Merge trees represent the topology of scalar functions. To assess the topo-logical similarity of functions, one can compare their merge trees. To do so, one needs a notion of a distance between merge trees, which we define. We provide examples of using our merge tree distance and compare this new measure to other ways used to characterize topological… (More)
We establish an analog of Faulhaber's theorem for a power sum of binomial coefficients. We study reciprocal power sums of binomial coefficients and Faulhaber coefficients for a power sum of triangular numbers.
Catalan numbers arise in many enumerative contexts as the counting sequence of combinatorial structures. In this work, we consider natural Markov chains on some of the realizations of the Catalan sequence. While our main result is in deriving an O(n 2 log n) bound on the mixing time in L2 (and hence total variation) distance for the random transposition… (More)
We describe the relation between graph decompositions into walks and the normal ordering of differential operators in the n-th Weyl algebra. Under several specifications , we study new types of restricted set partitions, and a generalization of Stirling numbers, which we call the λ-Stirling numbers.