Geodesic
Some statistics on the hypercubes of Catalan permutations
Journal of integer sequences, Tome 18 (2015) no. 2.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: For a permutation $\sigma $ of length 3, we define the oriented graph $Q_{n}(\sigma )$. The graph $Q_{n}(\sigma )$ is obtained by imposing edge constraints on the classical oriented hypercube $Q_{n}$, such that each path going from $0^{n}$ to $1^{n}$ in $Q_{n}(\sigma )$ bijectively encodes a permutation of size $n$ avoiding the pattern $\sigma $. The orientation of the edges in $Q_{n}(\sigma )$ naturally induces an order relation ?$_{\sigma }$ among its nodes. First, we characterize ?$_{\sigma }$. Next, we study several enumerative statistics on $Q_{n}(\sigma )$, including the number of intervals, the number of intervals of fixed length $k$, and the number of paths (or permutations) intersecting a given node.
Classification : 05A15
Keywords: permutations avoiding patterns of length 3, edge-constrained hypercube, number of intervals, number of paths through a node, Catalan number 
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     author = {Disanto, Filippo},
     title = {Some statistics on the hypercubes of {Catalan} permutations},
     journal = {Journal of integer sequences},
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     volume = {18},
     number = {2},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2015__18_2_a3/}
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Disanto, Filippo. Some statistics on the hypercubes of Catalan permutations. Journal of integer sequences, Tome 18 (2015) no. 2. http://geodesic.mathdoc.fr/item/JIS_2015__18_2_a3/