Your data matches 15 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000359
(load all 2 compositions to match this statistic)
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000359: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => 0
[.,[.,.]]
 => [2,1] => [2,1] => 0
[[.,.],.]
 => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [3,1,2] => 0
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => 0
[[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,1,2,3] => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,3,2] => 0
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [4,2,1,3] => 0
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,2,1] => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,4,2,3] => 0
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,3,2] => 0
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,1,2,4] => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => 0
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,1,2,3,4] => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,1,2,4,3] => 0
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5,1,3,2,4] => 0
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,1,4,3,2] => 0
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [5,1,3,4,2] => 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [5,2,1,3,4] => 0
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [5,2,1,4,3] => 0
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [5,3,2,1,4] => 0
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [5,2,3,1,4] => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,4,2,3,1] => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,4,3,2,1] => 0
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [5,2,4,3,1] => 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,3,2,4,1] => 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,2,3,4,1] => 2
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,5,2,3,4] => 0
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,5,2,4,3] => 0
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,5,3,2,4] => 0
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,5,4,3,2] => 0
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,5,3,4,2] => 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,5,3,4] => 0
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,5,4,3] => 0
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,5,3,4] => 0
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,5,4,3] => 0
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,1,2,5,4] => 0
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,2,1,5,4] => 0
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,5,4] => 0
Description
The number of occurrences of the pattern 23-1. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $23\!\!-\!\!1$.
Matching statistic: St000355
(load all 2 compositions to match this statistic)
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St000355: Permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 57%
Values
[.,.]
 => [1] => [1] => [1] => 0
[.,[.,.]]
 => [2,1] => [2,1] => [1,2] => 0
[[.,.],.]
 => [1,2] => [1,2] => [2,1] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [2,3,1] => [1,3,2] => 0
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => [1,2,3] => 0
[[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => [2,3,1] => 0
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [3,1,2] => 0
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [3,2,1] => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [2,3,4,1] => [1,4,3,2] => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [2,4,3,1] => [1,3,4,2] => 0
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [3,2,4,1] => [1,4,2,3] => 0
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,3,4,2] => [2,4,3,1] => 0
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 0
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,3,1,4] => [4,1,3,2] => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => [4,1,2,3] => 0
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 0
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [2,3,4,5,1] => [1,5,4,3,2] => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [2,3,5,4,1] => [1,4,5,3,2] => 0
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [2,4,3,5,1] => [1,5,3,4,2] => 0
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [2,5,4,3,1] => [1,3,4,5,2] => 0
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [2,5,3,4,1] => [1,4,3,5,2] => 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [3,2,4,5,1] => [1,5,4,2,3] => 0
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [3,2,5,4,1] => [1,4,5,2,3] => 0
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [4,3,2,5,1] => [1,5,2,3,4] => 0
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [4,2,3,5,1] => [1,5,3,2,4] => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,3,4,2,1] => [1,2,4,3,5] => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [5,2,4,3,1] => [1,3,4,2,5] => 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,3,2,4,1] => [1,4,2,3,5] => 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => 2
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,3,4,5,2] => [2,5,4,3,1] => 0
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,3,5,4,2] => [2,4,5,3,1] => 0
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,4,3,5,2] => [2,5,3,4,1] => 0
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,5,3,4,2] => [2,4,3,5,1] => 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,4,5,3] => [3,5,4,1,2] => 0
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,5,4,3] => [3,4,5,1,2] => 0
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,4,5,3] => [3,5,4,2,1] => 0
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => 0
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [2,3,1,5,4] => [4,5,1,3,2] => 0
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,2,1,5,4] => [4,5,1,2,3] => 0
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 0
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 0
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => [1,7,6,5,4,3,2] => ? = 0
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
 => [2,7,6,5,4,3,1] => [3,2,4,5,6,7,1] => [1,7,6,5,4,2,3] => ? = 0
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
 => [3,2,7,6,5,4,1] => [4,3,2,5,6,7,1] => [1,7,6,5,2,3,4] => ? = 0
[.,[[[.,.],.],[.,[.,[.,.]]]]]
 => [2,3,7,6,5,4,1] => [4,2,3,5,6,7,1] => [1,7,6,5,3,2,4] => ? = 1
[.,[[[.,[.,.]],.],[.,[.,.]]]]
 => [3,2,4,7,6,5,1] => [5,3,2,4,6,7,1] => [1,7,6,4,2,3,5] => ? = 1
[.,[[[[.,.],.],.],[.,[.,.]]]]
 => [2,3,4,7,6,5,1] => [5,2,3,4,6,7,1] => [1,7,6,4,3,2,5] => ? = 2
[.,[[[[[.,.],.],.],.],[.,.]]]
