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Physical Sciences and Mathematics Commons™

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Discrete Mathematics and Combinatorics

Series

2001

Articles 1 - 4 of 4

Full-Text Articles in Physical Sciences and Mathematics

Enumeration Of Equicolourable Trees, Nicholas Pippenger Jan 2001

Enumeration Of Equicolourable Trees, Nicholas Pippenger

All HMC Faculty Publications and Research

A tree, being a connected acyclic graph, can be bicolored in two ways, which differ from each other by exchange of the colors. We shall say that a tree is equicolorable if these bicolorings assign the two colors to equal numbers of vertices. Labelled equicolored trees have been enumerated several times in the literature, and from this result it is easy to enumerate labelled equicolorable trees. The result is that the probability that a randomly chosen n-vertex labelled tree is equicolorable is asymptotically just twice the probability that its vertices would be equicolored if they were assigned colors by …

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Wholes And Parts In General Systems Methodology, Martin Zwick Jan 2001

Wholes And Parts In General Systems Methodology, Martin Zwick

Systems Science Faculty Publications and Presentations

Reconstructability analysis (RA) decomposes wholes, namely data in the form either of set-theoretic relations or multivariate probability distributions, into parts, namely relations or distributions involving subsets of variables. Data is modeled and compressed by variablebased decomposition, by more general state-based decomposition, or by the use of latent variables. Models, which specify the interdependencies among the variables, are selected to minimize error and complexity.

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Nested Balanced Incomplete Block Designs, J. P. Morgan, D. A. Preece, D. H. Rees Jan 2001

Nested Balanced Incomplete Block Designs, J. P. Morgan, D. A. Preece, D. H. Rees

Mathematics & Statistics Faculty Publications

If the blocks of a balanced incomplete block design (BIBD) with v treatments and with parameters (v; b1;r;k1) are each partitioned into sub-blocks of size k2, and the b2 =b1k1=k2 sub-blocks themselves constitute a BIBD with parameters (v; b2;r;k2), then the system of blocks, sub-blocks and treatments is, by de4nition, a nested BIBD (NBIBD). Whist tournaments are special types of NBIBD with k1 =2k2= 4. Although NBIBDs were introduced in the statistical literature in 1967 and have subsequently received occasional attention there, …

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The Multiplicities Of A Dual-Thin Q-Polynomial Association Scheme, Bruce E. Sagen, John S. Caughman Iv Jan 2001

The Multiplicities Of A Dual-Thin Q-Polynomial Association Scheme, Bruce E. Sagen, John S. Caughman Iv

Mathematics and Statistics Faculty Publications and Presentations

Let Y=(X,{Ri}1≤i≤D) denote a symmetric association scheme, and assume that Y is Q-polynomial with respect to an ordering E0,...,ED of the primitive idempotents. Bannai and Ito conjectured that the associated sequence of multiplicities mi (0≤i≤D) of Yis unimodal. Talking to Terwilliger, Stanton made the related conjecture that mi≤mi+1 and mi≤mD−i for i<D/2. We prove that if Y is dual-thin in the sense of Terwilliger, then the …

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