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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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 2element chains. Let pn denote the number of free posets of size n. In a recent paper, BousquetMélou 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 2element chains. Let pn denote the number of free posets of size n. In a recent paper, BousquetMélou 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 2element chains. Let pn denote the number of free posets of size n. In a recent paper, BousquetMélou 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 2element chains. Let pn denote the number of free posets of size n. In a recent paper, BousquetMélou 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 2element chains. Let pn denote the number of free posets of size n. In a recent paper, BousquetMélou 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 2element chains. Let pn denote the number of free posets of size n. In a recent paper, BousquetMélou 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/0000000333241647 and Remmel, Jeffrey;

Item type: Article ID code: 34554 Dates: DateEvent28 October 2011Published4 August 2011Published OnlineKeywords: 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: 24 Nov 2022 14:40 URI: https://strathprints.strath.ac.uk/id/eprint/34554