 => [2,3,4,5,7,6,1] => [6,2,3,4,5,7,1] => [1,7,5,4,3,2,6] => ? = 3
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
 => [1,7,6,5,4,3,2] => [1,3,4,5,6,7,2] => [2,7,6,5,4,3,1] => ? = 0
[[.,.],[[.,[[.,[.,.]],.]],.]]
 => [1,5,4,6,3,7,2] => [1,7,6,5,4,3,2] => [2,3,4,5,6,7,1] => ? = 0
[[.,.],[[.,[[[.,.],.],.]],.]]
 => [1,4,5,6,3,7,2] => [1,7,6,4,5,3,2] => [2,3,5,4,6,7,1] => ? = 2
[[.,.],[[[.,.],[[.,.],.]],.]]
 => [1,3,5,6,4,7,2] => [1,7,3,6,5,4,2] => [2,4,5,6,3,7,1] => ? = 1
[[.,.],[[[.,[.,.]],[.,.]],.]]
 => [1,4,3,6,5,7,2] => [1,7,4,3,6,5,2] => [2,5,6,3,4,7,1] => ? = 1
[[.,.],[[[.,[[.,.],.]],.],.]]
 => [1,4,5,3,6,7,2] => [1,7,5,4,3,6,2] => [2,6,3,4,5,7,1] => ? = 1
[[.,[.,.]],[.,[.,[.,[.,.]]]]]
 => [2,1,7,6,5,4,3] => [2,1,4,5,6,7,3] => [3,7,6,5,4,1,2] => ? = 0
[[.,[.,.]],[[.,[[.,.],.]],.]]
 => [2,1,5,6,4,7,3] => [2,1,7,6,5,4,3] => [3,4,5,6,7,1,2] => ? = 0
[[[.,[.,.]],.],[.,[.,[.,.]]]]
 => [2,1,3,7,6,5,4] => [2,1,3,5,6,7,4] => [4,7,6,5,3,1,2] => ? = 0
[[[[.,[.,.]],.],.],[.,[.,.]]]
 => [2,1,3,4,7,6,5] => [2,1,3,4,6,7,5] => [5,7,6,4,3,1,2] => ? = 0
[[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => [5,4,3,2,1,7,6] => [2,3,4,5,1,7,6] => [6,7,1,5,4,3,2] => ? = 0
[[.,[[.,[[.,.],.]],.]],[.,.]]
 => [3,4,2,5,1,7,6] => [5,4,3,2,1,7,6] => [6,7,1,2,3,4,5] => ? = 0
[[[.,.],[.,[.,[.,.]]]],[.,.]]
 => [1,5,4,3,2,7,6] => [1,3,4,5,2,7,6] => [6,7,2,5,4,3,1] => ? = 0
[[[.,[.,[.,[.,.]]]],.],[.,.]]
 => [4,3,2,1,5,7,6] => [2,3,4,1,5,7,6] => [6,7,5,1,4,3,2] => ? = 0
[[[[.,[.,[.,.]]],.],.],[.,.]]
 => [3,2,1,4,5,7,6] => [2,3,1,4,5,7,6] => [6,7,5,4,1,3,2] => ? = 0
[[[[[[.,.],.],.],.],.],[.,.]]
 => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [6,7,5,4,3,2,1] => ? = 0
[[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => [6,5,4,3,2,1,7] => [2,3,4,5,6,1,7] => [7,1,6,5,4,3,2] => ? = 0
[[.,[[.,[[.,[.,.]],.]],.]],.]
 => [4,3,5,2,6,1,7] => [6,5,4,3,2,1,7] => [7,1,2,3,4,5,6] => ? = 0
[[.,[[.,[[[.,.],.],.]],.]],.]
 => [3,4,5,2,6,1,7] => [6,5,3,4,2,1,7] => [7,1,2,4,3,5,6] => ? = 2
[[.,[[[.,.],[[.,.],.]],.]],.]
 => [2,4,5,3,6,1,7] => [6,2,5,4,3,1,7] => [7,1,3,4,5,2,6] => ? = 1
[[.,[[[.,[.,.]],[.,.]],.]],.]
 => [3,2,5,4,6,1,7] => [6,3,2,5,4,1,7] => [7,1,4,5,2,3,6] => ? = 1
[[.,[[[.,[[.,.],.]],.],.]],.]
 => [3,4,2,5,6,1,7] => [6,4,3,2,5,1,7] => [7,1,5,2,3,4,6] => ? = 1
[[[.,[.,.]],[.,[.,[.,.]]]],.]
 => [2,1,6,5,4,3,7] => [2,1,4,5,6,3,7] => [7,3,6,5,4,1,2] => ? = 0
[[[[[[.,.],.],.],.],[.,.]],.]
 => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [7,5,6,4,3,2,1] => ? = 0
[[[.,[.,[.,[.,[.,.]]]]],.],.]
