Identifier
-
Mp00017:
Binary trees
—to 312-avoiding permutation⟶
Permutations
Mp00223: Permutations —runsort⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001880: Posets ⟶ ℤ
Values
[.,[.,[.,.]]] => [3,2,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[.,[[.,.],.]] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[[[.,.],.],.] => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 3
[.,[.,[.,[.,.]]]] => [4,3,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[.,[.,[[.,.],.]]] => [3,4,2,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[.,[[[.,.],.],.]] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[[[.,.],[.,.]],.] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
[[[[.,.],.],.],.] => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 4
[.,[.,[.,[.,[.,.]]]]] => [5,4,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[.,[.,[.,[[.,.],.]]]] => [4,5,3,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[.,[.,[[[.,.],.],.]]] => [3,4,5,2,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[.,[[[.,.],[.,.]],.]] => [2,4,3,5,1] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
[.,[[[[.,.],.],.],.]] => [2,3,4,5,1] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[[.,.],[[.,[.,.]],.]] => [1,4,3,5,2] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[[[.,.],[[.,.],.]],.] => [1,3,4,2,5] => [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[[[.,[.,.]],[.,.]],.] => [2,1,4,3,5] => [1,4,2,3,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 4
[[[[.,.],.],[.,.]],.] => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
[[[[.,.],[.,.]],.],.] => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
[[[[[.,.],.],.],.],.] => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 5
[.,[.,[.,[.,[.,[.,.]]]]]] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[.,[.,[.,[.,[[.,.],.]]]]] => [5,6,4,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[.,[.,[.,[[[.,.],.],.]]]] => [4,5,6,3,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[.,[.,[[[.,.],[.,.]],.]]] => [3,5,4,6,2,1] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 6
[.,[.,[[[[.,.],.],.],.]]] => [3,4,5,6,2,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[.,[[.,.],[[.,[.,.]],.]]] => [2,5,4,6,3,1] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 5
[.,[[[.,.],[[.,.],.]],.]] => [2,4,5,3,6,1] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 5
[.,[[[.,[.,.]],[.,.]],.]] => [3,2,5,4,6,1] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 5
[.,[[[[.,.],.],[.,.]],.]] => [2,3,5,4,6,1] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 6
[.,[[[[.,.],[.,.]],.],.]] => [2,4,3,5,6,1] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 6
[.,[[[[[.,.],.],.],.],.]] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[[.,.],[.,[[.,[.,.]],.]]] => [1,5,4,6,3,2] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 4
[[.,.],[[.,[[.,.],.]],.]] => [1,4,5,3,6,2] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => 4
[[.,.],[[[.,.],[.,.]],.]] => [1,3,5,4,6,2] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[[.,.],[[[.,[.,.]],.],.]] => [1,4,3,5,6,2] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 5
[[.,[.,.]],[[.,[.,.]],.]] => [2,1,5,4,6,3] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 4
[[[.,.],.],[[.,[.,.]],.]] => [1,2,5,4,6,3] => [1,2,5,3,4,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 5
[[[.,.],[[[.,.],.],.]],.] => [1,3,4,5,2,6] => [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 4
[[[.,[.,.]],[[.,.],.]],.] => [2,1,4,5,3,6] => [1,4,5,2,3,6] => ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6) => 4
[[[[.,.],.],[[.,.],.]],.] => [1,2,4,5,3,6] => [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6) => 5
[[[.,[.,[.,.]]],[.,.]],.] => [3,2,1,5,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 4
