:4szy53mgeva3jgsq4oeom3xt6y
*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)*k*ϵ−2 times. ... Special thanks to Ryan Babbush for describing the problem*and*the beautiful results he*and*collaborators found during a surfing trip. ...##
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Schröder Coloring and Applications
[article]

2019
*
arXiv
*
pre-print

On the other hand, we derive partial

arXiv:1908.08103v1
fatcat:6riyospcwzfplasqu4k6h6veru
*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. ...##
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The inversion enumerator for labeled trees

1968
*
Bulletin of the American Mathematical Society
*

•It is also worth noting that (1)

doi:10.1090/s0002-9904-1968-11888-9
fatcat:xq4r4caaezfbbggwmiaarjwfkm
*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*. ...##
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Generating restricted classes of involutions, Bell and Stirling permutations

2010
*
European journal of combinatorics (Print)
*

It generates permutations at each recursive step

doi:10.1016/j.ejc.2009.03.028
fatcat:tyh5ovhcaja7lgnov2ryl2tsri
*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 ]. ...##
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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*. ...##
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Bijections and the Riordan group

2003
*
Theoretical Computer Science
*

In many cases this leads to both a combinatorial interpretation

doi:10.1016/s0304-3975(03)00227-5
fatcat:pymkcufk5bbsvb4g5ksxlpsyk4
*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. ...##
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Error-correcting entanglement swapping using a practical logical photon encoding
[article]

2021
*
arXiv
*
pre-print

Our approach uses a

arXiv:2101.11082v4
fatcat:hz7g2je2a5brrpup2yueuxvlwi
*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 . ...##
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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. ...##
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Synchronization Properties of Trees in the Kuramoto Model

2013
*
SIAM Journal on Applied Dynamical Systems
*

For several classes of

doi:10.1137/120899728
fatcat:brqlced3a5c2hmih3x7icdei64
*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. ...##
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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. ...##
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Ranked Schröder Trees
[article]

2019
*
arXiv
*
pre-print

Through the paper, we present these links, exhibit some parameters in typical large

arXiv:1808.08376v5
fatcat:szhbdu3urvbepaf37iiy6j7mru
*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*). ...
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