Number Walls in Combinatorics

Michael Somos

Cleveland State University

September 20,
refreshments at 3:45pm


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 Somos-4 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.

Speaker's Contact Info: somos(at-sign)

Return to seminar home page

Combinatorics Seminar, Mathematics Department, MIT, sara(at-sign)

Page loaded on August 31, 2000 at 10:23 PM. Copyright © 1998-99, Sara C. Billey. All rights reserved.