[[[.,[[.,.],.]],[.,.]],.] => [2,3,1,5,4,6] => [1,5,2,3,4,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6) => 4
[[[[.,[.,.]],.],[.,.]],.] => [2,1,3,5,4,6] => [1,3,5,2,4,6] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 6
[[[[[.,.],.],.],[.,.]],.] => [1,2,3,5,4,6] => [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 6
[[[[.,.],[[.,.],.]],.],.] => [1,3,4,2,5,6] => [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 5
[[[[.,[.,.]],[.,.]],.],.] => [2,1,4,3,5,6] => [1,4,2,3,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6) => 5
[[[[[.,.],.],[.,.]],.],.] => [1,2,4,3,5,6] => [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 6
[[[[[.,.],[.,.]],.],.],.] => [1,3,2,4,5,6] => [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 6
[[[[[[.,.],.],.],.],.],.] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 6
[.,[.,[.,[.,[.,[.,[.,.]]]]]]] => [7,6,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[.,[.,[.,[.,[.,[[.,.],.]]]]]] => [6,7,5,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[.,[.,[.,[.,[[[.,.],.],.]]]]] => [5,6,7,4,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[.,[.,[.,[[[.,.],[.,.]],.]]]] => [4,6,5,7,3,2,1] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 7
[.,[.,[.,[[[[.,.],.],.],.]]]] => [4,5,6,7,3,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[.,[.,[[.,.],[[.,[.,.]],.]]]] => [3,6,5,7,4,2,1] => [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => 6
[.,[.,[[[.,.],[[.,.],.]],.]]] => [3,5,6,4,7,2,1] => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => 6
[.,[.,[[[.,[.,.]],[.,.]],.]]] => [4,3,6,5,7,2,1] => [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => 6
[.,[.,[[[[.,.],.],[.,.]],.]]] => [3,4,6,5,7,2,1] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 7
[.,[.,[[[[.,.],[.,.]],.],.]]] => [3,5,4,6,7,2,1] => [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 7
[.,[.,[[[[[.,.],.],.],.],.]]] => [3,4,5,6,7,2,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[.,[[.,.],[.,[[.,[.,.]],.]]]] => [2,6,5,7,4,3,1] => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 5
[.,[[.,.],[[.,[[.,.],.]],.]]] => [2,5,6,4,7,3,1] => [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7) => 5
[.,[[.,.],[[[.,.],[.,.]],.]]] => [2,4,6,5,7,3,1] => [1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 7
[.,[[.,.],[[[.,[.,.]],.],.]]] => [2,5,4,6,7,3,1] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => 6
[.,[[.,[.,.]],[[.,[.,.]],.]]] => [3,2,6,5,7,4,1] => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 5
[.,[[[.,.],.],[[.,[.,.]],.]]] => [2,3,6,5,7,4,1] => [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => 6
[.,[[[.,.],[[[.,.],.],.]],.]] => [2,4,5,6,3,7,1] => [1,2,4,5,6,3,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 5
[.,[[[.,[.,.]],[[.,.],.]],.]] => [3,2,5,6,4,7,1] => [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7) => 5
[.,[[[[.,.],.],[[.,.],.]],.]] => [2,3,5,6,4,7,1] => [1,2,3,5,6,4,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => 6
[.,[[[.,[.,[.,.]]],[.,.]],.]] => [4,3,2,6,5,7,1] => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 5
[.,[[[.,[[.,.],.]],[.,.]],.]] => [3,4,2,6,5,7,1] => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 5
[.,[[[[.,[.,.]],.],[.,.]],.]] => [3,2,4,6,5,7,1] => [1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 7
[.,[[[[[.,.],.],.],[.,.]],.]] => [2,3,4,6,5,7,1] => [1,2,3,4,6,5,7] => ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7) => 7
[.,[[[[.,.],[[.,.],.]],.],.]] => [2,4,5,3,6,7,1] => [1,2,4,5,3,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => 6
[.,[[[[.,[.,.]],[.,.]],.],.]] => [3,2,5,4,6,7,1] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => 6
[.,[[[[[.,.],.],[.,.]],.],.]] => [2,3,5,4,6,7,1] => [1,2,3,5,4,6,7] => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 7
