### Abstract

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 each row and column contains a nonzero entry, and whose entries sum to n. We derive a generating function counting such matrices, which confirms a conjecture of Jovovic, and we refine the generating function to count upper triangular 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.

Original language | English |
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Pages (from-to) | 139-163 |

Number of pages | 25 |

Journal | Journal of Combinatorics |

Volume | 2 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2011 |

### Keywords

- enumerating
- (2+2)-free posets
- indistinguishable elements

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## Cite this

Dukes, M., Kitaev, S., Remmel, J., & Steingrimsson, E. (2011). Enumerating (2+2)-free posets by indistinguishable elements.

*Journal of Combinatorics*,*2*(1), 139-163. https://doi.org/10.4310/JOC.2011.v2.n1.a6