Order and Random Sequences

Consider a set of k elements. Let the elements be the first k positive integers. Consider the permutation set that corresponds to the first k positive integers. Let each element's order equal that element's value. Let p designate element position such that 1 less than or equal p less than or equal k. Let s designate element order such that 1 less than or equal s less than or equal k.

At every line of the permutation set there exists an element or there exists the elements of the greatest order so far while doing a left to right scan. As a consequence, there exist permutation set properties named takeover, level, jump from, quiescence, jump to and route.

Takeover is an event. Jump from and quiescence and jump to are contingent events and they have corresponding expected values named jump from value and quiescence value and jump to value.

With respect to takeover and level and the points of a route , if s < p then there are p unique elements, the order of which are less than p, which is impossible. Therefore s greater than or equal to p. Therefore 1 less than or equal to p less than or equal to s less than or equal to k. The constraint 1 less than or equal to p less than or equal to s less than or equal to k applies to no equal order; it does not apply if an equal order constraint is applied.

There is symmetry with respect to the line s + p = k + 1. There is symmetry of the function ET(k, p, s). There is symmetry between the functions QPS(k, p, s) and JFPS(p, s) and between QS(k, s) and JFP(k, p) and between QP(k, p) and JFS(k, s). The Special Lines and Special Points figure is symmetrical.