Enumerating (2+2)-free posets by indistinguishable elements

Abstract. A poset is said to be (2 + 2)-free if it does not contain an in-duced subposet that is isomorphic to 2 + 2, the union of two disjoint 2-element chains. Two elements in a poset are indistinguishable if they have the same strict up-set and the same strict down-set. Being indistinguishable defines an equivalence relation on the elements of the poset. We introduce the statistic maxindist, the maximum size of a set of indistinguishable elements. We show that, under a bijection of Bousquet-Mélou et al. [1], indistinguishable elements correspond to letters that belong to the same run in the so-called ascent se-quence corresponding to the poset. We derive the generating function for the number of (2 + 2)-free posets with respect to both maxindist and the num-ber of different strict down-sets of elements in the poset. Moreover, we show that (2 + 2)-free posets P with maxindist(P) at most k are in bijection with upper triangular matrices of nonnegative integers not exceeding k, where each row and each column contains a nonzero entry. (Here we consider isomorphic posets to be equal.) In particular, (2 + 2)-free posets P on n elements with maxindist(P) = 1 correspond to upper triangular binary matrices where each row and column contains a nonzero entry, and whose entries sum to n. We de-rive a generating function counting such matrices, which confirms a conjecture of Jovovic [8], and we refine the generating function to count upper triangu-lar matrices consisting of nonnegative integers not exceeding k and having a nonzero entry in each row and column. That refined generating function also enumerates (2 + 2)-free posets according to maxindist. Finally, we link our enumerative results to certain restricted permutations and matrices. 1