Two Hypothesis on a Combinatorial Problem for Possible States on the Arrival Line for n Competitor Runners

In a very small t-time interval, several runners could occupy the same place on the arrival line (hypothesis 1). Each runner has his own name and a competition number (on the shirt). The number of runners is a natural number n. For each given n, the hypothesis creates a combinatorial problem having a lot of posible states. All notations are choose so that to indicate easily by name their meaning. The states are separated into two classes: non-nominal states and nominal states. The states are related with the place I, II, III etc on arrival line. It is necessary to generate the total number of non-nominal states (on arrival line) and the total number of nominal states. In order to generate the states the work uses some formulas and some specialised algorithms. For example, the consrtuction of all non-nominal states recommends that the string for the position I to use a decreasing string. The same rule is validly for position II, but for sub-strings etc. A lot of numerical examples ilustrate the states generation. An independent method verifies the correctitude of states generation. In order to continue the study of combinatorial problem, the work introduces two new notions in section 5. The notions of partial frequency and final frequency are defined for a nominal known runner in final classification, together with computational formulas. The section 6 constructs the random variables attached to final classification and the probability of each place on arrival line. Each runner receives a score (a number of points) related with his final classification. May be the runner is interested to know the probability to ocuppy the first place (place I) and to estimate the number of possible points. All the results could be written in a centralisation table (section 7). Section 8 contains several numerical examples with statistical computations. At the end of the work we replace hypothesis 1 by hypothesis 2: only one runner could ocuppay each place. All the above notions have a new specific form. The numerical examples ilustrates the theory.