- FindStat

Your data matches 2 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001568

Mapping

Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[[[.,.],.],.]
 => [1,2,3] => [1,1,1]
 => [1,1]
 => 2
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [2,2]
 => [2]
 => 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [2,1,1]
 => [1,1]
 => 2
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [2,2]
 => [2]
 => 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,1,1]
 => [1,1]
 => 2
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,1]
 => [1,1]
 => 2
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,1,1,1]
 => [1,1,1]
 => 2
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [2,2,1]
 => [2,1]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [3,2]
 => [2]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,2]
 => [2]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [3,2]
 => [2]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [2,2,1]
 => [2,1]
 => 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [2,1,1,1]
 => [1,1,1]
 => 2
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [3,2]
 => [2]
 => 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [3,1,1]
 => [1,1]
 => 2
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [2,2,1]
 => [2,1]
 => 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [3,2]
 => [2]
 => 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [3,2]
 => [2]
 => 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [3,2]
 => [2]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [3,2]
 => [2]
 => 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [3,1,1]
 => [1,1]
 => 2
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [2,2,1]
 => [2,1]
 => 1
[[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [3,1,1]
 => [1,1]
 => 2
[[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => [2,2,1]
 => [2,1]
 => 1
[[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => [3,1,1]
 => [1,1]
 => 2
[[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [3,1,1]
 => [1,1]
 => 2
[[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => [3,1,1]
 => [1,1]
 => 2
[[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [2,1,1,1]
 => [1,1,1]
 => 2
[[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [1,1,1,1,1]
 => [1,1,1,1]
 => 2
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => [2,2,2]
 => [2,2]
 => 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [5,6,4,3,2,1] => [4,2]
 => [2]
 => 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [6,4,5,3,2,1] => [4,2]
 => [2]
 => 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [4,5,6,3,2,1] => [4,2]
 => [2]
 => 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [6,5,3,4,2,1] => [2,2,1,1]
 => [2,1,1]
 => 2
[.,[.,[[.,.],[[.,.],.]]]]
 => [5,6,3,4,2,1] => [4,1,1]
 => [1,1]
 => 2
[.,[.,[[.,[.,.]],[.,.]]]]
 => [6,4,3,5,2,1] => [3,2,1]
 => [2,1]
 => 1
[.,[.,[[[.,.],.],[.,.]]]]
 => [6,3,4,5,2,1] => [4,2]
 => [2]
 => 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [4,5,3,6,2,1] => [3,2,1]
 => [2,1]
 => 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [4,3,5,6,2,1] => [3,3]
 => [3]
 => 1
[.,[[.,.],[.,[.,[.,.]]]]]
 => [6,5,4,2,3,1] => [4,2]
 => [2]
 => 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [6,4,5,2,3,1] => [2,2,2]
 => [2,2]
 => 1
[.,[[.,.],[[.,[.,.]],.]]]
 => [5,4,6,2,3,1] => [4,2]
 => [2]
 => 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => [6,5,3,2,4,1] => [3,2,1]
 => [2,1]
 => 1
[.,[[[.,.],.],[.,[.,.]]]]
 => [6,5,2,3,4,1] => [4,2]
 => [2]
 => 1
[.,[[.,[.,[.,.]]],[.,.]]]
 => [6,4,3,2,5,1] => [2,2,1,1]
 => [2,1,1]
 => 2
[.,[[.,[[.,.],.]],[.,.]]]
 => [6,3,4,2,5,1] => [3,2,1]
 => [2,1]
 => 1
[.,[[[.,.],[.,.]],[.,.]]]
 => [6,4,2,3,5,1] => [3,2,1]
 => [2,1]
 => 1
[.,[[[.,[.,.]],.],[.,.]]]
 => [6,3,2,4,5,1] => [2,2,1,1]
 => [2,1,1]
 => 2

Description

The smallest positive integer that does not appear twice in the partition.

