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 Π1≤i≤j≤ki+j+2n/i+j and related them to the Bender - Knuth conjecture. The similar determinant formula Π1≤i≤j≤ki+j- 1+2n/i+j-1 can be shown to hold for Hankel matrices whose entries are successive middle binomial coefficients (m2m+1). Generalizing the Catalan numbers in a different direction, it can be shown that determinants of Hankel matrices consisting of numbers 1/3m+1(m3m+1) 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.