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To any rooted tree, we associate a sequence of numbers that we call the logarithmic factorials of the tree. This provides a generalization of Bhargavas factorials to a natural combinatorial setting suitable for studying questions around generalized factorials. We discuss several basic aspects of the framework in this paper. In particular, we relate the growth of the sequence of logarithmic factorials associated to a tree to the transience of the random walk and the existence of a harmonic measure on the tree, obtain an equidistribution theorem for factorial-determining-sequences of subsets of local fields, and provide a factorial-based characterization of the branching number of infinite trees. Our treatment is based on a local weighting process in the tree which gives an effective way of constructing the factorial sequence.
We show that in any two-coloring of the positive integers there is a color for which the set of positive integers that can be represented as a sum of distinct elements with this color has upper logarithmic density at least $(2+sqrt{3})/4$ and this is
Recently, by the Riordans identity related to tree enumerations, begin{eqnarray*} sum_{k=0}^{n}binom{n}{k}(k+1)!(n+1)^{n-k} &=& (n+1)^{n+1}, end{eqnarray*} Sun and Xu derived another analogous one, begin{eqnarray*} sum_{k=0}^{n}binom{n}{k}D_{k+1}(n+1
Let the random variable $X, :=, e(mathcal{H}[B])$ count the number of edges of a hypergraph $mathcal{H}$ induced by a random $m$ element subset $B$ of its vertex set. Focussing on the case that $mathcal{H}$ satisfies some regularity condition we prov
We develop a theory of log adic spaces by combining the theories of adic spaces and log schemes, and study the Kummer etale and pro-Kummer etale topology for such spaces. We also establish the primitive comparison theorem in this context, and deduce