Coefficientwise total positivity of some matrices defined by linear recurrences

18.05.2021 15:15 - 16:45

Tomack Gilmore (University College, London)


A matrix with real entries is totally positive if all of its minors are nonnegative. Many interesting lower-triangular matrices that arise in combinatorics have been shown to be totally positive, and many more appear to be totally positive but have yet to be proven so. Foremost among the latter is the Eulerian triangle, the entries of which count permutations by excedances (or descents) and satisfy a simple linear recurrence.

The notion of total positivity can be extended to matrices with entries belonging to a polynomial ring equipped with the coefficientwise partial order; we then say that such a matrix is *coefficientwise* totally positive if all of its minors are polynomials with nonnegative coefficients. In this talk I will present a more general triangle, discovered together with my co-authors, with entries that are polynomials in *six* indeterminates. This triangle appears empirically to be coefficientwise totally positive, and under suitable specialisations yields a number of interesting combinatorial triangles (including the Eulerian triangle and the reversed Stirling subset triangle). Although we do not yet have a proof of (coefficientwise) total positivity in the most general case, I will discuss our proof of one specialisation which includes a generalisation of the reversed Stirling subset triangle.

Meeting-ID: 945 4121 9182, Kenncode: Let2Vh


Ch. Krattenthaler

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