Date: Friday, October 04, 2019
Location: 4096 East Hall (3:00 PM to 4:00 PM)
Title: Total nonnegativity of some combinatorial matrices
Abstract: Many combinatorial matrices  such as those of binomial coefficients, Stirling numbers of both kinds, and Lah numbers  are known to be totally nonnegative, meaning that all minors (determinants of square submatrices) are nonnegative.
The examples noted above can be placed in a common framework: for each one there is a nondecreasing sequence (a_1, a_2, ...), and a sequence (e_1, e_2, ...), such that the (m,k)entry of the matrix is the coefficient of the polynomial (xa_1)...(xa_k) in the expansion of (xe_1)...(xe_m) as a linear combination of the polynomials 1, xa_1, ..., (xa_1)...(xa_m).
We consider this general framework. For a nondecreasing sequence a_1, a_2, ... we establish necessary and sufficient conditions on the sequence e_1, e_2, ... for the corresponding matrix to be totally nonnegative.
This is joint work with Adrian Pacurar, Notre Dame.
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Speaker: David Galvin
Institution: University of Notre Dame
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