Geodesic
Maximal chains in bond lattices
Shreya Ahirwar1 ; Susanna Fishel2 ; Parikshita Gya1 ; Pamela Harris3 ; Nguyen Pham1 ; Andrés Vindas Meléndez4 ; Dan Khanh Vo1
1Mount Holyoke College
2Arizona State University
3Williams College
4Mathematical Sciences Research Institute
The electronic journal of combinatorics, Tome 29 (2022) no. 3
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Let $G$ be a graph with vertex set $\{1,2,\ldots,n\}$. Its bond lattice, $BL(G)$, is a sublattice of the set partition lattice. The elements of $BL(G)$ are the set partitions whose blocks induce connected subgraphs of $G$. In this article, we consider graphs $G$ whose bond lattice consists only of noncrossing partitions. We define a family of graphs, called triangulation graphs, with this property and show that any two produce isomorphic bond lattices. We then look at the enumeration of the maximal chains in the bond lattices of triangulation graphs. Stanley's map from maximal chains in the noncrossing partition lattice to parking functions was our motivation. We find the restriction of his map to the bond lattice of certain subgraphs of triangulation graphs. Finally, we show the number of maximal chains in the bond lattice of a triangulation graph is the number of ordered cycle decompositions.
DOI : 10.37236/10983
Classification : 05A10, 05C99, 06A07
Mots-clés : triangulation graphs, ordered cycle decompositions

Shreya Ahirwar  1   ; Susanna Fishel  2   ; Parikshita Gya  1   ; Pamela Harris  3   ; Nguyen Pham  1   ; Andrés Vindas Meléndez  4   ; Dan Khanh Vo  1

1 Mount Holyoke College
2 Arizona State University
3 Williams College
4 Mathematical Sciences Research Institute 
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     author = {Shreya Ahirwar and Susanna Fishel and Parikshita Gya and Pamela Harris and Nguyen Pham and Andr\'es  Vindas Mel\'endez and Dan Khanh Vo},
     title = {Maximal chains in bond lattices},
     journal = {The electronic journal of combinatorics},
     year = {2022},
     volume = {29},
     number = {3},
     doi = {10.37236/10983},
     zbl = {1494.05004},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/10983/}
}
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%A Dan Khanh Vo
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Shreya Ahirwar; Susanna Fishel; Parikshita Gya; Pamela Harris; Nguyen Pham; Andrés  Vindas Meléndez; Dan Khanh Vo. Maximal chains in bond lattices. The electronic journal of combinatorics, Tome 29 (2022) no. 3. doi: 10.37236/10983

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