[.,[[[[[.,.],[.,.]],.],.],.]] => [2,4,3,5,6,7,1] => [1,2,4,3,5,6,7] => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7) => 7
[.,[[[[[[.,.],.],.],.],.],.]] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7) => 7
[[.,.],[.,[.,[[.,[.,.]],.]]]] => [1,6,5,7,4,3,2] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 4
[[.,.],[.,[[.,[[.,.],.]],.]]] => [1,5,6,4,7,3,2] => [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => 4
[[.,.],[.,[[[.,.],[.,.]],.]]] => [1,4,6,5,7,3,2] => [1,4,6,2,3,5,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7) => 6
[[.,.],[.,[[[.,[.,.]],.],.]]] => [1,5,4,6,7,3,2] => [1,5,2,3,4,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 5
[[.,.],[[.,.],[[.,[.,.]],.]]] => [1,3,6,5,7,4,2] => [1,3,6,2,4,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => 6
[[.,.],[[.,[[[.,.],.],.]],.]] => [1,4,5,6,3,7,2] => [1,4,5,6,2,3,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => 4
[[.,.],[[[.,.],[[.,.],.]],.]] => [1,3,5,6,4,7,2] => [1,3,5,6,2,4,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(4,3),(4,5),(5,6)],7) => 6
[[.,.],[[[[.,.],.],[.,.]],.]] => [1,3,4,6,5,7,2] => [1,3,4,6,2,5,7] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,5),(4,3),(5,6)],7) => 6
[[.,.],[[[.,[[.,.],.]],.],.]] => [1,4,5,3,6,7,2] => [1,4,5,2,3,6,7] => ([(0,4),(0,5),(1,6),(2,6),(4,2),(5,1),(6,3)],7) => 5
[[.,.],[[[[.,.],[.,.]],.],.]] => [1,3,5,4,6,7,2] => [1,3,5,2,4,6,7] => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7) => 7
[[.,.],[[[[.,[.,.]],.],.],.]] => [1,4,3,5,6,7,2] => [1,4,2,3,5,6,7] => ([(0,3),(0,5),(2,6),(3,6),(4,1),(5,2),(6,4)],7) => 6
[[.,[.,.]],[.,[[.,[.,.]],.]]] => [2,1,6,5,7,4,3] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 4
[[.,[.,.]],[[.,[[.,.],.]],.]] => [2,1,5,6,4,7,3] => [1,5,6,2,3,4,7] => ([(0,4),(0,5),(1,6),(2,6),(3,2),(4,3),(5,1)],7) => 4
[[.,[.,.]],[[[.,.],[.,.]],.]] => [2,1,4,6,5,7,3] => [1,4,6,2,3,5,7] => ([(0,3),(0,4),(1,6),(2,5),(3,2),(4,1),(4,5),(5,6)],7) => 6
[[.,[.,.]],[[[.,[.,.]],.],.]] => [2,1,5,4,6,7,3] => [1,5,2,3,4,6,7] => ([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7) => 5
[[[.,.],.],[.,[[.,[.,.]],.]]] => [1,2,6,5,7,4,3] => [1,2,6,3,4,5,7] => ([(0,5),(1,6),(2,6),(3,4),(4,2),(5,1),(5,3)],7) => 5
[[[.,.],.],[[.,[[.,.],.]],.]] => [1,2,5,6,4,7,3] => [1,2,5,6,3,4,7] => ([(0,5),(1,6),(2,6),(3,2),(4,1),(5,3),(5,4)],7) => 5
[[[.,.],.],[[[.,.],[.,.]],.]] => [1,2,4,6,5,7,3] => [1,2,4,6,3,5,7] => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7) => 7
[[[.,.],.],[[[.,[.,.]],.],.]] => [1,2,5,4,6,7,3] => [1,2,5,3,4,6,7] => ([(0,5),(1,6),(2,6),(4,2),(5,1),(5,4),(6,3)],7) => 6
[[.,[.,[.,.]]],[[.,[.,.]],.]] => [3,2,1,6,5,7,4] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 4
[[.,[[.,.],.]],[[.,[.,.]],.]] => [2,3,1,6,5,7,4] => [1,6,2,3,4,5,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 4
[[[.,[.,.]],.],[[.,[.,.]],.]] => [2,1,3,6,5,7,4] => [1,3,6,2,4,5,7] => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,2),(4,6),(6,1)],7) => 6
[[[[.,.],.],.],[[.,[.,.]],.]] => [1,2,3,6,5,7,4] => [1,2,3,6,4,5,7] => ([(0,4),(1,6),(2,6),(3,2),(4,5),(5,1),(5,3)],7) => 6
[[[.,.],[[[[.,.],.],.],.]],.] => [1,3,4,5,6,2,7] => [1,3,4,5,6,2,7] => ([(0,2),(0,5),(1,6),(2,6),(3,4),(4,1),(5,3)],7) => 4
>>> Load all 129 entries. <<<
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
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
runsort
Description
The permutation obtained by sorting the increasing runs lexicographically.
Map
to 312-avoiding permutation
Description
Return a 312-avoiding permutation corresponding to a binary tree.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.
The linear extensions of a binary tree form an interval of the weak order called the Sylvester class of the tree. This permutation is the minimal element of this Sylvester class.
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