In this paper we consider two aspects of the inverse problem of how to construct merge trees realizing a given barcode. Much of our investigation exploits a recently discovered connection between the symmetric group and barcodes in general position, based on the simple observation that death order is a permutation of birth order. We show how to lift this combinatorial characterization of barcodes to an analogous combinatorialization of merge trees. As result of this study, we provide the first clear combinatorial distinction between the space of phylogenetic trees (as defined by Billera, Holmes and Vogtmann) and the space of merge trees: generic phylogenetic trees on n+1 leaf nodes fall into (2n−1)!! distinct equivalence classes, but the analogous number for merge trees is equal to the number of maximal chains in the lattice of partitions, i.e., (n+1)!n!2−n. The second aspect of our study is the derivation of precise formulas for the distribution of tree realization numbers (the number of merge trees realizing a given barcode) when we assume that barcodes are sampled using a uniform distribution on the symmetric group. We are able to characterize some of the higher moments of this distribution, thanks in part to a reformulation of our distribution in terms of Dirichlet convolution. This characterization provides a type of null hypothesis, apparently different from the distributions observed in real neuron data, which opens the door to doing more precise statistics and science.

Recent work in mathematical neuroscience has calculated the directed graph homology of the directed simplicial complex given by the brain's sparse adjacency graph, the so called connectome. These biological connectomes show an abundance of both high-dimensional directed simplices and Betti-numbers in all viable dimensions – in contrast to Erdős–Rényi-graphs of comparable size and density. An analysis of synthetically trained connectomes reveals similar findings, raising questions about the graphs comparability and the nature of origin of the simplices.We present a new method capable of delivering insight into the emergence of simplices and thus simplicial abundance. Our approach allows to easily distinguish simplex-rich connectomes of different origin. The method relies on the novel concept of an almost-d-simplex, that is, a simplex missing exactly one edge, and consequently the almost-d-simplex closing probability by dimension. We also describe a fast algorithm to identify almost-d-simplices in a given graph. Applying this method to biological and artificial data allows us to identify a mechanism responsible for simplex emergence, and suggests this mechanism is responsible for the simplex signature of the excitatory subnetwork of a statistical reconstruction of the mouse primary visual cortex. Our highly optimized code for this new method is publicly available.