Matching statistic: St001820

Mapping

Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001820: Lattices ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 67%

Values

[[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 2
[.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => 1
[.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => 2
[[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => 1
[[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => 2
[[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => 2
[[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 2
[.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 1
[.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 1
[.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 1
[.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
 => ? = 1
[.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 2
[.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 1
[.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 2
[[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [3,4,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
 => ? = 1
[[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [3,4,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [3,5,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1
[[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [4,1,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [3,1,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [3,1,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 2
[[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 1
[[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 2
[[.,[[.,[.,.]],.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 2
[[[.,.],[[.,.],.]],.]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,4,5,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,7),(4,6),(5,6),(6,9),(7,9),(9,8)],10)
 => ? = 1
[[[.,[.,.]],[.,.]],.]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,4,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 2
[[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 2
[[[.,[[.,.],.]],.],.]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 2
[[[[.,.],[.,.]],.],.]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,2,5,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 2
[[[[.,[.,.]],.],.],.]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 2
[[[[[.,.],.],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 2
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => [2,1,4,3,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,9),(7,11),(8,10),(9,10),(9,11),(10,12),(11,12)],13)
 => ? = 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [2,4,1,3,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
 => ? = 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,1,4,6,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
 => ? = 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [2,4,6,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,12),(4,12),(5,11),(6,11),(6,13),(8,9),(9,7),(10,7),(11,10),(12,8),(13,9),(13,10),(14,8),(14,13)],15)
 => ? = 2
[.,[.,[[.,.],[[.,.],.]]]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,4,1,3,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
 => ? = 2
[.,[.,[[.,[.,.]],[.,.]]]]
 => [1,0,1,0,1,1,0,1,0,0,1,0]
 => [2,1,4,5,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,12),(8,9),(9,10),(9,12),(10,11),(12,11)],13)
 => ? = 1
[.,[.,[[[.,.],.],[.,.]]]]
 => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [2,1,4,5,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
 => ? = 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [2,4,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,7),(5,7),(6,8),(7,9),(9,8)],10)
 => ? = 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,4,5,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
[.,[[.,.],[.,[.,[.,.]]]]]
 => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,5,3,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
 => ? = 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [2,1,5,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
 => ? = 1
[.,[[.,.],[[.,[.,.]],.]]]
 => [1,0,1,1,0,0,1,1,0,1,0,0]
 => [2,5,1,6,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => [2,1,5,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,12),(8,9),(9,10),(9,12),(10,11),(12,11)],13)
 => ? = 1
[.,[[[.,.],.],[.,[.,.]]]]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,6,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
 => ? = 1
[.,[[.,[.,[.,.]]],[.,.]]]
 => [1,0,1,1,0,1,0,1,0,0,1,0]
 => [2,1,3,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,12),(4,12),(5,11),(6,10),(6,13),(8,7),(9,7),(10,8),(11,9),(12,10),(13,8),(13,9),(14,11),(14,13)],15)
 => ? = 2
[.,[[.,[[.,.],.]],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0,1,0]
 => [2,1,3,5,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,11),(6,9),(6,10),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
 => ? = 1
[.,[[[.,.],[.,.]],[.,.]]]
