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{\bf Ulrich Tamm}
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{\bf Some Aspects of Hankel Matrices in Coding Theory and Combinatorics}
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Hankel matrices consisting of Catalan numbers have been
analyzed by various authors. Desainte-Catherine and Viennot found their
determinant to be $\prod_{1 \leq i \leq j \leq k}
{{i+j+2n}\over {i+j}}$ and related them to the \hfil\break 
Bender - Knuth conjecture.
The similar determinant formula \hfil\break
$\prod_{1 \leq i \leq j \leq k} {{i+j-1+2n}\over {i+j-1}}$
can be shown to hold for Hankel matrices \hfil\break
whose entries are successive middle
binomial coefficients ${{2m+1} \choose m}$. Generalizing the
Catalan numbers in a different direction, it can be shown that
determinants of Hankel matrices consisting of numbers ${{1}\over {3m+1}}
{{3m+1} \choose m}$ yield an alternate expression of two Mills -- \hfil\break 
Robbins --
Rumsey determinants important in the enumeration of plane partitions and
alternating sign matrices. Hankel matrices with determinant 1 were studied
by Aigner in the definition of Catalan -- like numbers. The well - known
relation of Hankel matrices to orthogonal polynomials further
yields a combinatorial application of the famous Berlekamp -- Massey algorithm
in Coding Theory, which can be applied in order to calculate the
coefficients in the three -- term recurrence of the family of
orthogonal polynomials related to the sequence of Hankel matrices.


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