Abstract
AbstractWe 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 $$m! \, \phi _m$$
m
!
ϕ
m
for each vertex with m proper children. We show that if the weight sequence $$\varvec{\phi }$$
ϕ
is Toeplitz-totally positive, then the two foregoing total-positivity results continue to hold. Our proofs use production matrices and exponential Riordan arrays.
Funder
Engineering and Physical Sciences Research Council
Publisher
Springer Science and Business Media LLC
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