Denombrement des ordres etages

AbstractWe define a ‘tiered poset’ N of cardinality n by the condition that all the maximal chains have the same length e. Each element has its ‘tier number’; the elements with tier number e are the maximal elements.The paper is devoted to the enumeration of the tiered orders. The method is a recursive calculation of the integers εn(e, k), which enumerate the tiered orders on a given set of n elements, with e tiers and k maximal elements; these numbers are summed up with respect to e and k. A special treatment is necessary for e = 2 to begin the procedure.The numerical results are exhibited for n ⩽ 11 and compared with the total number of posets for given n. The paper concludes with a common arithmetical conjecture for both sequences