Some Aspects of Hankel Matrices in Coding Theory and Combinatorics

Abstract

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 Bender - Knuth conjecture. The similar determinant formula $\prod_{1 \leq i \leq j \leq k} {{i+j-1+2n}\over {i+j-1}}$ can be shown to hold for Hankel matrices 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 – 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.