 => [5,4,3,2,1,6,7] => [2,3,4,5,1,6,7] => [7,6,1,5,4,3,2] => ? = 0
[[[.,[[.,[[.,.],.]],.]],.],.]
 => [3,4,2,5,1,6,7] => [5,4,3,2,1,6,7] => [7,6,1,2,3,4,5] => ? = 0
[[[[.,[.,[.,[.,.]]]],.],.],.]
 => [4,3,2,1,5,6,7] => [2,3,4,1,5,6,7] => [7,6,5,1,4,3,2] => ? = 0
[[[[.,[[.,[.,.]],.]],.],.],.]
 => [3,2,4,1,5,6,7] => [4,3,2,1,5,6,7] => [7,6,5,1,2,3,4] => ? = 0
[[[[[.,[.,[.,.]]],.],.],.],.]
 => [3,2,1,4,5,6,7] => [2,3,1,4,5,6,7] => [7,6,5,4,1,3,2] => ? = 0
[[[[[.,[[.,.],.]],.],.],.],.]
 => [2,3,1,4,5,6,7] => [3,2,1,4,5,6,7] => [7,6,5,4,1,2,3] => ? = 0
[[[[[[.,.],[.,.]],.],.],.],.]
 => [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [7,6,5,4,2,3,1] => ? = 0
[[[[[[.,[.,.]],.],.],.],.],.]
 => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [7,6,5,4,3,1,2] => ? = 0
[[[[[[[.,.],.],.],.],.],.],.]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [7,6,5,4,3,2,1] => ? = 0
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => [8,7,6,5,4,3,2,1] => [2,3,4,5,6,7,8,1] => [1,8,7,6,5,4,3,2] => ? = 0
[.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
 => [4,5,6,8,7,3,2,1] => [2,3,7,4,5,6,8,1] => [1,8,6,5,4,7,3,2] => ? = 2
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
 => [2,8,7,6,5,4,3,1] => [3,2,4,5,6,7,8,1] => [1,8,7,6,5,4,2,3] => ? = 0
[.,[[.,.],[[.,[.,.]],[[.,.],.]]]]
 => [2,5,4,7,8,6,3,1] => [3,2,6,5,4,8,7,1] => [1,7,8,4,5,6,2,3] => ? = 0
[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
 => [3,2,8,7,6,5,4,1] => [4,3,2,5,6,7,8,1] => [1,8,7,6,5,2,3,4] => ? = 0
[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
 => [2,3,8,7,6,5,4,1] => [4,2,3,5,6,7,8,1] => [1,8,7,6,5,3,2,4] => ? = 1
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
 => [2,3,4,8,7,6,5,1] => [5,2,3,4,6,7,8,1] => [1,8,7,6,4,3,2,5] => ? = 2
[.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
 => [4,5,3,6,2,8,7,1] => [7,6,5,4,3,2,8,1] => [1,8,2,3,4,5,6,7] => ? = 0
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
 => [5,4,6,3,7,2,8,1] => [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => ? = 0
[.,[[.,[[[.,[.,.]],[.,.]],.]],.]]
 => [4,3,6,5,7,2,8,1] => [8,7,4,3,6,5,2,1] => [1,2,5,6,3,4,7,8] => ? = 2
Description
The number of occurrences of the pattern 21-3. See [[Permutations/#Pattern-avoiding_permutations]] for the definition of the pattern $21\!\!-\!\!3$.
Matching statistic: St000317
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 36% ●values known / values provided: 36%●distinct values known / distinct values provided: 57%
Values
[.,.]
 => [1] => [1] => [1] => 0
[.,[.,.]]
 => [2,1] => [2,1] => [2,1] => 0
[[.,.],.]
 => [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [2,3,1] => [1,3,2] => 0
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => [3,2,1] => 0
[[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [2,3,4,1] => [1,2,4,3] => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [2,4,3,1] => [1,4,3,2] => 0
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [3,2,4,1] => [2,1,4,3] => 0
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => [4,1,3,2] => 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,3,4,2] => [1,2,4,3] => 0
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,3,1,4] => [1,3,2,4] => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [2,3,4,5,1] => [1,2,3,5,4] => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [2,3,5,4,1] => [1,2,5,4,3] => 0
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [2,4,3,5,1] => [1,3,2,5,4] => 0
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [2,5,4,3,1] => [1,5,4,3,2] => 0
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [2,5,3,4,1] => [1,5,2,4,3] => 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [3,2,4,5,1] => [2,1,3,5,4] => 0
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [3,2,5,4,1] => [2,1,5,4,3] => 0
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [4,3,2,5,1] => [3,2,1,5,4] => 0
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [4,2,3,5,1] => [3,1,2,5,4] => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,3,4,2,1] => [5,1,4,3,2] => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [5,2,4,3,1] => [5,1,4,3,2] => 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,3,2,4,1] => [5,2,1,4,3] => 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,2,3,4,1] => [5,1,2,4,3] => 2
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,3,4,5,2] => [1,2,3,5,4] => 0
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,3,5,4,2] => [1,2,5,4,3] => 0
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,4,3,5,2] => [1,3,2,5,4] => 0
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,2,4,3] => 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,4,5,3] => [2,1,3,5,4] => 0
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 0
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,4,5,3] => [1,2,3,5,4] => 0
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [2,3,1,5,4] => [1,3,2,5,4] => 0
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 0
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 0
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[.,[.,[.,[.,[.,[.,[.,.]]]]]]]
