Enumeration of chains and saturated chains in Dyck lattices

Ferrari, Luca; Munarini, Emanuele

doi:10.1016/j.aam.2014.09.003

Cited by 7 publications

(5 citation statements)

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“…From a combinatorial point of view, typical questions on intervals of a poset concern the counting of elements or, more generally, the enumeration of (saturated) chains of a given interval. These are problems that have been classically studied for many combinatorial posets, such as Bruhat orders [31], Tamari lattices [9,15], and Stanley (or Dyck) lattices [16,17]. In this section, we just scratch the surface of this vast subject, by proposing a couple of relatively simple results concerning the enumeration of intervals of the form [ , τ], when τ has a specific form.…”

Section: Combinatorics Of Intervals: Preliminary Resultsmentioning

confidence: 99%

Enumeration of Some Classes of Pattern Avoiding Matchings, with a Glimpse into the Matching Pattern Poset

Cervetti

Ferrari

2022

Ann. Comb.

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A matching of the set $$[2n]=\{1,2,\ldots ,2n\}$$ [ 2 n ] = { 1 , 2 , … , 2 n } is a partition of [2n] into blocks with two elements, i.e. a graph on [2n], such that every vertex has degree one. Given two matchings $$\sigma $$ σ and $$\tau $$ τ , we say that $$\sigma $$ σ is a $$pattern $$ pattern of $$\tau $$ τ when $$\sigma $$ σ can be obtained from $$\tau $$ τ by deleting some of its edges and consistently relabelling the remaining vertices. This is a partial order relation turning the set of all matchings into a poset, which will be called the matching pattern poset. In this paper, we continue the study of classes of pattern avoiding matchings (see below for previous work on this subject). In particular, we work out explicit formulas to enumerate the class of matchings avoiding two new patterns, obtained by juxtaposition of smaller patterns, and we describe a recursive formula for the generating function of the class of matchings avoiding the lifting of a pattern and two additional patterns. Moreover, we introduce the notion of unlabeled pattern, as a combinatorial way to collect patterns, and we provide enumerative formulas for two classes of matchings avoiding an unlabeled pattern of order three. In one case, the enumeration follows from an interesting bijection between the matchings of the class and ternary trees. The last part of the paper initiates the study of the matching pattern poset, by providing some preliminary results about its Möbius functions and the structure of some simple intervals.

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Section: Combinatorics Of Intervals: Preliminary Resultsmentioning

confidence: 99%

Enumeration of Some Classes of Pattern Avoiding Matchings, with a Glimpse into the Matching Pattern Poset

Cervetti

Ferrari

2022

Ann. Comb.

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“…We denote (3) this relation by P 1 ddom P 2 . The partially ordered set D n;t , ddom is isomorphic to NN (1) n;t , ⊆ via the bijection from Lemma 5.8, and its dual was for instance studied in [5,7,17,27,28]. Figure 5 shows D 4;2 , ddom .…”

Section: The Rank Enumeration Of Certain Parabolic Non-crossing Parti...mentioning

confidence: 99%

The rank enumeration of certain parabolic non-crossing partitions

Krattenthaler¹,

Mühle²

2022

Algebraic Combinatorics

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We consider m-divisible non-crossing partitions of {1, 2, . . . , mn} with the property that for some t n no block contains more than one of the integers 1, 2, . . . , t. We give a closed formula for the number of multi-chains of such non-crossing partitions with prescribed number of blocks. Building on this result, we compute Chapoton's M -triangle in this setting and conjecture a combinatorial interpretation for the H-triangle. This conjecture is proved for m = 1.

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“…The poset (D k , ≤ S ) turns out to be a distributive lattice; it is known as the k th Stanley lattice and is denoted by L S k . See [4,[21][22][23] for more information about these fascinating lattices. The upper left image in Figure 2 shows the Hasse diagram of L S 3 .…”

Section: Introductionmentioning

confidence: 99%