Identifier
-
Mp00150:
Perfect matchings
—to Dyck path⟶
Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤ
Values
[(1,6),(2,4),(3,5)] => [1,1,1,0,0,0] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[(1,5),(2,4),(3,6)] => [1,1,1,0,0,0] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[(1,4),(2,5),(3,6)] => [1,1,1,0,0,0] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[(1,4),(2,6),(3,5)] => [1,1,1,0,0,0] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[(1,5),(2,6),(3,4)] => [1,1,1,0,0,0] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[(1,6),(2,5),(3,4)] => [1,1,1,0,0,0] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[(1,8),(2,7),(3,5),(4,6)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,7),(2,8),(3,5),(4,6)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,6),(2,8),(3,5),(4,7)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,5),(2,8),(3,6),(4,7)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,5),(2,7),(3,6),(4,8)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,6),(2,7),(3,5),(4,8)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,7),(2,6),(3,5),(4,8)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,8),(2,6),(3,5),(4,7)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,8),(2,5),(3,6),(4,7)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,7),(2,5),(3,6),(4,8)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,6),(2,5),(3,7),(4,8)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,5),(2,6),(3,7),(4,8)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,5),(2,6),(3,8),(4,7)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,6),(2,5),(3,8),(4,7)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,7),(2,5),(3,8),(4,6)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,8),(2,5),(3,7),(4,6)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,8),(2,6),(3,7),(4,5)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,7),(2,6),(3,8),(4,5)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,6),(2,7),(3,8),(4,5)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,5),(2,7),(3,8),(4,6)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,5),(2,8),(3,7),(4,6)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,6),(2,8),(3,7),(4,5)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,7),(2,8),(3,6),(4,5)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,8),(2,7),(3,6),(4,5)] => [1,1,1,1,0,0,0,0] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[(1,10),(2,9),(3,8),(4,6),(5,7)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,9),(2,10),(3,8),(4,6),(5,7)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,8),(2,10),(3,9),(4,6),(5,7)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,7),(2,10),(3,9),(4,6),(5,8)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,6),(2,10),(3,9),(4,7),(5,8)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,6),(2,9),(3,10),(4,7),(5,8)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,7),(2,9),(3,10),(4,6),(5,8)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,8),(2,9),(3,10),(4,6),(5,7)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,9),(2,8),(3,10),(4,6),(5,7)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,10),(2,8),(3,9),(4,6),(5,7)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,10),(2,7),(3,9),(4,6),(5,8)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,9),(2,7),(3,10),(4,6),(5,8)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,8),(2,7),(3,10),(4,6),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,7),(2,8),(3,10),(4,6),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,6),(2,8),(3,10),(4,7),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,6),(2,7),(3,10),(4,8),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,7),(2,6),(3,10),(4,8),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,8),(2,6),(3,10),(4,7),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,9),(2,6),(3,10),(4,7),(5,8)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,10),(2,6),(3,9),(4,7),(5,8)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,10),(2,6),(3,8),(4,7),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,9),(2,6),(3,8),(4,7),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,8),(2,6),(3,9),(4,7),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,7),(2,6),(3,9),(4,8),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,6),(2,7),(3,9),(4,8),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,6),(2,8),(3,9),(4,7),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,7),(2,8),(3,9),(4,6),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,8),(2,7),(3,9),(4,6),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,9),(2,7),(3,8),(4,6),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,10),(2,7),(3,8),(4,6),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,10),(2,8),(3,7),(4,6),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,9),(2,8),(3,7),(4,6),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,8),(2,9),(3,7),(4,6),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,7),(2,9),(3,8),(4,6),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,6),(2,9),(3,8),(4,7),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,6),(2,10),(3,8),(4,7),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,7),(2,10),(3,8),(4,6),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,8),(2,10),(3,7),(4,6),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,9),(2,10),(3,7),(4,6),(5,8)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,10),(2,9),(3,7),(4,6),(5,8)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,10),(2,9),(3,6),(4,7),(5,8)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,9),(2,10),(3,6),(4,7),(5,8)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,8),(2,10),(3,6),(4,7),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,7),(2,10),(3,6),(4,8),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,6),(2,10),(3,7),(4,8),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,6),(2,9),(3,7),(4,8),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,7),(2,9),(3,6),(4,8),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,8),(2,9),(3,6),(4,7),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,9),(2,8),(3,6),(4,7),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,10),(2,8),(3,6),(4,7),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,10),(2,7),(3,6),(4,8),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,9),(2,7),(3,6),(4,8),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,8),(2,7),(3,6),(4,9),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,7),(2,8),(3,6),(4,9),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,6),(2,8),(3,7),(4,9),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,6),(2,7),(3,8),(4,9),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,7),(2,6),(3,8),(4,9),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,8),(2,6),(3,7),(4,9),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,9),(2,6),(3,7),(4,8),(5,10)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,10),(2,6),(3,7),(4,8),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,10),(2,6),(3,7),(4,9),(5,8)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,9),(2,6),(3,7),(4,10),(5,8)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,8),(2,6),(3,7),(4,10),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,7),(2,6),(3,8),(4,10),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,6),(2,7),(3,8),(4,10),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,6),(2,8),(3,7),(4,10),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,7),(2,8),(3,6),(4,10),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,8),(2,7),(3,6),(4,10),(5,9)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,9),(2,7),(3,6),(4,10),(5,8)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,10),(2,7),(3,6),(4,9),(5,8)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[(1,10),(2,8),(3,6),(4,9),(5,7)] => [1,1,1,1,1,0,0,0,0,0] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
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search for individual values
searching the database for the individual values of this statistic
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Map
to Dyck path
Description
The Dyck path corresponding to the opener-closer sequence of the perfect matching.
Map
permutation poset
Description
Sends a permutation to its permutation poset.
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
For a permutation $\pi$ of length $n$, this poset has vertices
$$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$$
and the cover relation is given by $(w, x) \leq (y, z)$ if $w \leq y$ and $x \leq z$.
For example, the permutation $[3,1,5,4,2]$ is mapped to the poset with cover relations
$$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$$
Map
to 132-avoiding permutation
Description
Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].
This bijection is defined in [1, Section 2].
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