 => [7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => [1,2,3,4,5,7,6] => ? = 0
[.,[[.,.],[.,[.,[.,[.,.]]]]]]
 => [2,7,6,5,4,3,1] => [3,2,4,5,6,7,1] => [2,1,3,4,5,7,6] => ? = 0
[.,[[.,[.,.]],[.,[.,[.,.]]]]]
 => [3,2,7,6,5,4,1] => [4,3,2,5,6,7,1] => [3,2,1,4,5,7,6] => ? = 0
[.,[[[.,.],.],[.,[.,[.,.]]]]]
 => [2,3,7,6,5,4,1] => [4,2,3,5,6,7,1] => [3,1,2,4,5,7,6] => ? = 1
[.,[[[.,[.,.]],.],[.,[.,.]]]]
 => [3,2,4,7,6,5,1] => [5,3,2,4,6,7,1] => [4,2,1,3,5,7,6] => ? = 1
[.,[[[[.,.],.],.],[.,[.,.]]]]
 => [2,3,4,7,6,5,1] => [5,2,3,4,6,7,1] => [4,1,2,3,5,7,6] => ? = 2
[.,[[[[[.,.],.],.],.],[.,.]]]
 => [2,3,4,5,7,6,1] => [6,2,3,4,5,7,1] => [5,1,2,3,4,7,6] => ? = 3
[.,[[.,[[.,[[.,.],.]],.]],.]]
 => [4,5,3,6,2,7,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 0
[[.,.],[.,[.,[.,[.,[.,.]]]]]]
 => [1,7,6,5,4,3,2] => [1,3,4,5,6,7,2] => [1,2,3,4,5,7,6] => ? = 0
[[.,.],[[.,[[.,[.,.]],.]],.]]
 => [1,5,4,6,3,7,2] => [1,7,6,5,4,3,2] => [1,7,6,5,4,3,2] => ? = 0
[[.,.],[[.,[[[.,.],.],.]],.]]
 => [1,4,5,6,3,7,2] => [1,7,6,4,5,3,2] => [1,7,6,2,5,4,3] => ? = 2
[[.,.],[[[.,.],[[.,.],.]],.]]
 => [1,3,5,6,4,7,2] => [1,7,3,6,5,4,2] => [1,7,2,6,5,4,3] => ? = 1
[[.,.],[[[.,[.,.]],[.,.]],.]]
 => [1,4,3,6,5,7,2] => [1,7,4,3,6,5,2] => [1,7,3,2,6,5,4] => ? = 1
[[.,.],[[[.,[[.,.],.]],.],.]]
 => [1,4,5,3,6,7,2] => [1,7,5,4,3,6,2] => [1,7,4,3,2,6,5] => ? = 1
[[.,[.,.]],[.,[.,[.,[.,.]]]]]
 => [2,1,7,6,5,4,3] => [2,1,4,5,6,7,3] => [2,1,3,4,5,7,6] => ? = 0
[[.,[.,.]],[[.,[[.,.],.]],.]]
 => [2,1,5,6,4,7,3] => [2,1,7,6,5,4,3] => [2,1,7,6,5,4,3] => ? = 0
[[[.,[.,.]],.],[.,[.,[.,.]]]]
 => [2,1,3,7,6,5,4] => [2,1,3,5,6,7,4] => [2,1,3,4,5,7,6] => ? = 0
[[[[.,[.,.]],.],.],[.,[.,.]]]
 => [2,1,3,4,7,6,5] => [2,1,3,4,6,7,5] => [2,1,3,4,5,7,6] => ? = 0
[[.,[.,[.,[.,[.,.]]]]],[.,.]]
 => [5,4,3,2,1,7,6] => [2,3,4,5,1,7,6] => [1,2,3,5,4,7,6] => ? = 0
[[.,[[.,[[.,.],.]],.]],[.,.]]
 => [3,4,2,5,1,7,6] => [5,4,3,2,1,7,6] => [5,4,3,2,1,7,6] => ? = 0
[[[.,.],[.,[.,[.,.]]]],[.,.]]
 => [1,5,4,3,2,7,6] => [1,3,4,5,2,7,6] => [1,2,3,5,4,7,6] => ? = 0
[[[.,[.,[.,[.,.]]]],.],[.,.]]
