Speaker
Description
A matrix of polynomials is called coefficientwise totally positive (CTP)
if all its minors are polynomials with positive coefficients. We verify
this property for a few families of infinite lower-triangular matrices.
During the talk we will, in particular, touch upon CTP triangular
matrices stemming from orthogonal and multiple orthogonal polynomials.
It is also intriguing to consider triangles generated by other types of
recurrence relations. Almost 30 years ago Brenti conjectured that the
Eulerian triangle (the lower-triangular matrix of Eulerian numbers,
A008292 in OEIS) is totally positive. The Eulerian numbers appear in
polylogarithms of negative integer orders and count the number of
permutations of $1,2,...,n+1$ with $k$ excedances. We introduce a more
general family of matrices that experimentally appear to be CTP. Then we
prove that its special subfamily including the reversed Stirling subset
triangle (A008278) is indeed CTP. This result is new and required a more
delicate approach, than total positivity in the non-reversed case (cf.
A008277 in OEIS).
