# Enumerating (2+2) -free posets by the number of minimal elements and other statistics

Kitaev, Sergey and Remmel, Jeffrey
(2011)
*Enumerating (2+2) -free posets by the number of minimal elements and other statistics.*
Discrete Mathematics, 159 (17).
2098 - 2108.

## Abstract

An unlabeled poset is said to be -free if it does not contain an induced subposet that is isomorphic to , the union of two disjoint 2-element chains. Let pn denote the number of -free posets of size n. In a recent paper, Bousquet-Mélou et al. [1] found, using the so called ascent sequences, the generating function for the number of -free posets of size n: . We extend this result in two ways. First, we find the generating function for -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 pn,k equals the number of -free posets of size n with k minimal elements, then . 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 [1] and [2]. Finally, we define a subset of ascent sequences counted by the Catalan numbers and we discuss its relations with - and -free posets. An unlabeled poset is said to be -free if it does not contain an induced subposet that is isomorphic to , the union of two disjoint 2-element chains. Let pn denote the number of -free posets of size n. In a recent paper, Bousquet-Mélou et al. [1] found, using the so called ascent sequences, the generating function for the number of -free posets of size n: . We extend this result in two ways. First, we find the generating function for -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 pn,k equals the number of -free posets of size n with k minimal elements, then . 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 [1] and [2]. Finally, we define a subset of ascent sequences counted by the Catalan numbers and we discuss its relations with - and -free posets. An unlabeled poset is said to be -free if it does not contain an induced subposet that is isomorphic to , the union of two disjoint 2-element chains. Let pn denote the number of -free posets of size n. In a recent paper, Bousquet-Mélou et al. [1] found, using the so called ascent sequences, the generating function for the number of -free posets of size n: . We extend this result in two ways. First, we find the generating function for -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 pn,k equals the number of -free posets of size n with k minimal elements, then . 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 [1] and [2]. Finally, we define a subset of ascent sequences counted by the Catalan numbers and we discuss its relations with - and -free posets. An unlabeled poset is said to be -free if it does not contain an induced subposet that is isomorphic to , the union of two disjoint 2-element chains. Let pn denote the number of -free posets of size n. In a recent paper, Bousquet-Mélou et al. [1] found, using the so called ascent sequences, the generating function for the number of -free posets of size n: . We extend this result in two ways. First, we find the generating function for -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 pn,k equals the number of -free posets of size n with k minimal elements, then . 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 [1] and [2]. Finally, we define a subset of ascent sequences counted by the Catalan numbers and we discuss its relations with - and -free posets. An unlabeled poset is said to be -free if it does not contain an induced subposet that is isomorphic to , the union of two disjoint 2-element chains. Let pn denote the number of -free posets of size n. In a recent paper, Bousquet-Mélou et al. [1] found, using the so called ascent sequences, the generating function for the number of -free posets of size n: . We extend this result in two ways. First, we find the generating function for -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 pn,k equals the number of -free posets of size n with k minimal elements, then . 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 [1] and [2]. Finally, we define a subset of ascent sequences counted by the Catalan numbers and we discuss its relations with - and -free posets. An unlabeled poset is said to be -free if it does not contain an induced subposet that is isomorphic to , the union of two disjoint 2-element chains. Let pn denote the number of -free posets of size n. In a recent paper, Bousquet-Mélou et al. [1] found, using the so called ascent sequences, the generating function for the number of -free posets of size n: . We extend this result in two ways. First, we find the generating function for -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 pn,k equals the number of -free posets of size n with k minimal elements, then . 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 [1] and [2]. Finally, we define a subset of ascent sequences counted by the Catalan numbers and we discuss its relations with - and -free posets.

Item type: | Article |
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ID code: | 34554 |

Keywords: | computer science, enumeration, generating functions, multiple statistics, minimal elements, posets, Mathematics, Discrete Mathematics and Combinatorics, Theoretical Computer Science |

Subjects: | Science > Mathematics |

Department: | Faculty of Science > Computer and Information Sciences |

Depositing user: | Pure Administrator |

Date deposited: | 26 Oct 2011 10:48 |

Last modified: | 05 Jan 2018 11:12 |

URI: | https://strathprints.strath.ac.uk/id/eprint/34554 |

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