Interlacing and smoothing: Combinatorial aspects

We study functional consequences of the interlacing property consisting in that a new configuration of “particles” occurs in gaps between elements of a previous configuration. This property was introduced by I.M. Gelfand in terms of spectra of sequences of matrices of increasing dimensions and turned out to be highly needed in many areas of modern mathematics. We examine conditions under which the next generation is “on average smoother” than the previous one and discuss issues related to “complexity” of the set of pairs of interlacing functions.