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The purpose of this paper is to enumerate various classes of cyclically colored m-gonal plane cacti, called m-ary cacti. This combinatorial problem is motivated by the topological classification of complex polynomials having at most m critical values, studied by Zvonkin and others. We obtain explicit formulae for both Ž . labelled and unlabelled m-ary cacti, according to i the number of polygons, Ž .Ž . ii the vertex-color distribution, iii the vertex-degree distribution of each color. We also enumerate m-ary cacti according to the order of their automorphism group. Using a generalization of Otter's formula, we express the species of m-ary cacti in terms of rooted and of pointed cacti. A variant of the m-dimensional Lagrange inversion is then used to enumerate these structures. The method of Liskovets for the enumeration of unrooted planar maps can also be adapted to m-ary cacti.

SUMMARY To reach the terabit per second of throughput, telecommunication satellites cannot make use of frequency below Ka band only. Therefore, the use of broad portion of the spectrum available at Q/V (40/50 GHz) band is foreseen for the feeder link. This study presents the evaluation of performances of different macro‐diversity schemes that may allow mitigating the deep fades experienced at Q/V bands by introducing cooperation and a limited redundancy between the different gateways of the system. Two different solutions are firstly described. The performances resulting from the use of those assumptions are derived in a second stage. Copyright © 2014 John Wiley & Sons, Ltd.