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Sokal, Alan D.
Monatshefte für Mathematik, 02/2023, Volume: 200, Issue: 2Journal Article
We consider the lower-triangular matrix of generating polynomials that enumerate k -component forests of rooted trees on the vertex set n according to the number of improper edges (generalizations of the Ramanujan polynomials). We show that this matrix is coefficientwise totally positive and that the sequence of its row-generating polynomials is coefficientwise Hankel-totally positive. More generally, we define the generic rooted-forest polynomials by introducing also a weight m ! ϕ m for each vertex with m proper children. We show that if the weight sequence ϕ is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.