 => [4,3,2,1,5,7,6] => [2,3,4,1,5,7,6] => [1,2,4,3,5,7,6] => ? = 0
[[[[.,[.,[.,.]]],.],.],[.,.]]
 => [3,2,1,4,5,7,6] => [2,3,1,4,5,7,6] => [1,3,2,4,5,7,6] => ? = 0
[[[[[[.,.],.],.],.],.],[.,.]]
 => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => ? = 0
[[.,[.,[.,[.,[.,[.,.]]]]]],.]
 => [6,5,4,3,2,1,7] => [2,3,4,5,6,1,7] => [1,2,3,4,6,5,7] => ? = 0
[[.,[[.,[[.,[.,.]],.]],.]],.]
 => [4,3,5,2,6,1,7] => [6,5,4,3,2,1,7] => [6,5,4,3,2,1,7] => ? = 0
[[.,[[.,[[[.,.],.],.]],.]],.]
 => [3,4,5,2,6,1,7] => [6,5,3,4,2,1,7] => [6,5,1,4,3,2,7] => ? = 2
[[.,[[[.,.],[[.,.],.]],.]],.]
 => [2,4,5,3,6,1,7] => [6,2,5,4,3,1,7] => [6,1,5,4,3,2,7] => ? = 1
[[.,[[[.,[.,.]],[.,.]],.]],.]
 => [3,2,5,4,6,1,7] => [6,3,2,5,4,1,7] => [6,2,1,5,4,3,7] => ? = 1
[[.,[[[.,[[.,.],.]],.],.]],.]
 => [3,4,2,5,6,1,7] => [6,4,3,2,5,1,7] => [6,3,2,1,5,4,7] => ? = 1
[[[.,[.,.]],[.,[.,[.,.]]]],.]
 => [2,1,6,5,4,3,7] => [2,1,4,5,6,3,7] => [2,1,3,4,6,5,7] => ? = 0
[[[[[[.,.],.],.],.],[.,.]],.]
 => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => [1,2,3,4,6,5,7] => ? = 0
[[[.,[.,[.,[.,[.,.]]]]],.],.]
 => [5,4,3,2,1,6,7] => [2,3,4,5,1,6,7] => [1,2,3,5,4,6,7] => ? = 0
[[[.,[[.,[[.,.],.]],.]],.],.]
 => [3,4,2,5,1,6,7] => [5,4,3,2,1,6,7] => [5,4,3,2,1,6,7] => ? = 0
[[[[.,[.,[.,[.,.]]]],.],.],.]
 => [4,3,2,1,5,6,7] => [2,3,4,1,5,6,7] => [1,2,4,3,5,6,7] => ? = 0
[[[[.,[[.,[.,.]],.]],.],.],.]
 => [3,2,4,1,5,6,7] => [4,3,2,1,5,6,7] => [4,3,2,1,5,6,7] => ? = 0
[[[[[.,[.,[.,.]]],.],.],.],.]
 => [3,2,1,4,5,6,7] => [2,3,1,4,5,6,7] => [1,3,2,4,5,6,7] => ? = 0
[[[[[.,[[.,.],.]],.],.],.],.]
 => [2,3,1,4,5,6,7] => [3,2,1,4,5,6,7] => [3,2,1,4,5,6,7] => ? = 0
[[[[[[.,.],[.,.]],.],.],.],.]
 => [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => [1,3,2,4,5,6,7] => ? = 0
[[[[[[.,[.,.]],.],.],.],.],.]
 => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => [2,1,3,4,5,6,7] => ? = 0
[[[[[[[.,.],.],.],.],.],.],.]
 => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => ? = 0
[.,[.,[.,[.,[.,[.,[.,[.,.]]]]]]]]
 => [8,7,6,5,4,3,2,1] => [2,3,4,5,6,7,8,1] => [1,2,3,4,5,6,8,7] => ? = 0
[.,[.,[.,[[[[.,.],.],.],[.,.]]]]]
 => [4,5,6,8,7,3,2,1] => [2,3,7,4,5,6,8,1] => ? => ? = 2
[.,[[.,.],[.,[.,[.,[.,[.,.]]]]]]]
 => [2,8,7,6,5,4,3,1] => [3,2,4,5,6,7,8,1] => ? => ? = 0
[.,[[.,.],[[.,[.,.]],[[.,.],.]]]]
 => [2,5,4,7,8,6,3,1] => [3,2,6,5,4,8,7,1] => ? => ? = 0
[.,[[.,[.,.]],[.,[.,[.,[.,.]]]]]]
 => [3,2,8,7,6,5,4,1] => [4,3,2,5,6,7,8,1] => ? => ? = 0
[.,[[[.,.],.],[.,[.,[.,[.,.]]]]]]
