scholarly journals DEGREE OF THE -OPERATOR AND NONCROSSING PARTITIONS

2019 ◽  
Vol 101 (2) ◽  
pp. 186-200
Author(s):  
HAO SUN
Keyword(s):  
Maximal Degree ◽  
Noncrossing Partitions

The $W$-operator, $W([n])$, generalises the cut-and-join operator. We prove that $W([n])$ can be written as the sum of $n!$ terms, each term corresponding uniquely to a permutation in $S_{\!n}$. We also prove that there is a correspondence between the terms of $W([n])$ with maximal degree and noncrossing partitions.

scholarly journals Non-defectivity of Grassmannian bundles over a curve

2016 ◽  
Vol 27 (07) ◽  
pp. 1640002 ◽  
Author(s):  
Insong Choe ◽  
George H. Hitching
Keyword(s):  
Vector Bundles ◽  
Manuscripta Math ◽  
Maximal Degree ◽  
Geometric Proof ◽  
Secant Varieties ◽  
Grassmann Bundle ◽  
Positive Rank

Let [Formula: see text] be the Grassmann bundle of two-planes associated to a general bundle [Formula: see text] over a curve [Formula: see text]. We prove that an embedding of [Formula: see text] by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the isotropic Segre invariant for maximal isotropic sub-bundles of orthogonal bundles over [Formula: see text], analogous to those given for vector bundles and symplectic bundles in [I. Choe and G. H. Hitching, Secant varieties and Hirschowitz bound on vector bundles over a curve, Manuscripta Math. 133 (2010) 465–477, I. Choe and G. H. Hitching, Lagrangian sub-bundles of symplectic vector bundles over a curve, Math. Proc. Cambridge Phil. Soc. 153 (2012) 193–214]. From the non-defectivity, we also deduce an interesting feature of a general orthogonal bundle of even rank over [Formula: see text], contrasting with the classical and symplectic cases: a general maximal isotropic sub-bundle of maximal degree intersects at least one other such sub-bundle in positive rank.

scholarly journals Combinatorics of Exceptional Sequences in Type A

10.37236/6251 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Alexander Garver ◽  
Kiyoshi Igusa ◽  
Jacob P. Matherne ◽  
Jonah Ostroff
Keyword(s):  
Dynkin Diagram ◽  
Type A ◽  
Linear Extensions ◽  
Quiver Representations ◽  
Noncrossing Partitions ◽  
Underlying Graph ◽  
Exceptional Sequences

Exceptional sequences are certain sequences of quiver representations.  We introduce a class of objects called strand diagrams and use these to classify exceptional sequences of representations of a quiver whose underlying graph is a type $\mathbb{A}_n$ Dynkin diagram. We also use variations of these objects to classify $c$-matrices of such quivers, to interpret exceptional sequences as linear extensions of explicitly constructed posets, and to give a simple bijection between exceptional sequences and certain saturated chains in the lattice of noncrossing partitions. 

scholarly journals Euler Characteristic of the Truncated Order Complex of Generalized Noncrossing Partitions

10.37236/232 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
D. Armstrong ◽  
C. Krattenthaler
Keyword(s):  
Euler Characteristic ◽  
Order Complex ◽  
Reflection Groups ◽  
Noncrossing Partitions ◽  
Minimal Elements ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Phd Thesis

The purpose of this paper is to complete the study, begun in the first author's PhD thesis, of the topology of the poset of generalized noncrossing partitions associated to real reflection groups. In particular, we calculate the Euler characteristic of this poset with the maximal and minimal elements deleted. As we show, the result on the Euler characteristic extends to generalized noncrossing partitions associated to well-generated complex reflection groups.

scholarly journals Pairs of noncrossing free Dyck paths and noncrossing partitions

2009 ◽  
Vol 309 (9) ◽  
pp. 2834-2838 ◽  
Author(s):  
William Y.C. Chen ◽  
Sabrina X.M. Pang ◽  
Ellen X.Y. Qu ◽  
Richard P. Stanley
Keyword(s):  
Noncrossing Partitions ◽  
Dyck Paths

scholarly journals Clique coverings of graphs V: maximal-clique partitions

1982 ◽  
Vol 25 (3) ◽  
pp. 337-356 ◽  
Author(s):  
N.J. Pullman ◽  
H. Shank ◽  
W.D. Wallis
Keyword(s):  
Maximal Clique ◽  
Maximal Degree ◽  
Open Problems ◽  
Partition Number

A maximal-clique partition of a graph G is a way of covering G with maximal complete subgraphs, such that every edge belongs to exactly one of the subgraphs. If G has a maximal-clique partition, the maximal-clique partition number of G is the smallest cardinality of such partitions. In this paper the existence of maximal-clique partitions is discussed – for example, we explicitly describe all graphs with maximal degree at most four which have maximal-clique partitions - and discuss the maximal-clique partition number and its relationship to other clique covering and partition numbers. The number of different maximal-clique partitions of a given graph is also discussed. Several open problems are presented.

scholarly journals From m-clusters to m-noncrossing partitions via exceptional sequences

2011 ◽  
Vol 271 (3-4) ◽  
pp. 1117-1139 ◽  
Author(s):  
Aslak Bakke Buan ◽  
Idun Reiten ◽  
Hugh Thomas
Keyword(s):  
Noncrossing Partitions ◽  
Exceptional Sequences

scholarly journals Improved Resource State for Verifiable Blind Quantum Computation

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 996
Author(s):  
Qingshan Xu ◽  
Xiaoqing Tan ◽  
Rui Huang
Keyword(s):  
Quantum Computing ◽  
Quantum Computation ◽  
Fault Tolerant ◽  
Maximal Degree ◽  
Recent Advances ◽  
Quantum Computations ◽  
Blind Quantum Computation ◽  
Computation Graph ◽  
Resource State

Recent advances in theoretical and experimental quantum computing raise the problem of verifying the outcome of these quantum computations. The recent verification protocols using blind quantum computing are fruitful for addressing this problem. Unfortunately, all known schemes have relatively high overhead. Here we present a novel construction for the resource state of verifiable blind quantum computation. This approach achieves a better verifiability of 0.866 in the case of classical output. In addition, the number of required qubits is 2N+4cN, where N and c are the number of vertices and the maximal degree in the original computation graph, respectively. In other words, our overhead is less linear in the size of the computational scale. Finally, we utilize the method of repetition and fault-tolerant code to optimise the verifiability.

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