Geodesic
Some statistics on the hypercubes of Catalan permutations
Journal of integer sequences, Tome 18 (2015) no. 2
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 
@article{JIS_2015__18_2_a3,
     author = {Disanto,  Filippo},
     title = {Some statistics on the hypercubes of {Catalan} permutations},
     journal = {Journal of integer sequences},
     year = {2015},
     volume = {18},
     number = {2},
     zbl = {1309.05100},
     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/