Kostka polynomial: Difference between revisions

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In mathematics, the '''Johnson scheme''', named after [[Selmer M. Johnson]], is also known as the triangular [[association scheme]]. It consists of the set of all binary vectors ''X'' of length ''ℓ'' and weight&nbsp;''n'', such that <math>v=\left|X\right|=\binom{\ell}{n}</math>.<ref>P. Delsarte and V. I. Levenshtein, “Association schemes and coding theory,“  ''IEEE Trans. Inform. Theory'', vol. 44, no. 6, pp. 2477&ndash;2504, 1998.</ref><ref>P. Camion, "Codes and Association Schemes: Basic Properties of Association Schemes Relevant to Coding," in ''Handbook of Coding Theory'', V. S. Pless and W. C. Huffman, Eds., Elsevier, The Netherlands, 1998.</ref><ref>F. J. MacWilliams and N. J. A. Sloane, ''The Theory of Error-Correcting Codes'', Elsevier, New York, 1978.</ref>  Two vectors&nbsp;''x'',&nbsp;''y''&nbsp;&isin;&nbsp;''X'' are called ''i''th associates if dist(''x'',&nbsp;''y'')&nbsp;=&nbsp;2''i'' for ''i''&nbsp;=&nbsp;0,&nbsp;1,&nbsp;...,&nbsp;''n''. The [[eigenvalues]] are given by
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: <math>
p_{i}\left(k\right)=E_{i}\left(k\right),
</math>
 
: <math>
q_{k}\left(i\right)=\frac{\mu_{k}}{v_{i}}E_{i}\left(k\right),
</math>
 
where
 
: <math>
\mu_{i}=\frac{\ell-2i+1}{\ell-i+1}\binom{\ell}{i},
</math>
 
and ''E''<sub>''k''</sub>(''x'') is an '''Eberlein polynomial''' defined by
 
: <math>E_{k}\left(x\right)=\sum_{j=0}^{k}(-1)^{j}\binom{x}{j} \binom{n-x}{k-j}\binom{\ell-n-x}{k-j},\qquad k=0,\ldots,n.</math>
 
==References==
{{reflist}}
 
[[Category:Combinatorics]]

Latest revision as of 20:47, 5 June 2014

Im Kathlene and was born on 2 December 1983. My hobbies are Coloring and Games Club - Dungeons and Dragons, Monopoly, Etc..

Feel free to surf to my webpage зеленый чай для похудения

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