 => [2,3,8,7,6,5,4,1] => [4,2,3,5,6,7,8,1] => ? => ? = 1
[.,[[[[.,.],.],.],[.,[.,[.,.]]]]]
 => [2,3,4,8,7,6,5,1] => [5,2,3,4,6,7,8,1] => ? => ? = 2
[.,[[.,[[.,[[.,.],.]],.]],[.,.]]]
 => [4,5,3,6,2,8,7,1] => [7,6,5,4,3,2,8,1] => ? => ? = 0
[.,[[.,[[.,[[.,[.,.]],.]],.]],.]]
 => [5,4,6,3,7,2,8,1] => [8,7,6,5,4,3,2,1] => [8,7,6,5,4,3,2,1] => ? = 0
Description
The cycle descent number of a permutation. Let $(i_1,\ldots,i_k)$ be a cycle of a permutation $\pi$ such that $i_1$ is its smallest element. A **cycle descent** of $(i_1,\ldots,i_k)$ is an $i_a$ for $1 \leq a < k$ such that $i_a > i_{a+1}$. The **cycle descent set** of $\pi$ is then the set of descents in all the cycles of $\pi$, and the **cycle descent number** is its cardinality.
Matching statistic: St001330
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 15%●distinct values known / distinct values provided: 14%
Values
[.,.]
 => [1,0]
 => [2,1] => ([(0,1)],2)
 => 2 = 0 + 2
[.,[.,.]]
 => [1,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[[.,.],.]
 => [1,0,1,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => 2 = 0 + 2
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 0 + 2
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 0 + 2
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 0 + 2
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 0 + 2
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => ? = 0 + 2
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 2
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 0 + 2
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 0 + 2
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 0 + 2
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 0 + 2
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 0 + 2
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 2 = 0 + 2
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 0 + 2
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [6,3,4,5,1,2] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 2
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [5,3,4,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [2,6,4,5,1,3] => ([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [6,5,4,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [2,5,4,1,6,3] => ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 0 + 2
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [6,3,5,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 1 + 2
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 2 + 2
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 0 + 2
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 2
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 0 + 2
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 2
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 0 + 2
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 2
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 0 + 2
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 0 + 2
[[.,[.,[[.,.],.]]],.]
 => [1,1,1,0,1,0,0,0,1,0]
 => [6,3,4,1,2,5] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
 => ? = 0 + 2
[[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[[.,[[.,[.,.]],.]],.]
 => [1,1,1,0,0,1,0,0,1,0]
 => [2,6,4,1,3,5] => ([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[[.,[[[.,.],.],.]],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 2
[[[.,.],[.,[.,.]]],.]
 => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 2 = 0 + 2
[[[.,.],[[.,.],.]],.]
 => [1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[[[.,[.,.]],[.,.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 2 = 0 + 2
[[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 0 + 2
[[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 0 + 2
[[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 2
[[[[.,.],[.,.]],.],.]
 => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 2 = 0 + 2
[[[[.,[.,.]],.],.],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 2 = 0 + 2
[[[[[.,.],.],.],.],.]
 => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 0 + 2
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 0 + 2
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [7,3,4,5,6,1,2] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 0 + 2
[.,[.,[.,[[.,.],[.,.]]]]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [6,3,4,5,1,7,2] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
 => ? = 0 + 2
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [2,7,4,5,6,1,3] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
 => ? = 0 + 2
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [7,6,4,5,1,2,3] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 2
[.,[.,[[.,.],[.,[.,.]]]]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [5,3,4,1,6,7,2] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 2
[.,[.,[[.,.],[[.,.],.]]]]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [7,5,4,1,6,2,3] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 2
[.,[.,[[.,[.,.]],[.,.]]]]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [2,6,4,5,1,7,3] => ([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)
 => ? = 0 + 2
[.,[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [6,5,4,1,2,7,3] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 2
[.,[.,[[.,[.,[.,.]]],.]]]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [2,3,7,5,6,1,4] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 2
[.,[.,[[.,[[.,.],.]],.]]]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [7,3,6,5,1,2,4] => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 2
[.,[.,[[[.,.],[.,.]],.]]]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [7,3,5,1,6,2,4] => ([(0,3),(0,5),(0,6),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 2
[.,[.,[[[.,[.,.]],.],.]]]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [2,7,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1 + 2
[.,[.,[[[[.,.],.],.],.]]]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [6,7,5,1,2,3,4] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
 => ? = 2 + 2
[.,[[.,.],[.,[.,[.,.]]]]]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [4,3,1,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 2
[.,[[.,.],[.,[[.,.],.]]]]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [7,4,1,5,6,2,3] => ([(0,5),(0,6),(1,3),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 2
[.,[[.,.],[[.,.],[.,.]]]]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [6,4,1,5,2,7,3] => ([(0,5),(1,4),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 2
[.,[[.,.],[[.,[.,.]],.]]]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [7,3,1,5,6,2,4] => ([(0,4),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0 + 2
[[.,.],[.,[.,[.,[.,.]]]]]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 2 = 0 + 2
[[.,[.,.]],[.,[.,[.,.]]]]
 => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [2,4,1,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 2 = 0 + 2
[[[.,.],.],[.,[.,[.,.]]]]
 => [1,0,1,0,1,1,1,1,0,0,0,0]
 => [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 2 = 0 + 2
[[.,[.,[.,.]]],[.,[.,.]]]
 => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,3,5,1,6,7,4] => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => 2 = 0 + 2
[[[.,.],[.,.]],[.,[.,.]]]
 => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [3,1,5,2,6,7,4] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 2 = 0 + 2
[[[.,[.,.]],.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [2,5,1,3,6,7,4] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[[[.,.],.],.],[.,[.,.]]]
 => [1,0,1,0,1,0,1,1,1,0,0,0]
 => [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => 2 = 0 + 2
[[.,[.,[.,[.,.]]]],[.,.]]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,3,4,6,1,7,5] => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => 2 = 0 + 2
[[[.,.],[.,[.,.]]],[.,.]]
 => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [3,1,4,6,2,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 2 = 0 + 2
[[[.,[.,.]],[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,4,1,6,3,7,5] => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
 => 2 = 0 + 2
[[[[.,.],.],[.,.]],[.,.]]
 => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => 2 = 0 + 2
[[[.,[.,[.,.]]],.],[.,.]]
 => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,6,1,4,7,5] => ([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[[[.,.],[.,.]],.],[.,.]]
 => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 2 = 0 + 2
[[[[.,[.,.]],.],.],[.,.]]
 => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[[[[.,.],.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,0,1,1,0,0]
 => [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 2 = 0 + 2
[[.,[.,[.,[.,[.,.]]]]],.]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => 2 = 0 + 2
[[[.,.],[.,[.,[.,.]]]],.]
 => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [3,1,4,5,7,2,6] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => 2 = 0 + 2
[[[.,[.,.]],[.,[.,.]]],.]
 => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,4,1,5,7,3,6] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
 => 2 = 0 + 2
Description
The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St001811
(load all 3 compositions to match this statistic)
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 12% ●values known / values provided: 12%●distinct values known / distinct values provided: 43%
Values
[.,.]
 => [1] => [1] => [1] => ? = 0
[.,[.,.]]
 => [2,1] => [2,1] => [1,2] => 0
[[.,.],.]
 => [1,2] => [1,2] => [2,1] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [3,1,2] => [2,1,3] => 0
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => [1,2,3] => 0
[[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => [2,3,1] => 0
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [3,1,2] => 0
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [3,2,1] => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,1,2,3] => [3,2,1,4] => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,3,2] => [2,3,1,4] => 0
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [4,2,1,3] => [3,1,2,4] => 0
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,2,1] => [1,2,3,4] => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => [1,3,2,4] => 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,4,2,3] => [3,2,4,1] => 0
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,3,2] => [2,3,4,1] => 0
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 0
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,4,3] => [3,4,2,1] => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,1,2,4] => [4,2,1,3] => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => [4,1,2,3] => 0
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 0
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 0
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,1,2,3,4] => [4,3,2,1,5] => 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,1,2,4,3] => [3,4,2,1,5] => 0
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5,1,3,2,4] => [4,2,3,1,5] => 0
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,1,4,3,2] => [2,3,4,1,5] => 0
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [5,1,3,4,2] => [2,4,3,1,5] => 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [5,2,1,3,4] => [4,3,1,2,5] => 0
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [5,2,1,4,3] => [3,4,1,2,5] => 0
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [5,3,2,1,4] => [4,1,2,3,5] => 0
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [5,2,3,1,4] => [4,1,3,2,5] => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,4,2,3,1] => [1,3,2,4,5] => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,4,3,2,1] => [1,2,3,4,5] => 0
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [5,2,4,3,1] => [1,3,4,2,5] => 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,3,2,4,1] => [1,4,2,3,5] => 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,2,3,4,1] => [1,4,3,2,5] => 2
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,5,2,3,4] => [4,3,2,5,1] => 0
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,5,2,4,3] => [3,4,2,5,1] => 0
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,5,3,2,4] => [4,2,3,5,1] => 0
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,5,4,3,2] => [2,3,4,5,1] => 0
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,5,3,4,2] => [2,4,3,5,1] => 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,5,3,4] => [4,3,5,1,2] => 0
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,5,4,3] => [3,4,5,1,2] => 0
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,5,3,4] => [4,3,5,2,1] => 0
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,5,4,3] => [3,4,5,2,1] => 0
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,1,2,5,4] => [4,5,2,1,3] => 0
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,2,1,5,4] => [4,5,1,2,3] => 0
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,5,4] => [4,5,2,3,1] => 0
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 0
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,5,4] => [4,5,3,2,1] => 0
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,1,2,3,5] => [5,3,2,1,4] => 0
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => [6,1,2,3,4,5] => [5,4,3,2,1,6] => ? = 0
[.,[.,[.,[.,[[.,.],.]]]]]
