# 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. (https://doi.org/10.1016/j.dam.2011.07.010)

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## 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.

#### ORCID iDs

Kitaev, Sergey ORCID: https://orcid.org/0000-0003-3324-1647 and Remmel, Jeffrey;
• Item type: Article 34554 DateEvent28 October 2011Published4 August 2011Published Online computer science, enumeration, generating functions, multiple statistics, minimal elements, posets, Mathematics, Discrete Mathematics and Combinatorics, Theoretical Computer Science Science > Mathematics Faculty of Science > Computer and Information Sciences Pure Administrator 26 Oct 2011 10:48 24 Nov 2022 14:40 https://strathprints.strath.ac.uk/id/eprint/34554