Number Walls in CombinatoricsMichael SomosCleveland State University
September 20,

ABSTRACT


Number walls of Toeplitz determinants have recently appeared by name in books by Sloane & Plouffe and Conway & Guy. Similar arrays of Hankel determinants can be used to make unexpected connections between sequences of integers with combinatorial interpretations. For this see recent articles by Aigner, Ehrenborg, or Dumont & Zeng, although the study of Hankel determinants of combinatorial sequences goes back to Radoux in 1979 and probably earlier. The work of Gessel and Viennot with nonintersecting lattice paths is also related. Several Hankel number walls have nice symmetry properties with respect to their diagonal or are otherwise noteworthy. I have many examples. In July I made a simple observation using number walls to connect a sequence related to Catalan numbers and Motzkin numbers to the Somos4 sequence and also to some alternating sign matrix enumeration sequences. Some of this is partly based on an article by David Cantor which relates hyperelliptic curves to Hankel determinants via Pade approximants which go back to Jacobi. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

