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Enumerating (2+2)-Free Posets by Indistinguishable Elements by Dukes, Mark; Kitaev, Sergey; Remmel, Jeffrey; Steingrimsson, Einar is a scholarly article available to read on EtoBox.

What is Enumerating (2+2)-Free Posets by Indistinguishable Elements about?

A poset is said to be (2+2)-free if it does not contain an induced 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-Melou et al., indistinguishable elements correspond to letters that belong to the same run in the so-called ascent sequence corresponding to the poset. We derive the generating function for the number of (2+2)-free posets with respect to both maxindist and the number 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 eac

Author

Dukes, Mark; Kitaev, Sergey; Remmel, Jeffrey; Steingrimsson, Einar

Published

2010

Language

EN