 => [1,0,1,1,1,0,0,1,0,0,1,0]
 => [2,1,3,6,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,11),(6,9),(6,10),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
 => ? = 1
[.,[[[.,[.,.]],.],[.,.]]]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => [2,1,3,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,12),(3,13),(4,13),(5,11),(5,14),(6,10),(6,14),(8,7),(9,7),(10,8),(11,9),(12,10),(13,11),(14,8),(14,9)],15)
 => ? = 2
[.,[[[[.,.],.],.],[.,.]]]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,3,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,14),(3,13),(4,12),(4,16),(5,13),(5,17),(6,16),(6,17),(8,10),(9,11),(10,7),(11,7),(12,8),(13,9),(14,12),(15,10),(15,11),(16,8),(16,15),(17,9),(17,15)],18)
 => ? = 2
[.,[[.,[.,[.,[.,.]]]],.]]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,3,1,5,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,10),(8,9),(8,11),(9,12),(10,11),(11,12)],13)
 => ? = 1
[.,[[.,[[.,.],[.,.]]],.]]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => [2,3,1,5,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
 => ? = 1
[.,[[.,[[[.,.],.],.]],.]]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,3,5,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,9),(8,9)],10)
 => ? = 1
[.,[[[.,.],[.,[.,.]]],.]]
 => [1,0,1,1,1,0,0,1,0,1,0,0]
 => [2,3,1,6,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
 => ? = 1
[.,[[[.,[.,.]],[.,.]],.]]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => [2,3,1,4,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,8),(4,8),(5,11),(6,9),(6,10),(7,10),(8,11),(9,12),(10,12),(11,9)],13)
 => ? = 1
[.,[[[[.,.],.],[.,.]],.]]
 => [1,0,1,1,1,1,0,0,0,1,0,0]
 => [2,3,1,4,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,13),(3,13),(4,11),(5,12),(5,14),(6,10),(6,14),(8,7),(9,7),(10,9),(11,10),(12,8),(13,11),(14,8),(14,9)],15)
 => ? = 2
[.,[[[.,[.,[.,.]]],.],.]]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,3,4,1,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
 => ? = 1
[.,[[[[.,.],[.,.]],.],.]]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => [2,3,4,1,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,10),(5,11),(5,12),(6,9),(6,13),(8,10),(9,7),(10,9),(11,8),(12,8),(13,7)],14)
 => ? = 2
[[.,.],[.,[.,[.,[.,.]]]]]
 => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [3,1,4,2,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,9),(4,9),(5,7),(6,7),(7,8),(9,8)],10)
 => ? = 1
[[.,.],[.,[.,[[.,.],.]]]]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [3,4,1,2,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(7,11),(8,10),(9,10),(10,11)],12)
 => ? = 1
[[.,.],[.,[[.,[.,.]],.]]]
 => [1,1,0,0,1,0,1,1,0,1,0,0]
 => [3,4,1,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
[[.,.],[[.,.],[.,[.,.]]]]
 => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [3,1,4,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,10),(5,7),(6,7),(6,8),(7,9),(8,9),(10,8)],11)
 => ? = 2
[[.,.],[[.,.],[[.,.],.]]]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [3,4,1,2,5,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,9),(3,8),(4,8),(5,7),(6,7),(6,10),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
 => ? = 2
[[.,[.,.]],[.,[[.,.],.]]]
 => [1,1,0,1,0,0,1,0,1,1,0,0]
 => [3,5,1,2,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
[[.,[.,.]],[[.,.],[.,.]]]
 => [1,1,0,1,0,0,1,1,0,0,1,0]
 => [3,1,5,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
[[.,[.,.]],[[.,[.,.]],.]]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => [3,5,1,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
[[[.,.],.],[.,[.,[.,.]]]]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
 => 1
[[[.,[.,.]],.],[.,[.,.]]]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => [4,1,6,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
 => 1
[.,[.,[.,[[.,[[.,.],.]],.]]]]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [2,4,6,1,3,7,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => 1
[.,[[.,.],[[.,[.,[.,.]]],.]]]
 => [1,0,1,1,0,0,1,1,0,1,0,1,0,0]
 => [2,5,1,6,3,7,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => 1
[[.,[.,.]],[[.,[.,[.,.]]],.]]
 => [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
 => [3,5,1,6,2,7,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => 1
[[.,[.,[.,.]]],[[.,.],[.,.]]]
 => [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
 => [3,1,5,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => 1
[[.,[.,[.,.]]],[[.,[.,.]],.]]
 => [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
 => [3,5,1,7,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => 1
[[[.,.],[.,.]],[.,[.,[.,.]]]]
 => [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
 => [4,1,5,2,7,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
 => 1

Description

The size of the image of the pop stack sorting operator. The pop stack sorting operator is defined by

$Pop_L^\downarrow(x) = x\wedge\bigwedge\{y\in L\mid y\lessdot x\}$ . This statistic returns the size of

$Pop_L^\downarrow(L)\}$ .