Department Angewandte Mathematik und Physik
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In the present work, using the recently introduced framework of local geometric deformations, special types of vector fields – so-called hidden Killing vector fields – are constructed, which solve the Killing equation not globally, but only locally, i.e. in local subregions of spacetime. Taking advantage of the fact that the vector fields coincide locally with Killing fields and therefore allow the consideration of integral laws that convert into exact physical conservation laws on local scales, balance laws in dynamical systems without global Killing symmetries are derived that mimic as closely as possible the conservation laws for energy and angular momentum of highly symmetric models. The utility of said balance laws is demonstrated by a concrete geometric example, namely a toy model for the binary merger of two extremal Reissner–Nordström black holes.
A permutation can be locally classified according to the four local types: peaks, valleys, double rises and double falls. The corresponding classification of binary increasing trees uses four different types of nodes. Flajolet demonstrated the continued fraction representation of the generating function of local types, using a classical bijection between permutations, binary increasing trees, and suitably defined path diagrams induced by Motzkin paths.
The aim of this article is to extend the notion of local types from permutations to k-Stirling permutations (also known as k-multipermutations). We establish a bijection of these local types to node types of (k+1)-ary increasing trees. We present a branched continued fraction representation of the generating function of these local types through a bijection with path diagrams induced by Łukasiewicz paths, generalizing the results from permutations to arbitrary k-Stirling permutations.
We further show that the generating function of ordinary Stirling permutation has at least three branched continued fraction representations, using correspondences between non-standard increasing trees, k-Stirling permutations and path diagrams.