Combinatorial characterization of inrankings as weak orders induced by intervals

The problem of reliable processing of heteroscedastic interval data occupies an important niche among urgent topics of measurement science. The paper is devoted to a combinatorial characterization of so called `inrankings' which are weak orders induced by input intervals of the interval fusion with preference aggregation (IF&PA) procedure. The procedure transforms the given m initial real line intervals into inrankings, which are a specific case of weak order relations (or rankings) over a set of n discrete values belonging to these intervals. The new notation of inranking appears as a result of restrictions imposed on the ordinary rankings by interval character of the initial data. In the paper, the inranking spaces properties are investigated from the combinatorial theory point of view. It is shown that the inranking space is a subset of the set of all weak orders with a single symbol of strict order. The cardinality of inranking space is defined by the triangle number for the given number n of the discrete elements. Cardinalities of other adjacent spaces are considered.