Enumerating image-free posets by the number of minimal elements and other statistics Original Research Article

An unlabeled poset is said to be image-free if it does not contain an induced subposet that is isomorphic to image, the union of two disjoint 2-element chains. Let image denote the number of image-free posets of size image. In a recent paper, Bousquet-Mélou et al. found, using the so called ascent sequences, the generating function for the number of image-free posets of size image: image. We extend this result in two ways. First, we find the generating function for image-free posets when four statistics are taken into account, one of which is the number of minimal elements in a poset. Second, we show that if image equals the number of image-free posets of size image with image minimal elements, then image. The second result cannot be derived from the first one by a substitution. Our enumeration results are extended to certain restricted permutations and to regular linearized chord diagrams through bijections in . Finally, we define a subset of ascent sequences counted by the Catalan numbers and we discuss its relations with image- and image-free posets.