Higher-order Stirling cycle and subset triangles: Total positivity, continued fractions and real-rootedness

Given a lower-triangular matrix of real numbers, one can ask the following four total-positivity questions: total positivity of the triangle itself; total positivity of its row-reversal; Toeplitz-total positivity of its row sequences (equivalent to negative-real-rootedness of the row-generating polynomials); and coefficientwise Hankel-total positivity of the sequence of row-generating polynomials. In this paper, we introduce two infinite families of lower-triangular matrices generalising the Stirling cycle and subset triangles, parametrised by an integer $r \ge 1$; we call these the $r$th-order Stirling cycle and subset numbers. We then ask the foregoing four questions for each of these triangles, leading us to several conjectures. We then prove some of these conjectures for the case $r=2$.