 => [5,6,4,3,2,1] => [6,1,2,3,5,4] => [4,5,3,2,1,6] => ? = 0
[.,[.,[.,[[.,.],[.,.]]]]]
 => [4,6,5,3,2,1] => [6,1,2,4,3,5] => [5,3,4,2,1,6] => ? = 0
[.,[.,[.,[[.,[.,.]],.]]]]
 => [5,4,6,3,2,1] => [6,1,2,5,4,3] => [3,4,5,2,1,6] => ? = 0
[.,[.,[.,[[[.,.],.],.]]]]
 => [4,5,6,3,2,1] => [6,1,2,4,5,3] => [3,5,4,2,1,6] => ? = 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [3,6,5,4,2,1] => [6,1,3,2,4,5] => [5,4,2,3,1,6] => ? = 0
[.,[.,[[.,.],[[.,.],.]]]]
 => [3,5,6,4,2,1] => [6,1,3,2,5,4] => [4,5,2,3,1,6] => ? = 0
[.,[.,[[.,[.,.]],[.,.]]]]
 => [4,3,6,5,2,1] => [6,1,4,3,2,5] => [5,2,3,4,1,6] => ? = 0
[.,[.,[[[.,.],.],[.,.]]]]
 => [3,4,6,5,2,1] => [6,1,3,4,2,5] => [5,2,4,3,1,6] => ? = 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [5,4,3,6,2,1] => [6,1,5,3,4,2] => [2,4,3,5,1,6] => ? = 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [4,5,3,6,2,1] => [6,1,5,4,3,2] => [2,3,4,5,1,6] => ? = 0
[.,[.,[[[.,.],[.,.]],.]]]
 => [3,5,4,6,2,1] => [6,1,3,5,4,2] => [2,4,5,3,1,6] => ? = 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [4,3,5,6,2,1] => [6,1,4,3,5,2] => [2,5,3,4,1,6] => ? = 1
[.,[.,[[[[.,.],.],.],.]]]
 => [3,4,5,6,2,1] => [6,1,3,4,5,2] => [2,5,4,3,1,6] => ? = 2
[.,[[.,.],[.,[.,[.,.]]]]]
 => [2,6,5,4,3,1] => [6,2,1,3,4,5] => [5,4,3,1,2,6] => ? = 0
[.,[[.,.],[.,[[.,.],.]]]]
 => [2,5,6,4,3,1] => [6,2,1,3,5,4] => [4,5,3,1,2,6] => ? = 0
[.,[[.,.],[[.,.],[.,.]]]]
 => [2,4,6,5,3,1] => [6,2,1,4,3,5] => [5,3,4,1,2,6] => ? = 0
[.,[[.,.],[[.,[.,.]],.]]]
 => [2,5,4,6,3,1] => [6,2,1,5,4,3] => [3,4,5,1,2,6] => ? = 0
[.,[[.,.],[[[.,.],.],.]]]
 => [2,4,5,6,3,1] => [6,2,1,4,5,3] => [3,5,4,1,2,6] => ? = 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => [3,2,6,5,4,1] => [6,3,2,1,4,5] => [5,4,1,2,3,6] => ? = 0
[.,[[.,[.,.]],[[.,.],.]]]
 => [3,2,5,6,4,1] => [6,3,2,1,5,4] => [4,5,1,2,3,6] => ? = 0
[.,[[[.,.],.],[.,[.,.]]]]
 => [2,3,6,5,4,1] => [6,2,3,1,4,5] => [5,4,1,3,2,6] => ? = 1
[.,[[[.,.],.],[[.,.],.]]]
 => [2,3,5,6,4,1] => [6,2,3,1,5,4] => [4,5,1,3,2,6] => ? = 1
[.,[[.,[.,[.,.]]],[.,.]]]
 => [4,3,2,6,5,1] => [6,4,2,3,1,5] => [5,1,3,2,4,6] => ? = 1
[.,[[.,[[.,.],.]],[.,.]]]
 => [3,4,2,6,5,1] => [6,4,3,2,1,5] => [5,1,2,3,4,6] => ? = 0
[.,[[[.,.],[.,.]],[.,.]]]
 => [2,4,3,6,5,1] => [6,2,4,3,1,5] => [5,1,3,4,2,6] => ? = 1
[.,[[[.,[.,.]],.],[.,.]]]
 => [3,2,4,6,5,1] => [6,3,2,4,1,5] => [5,1,4,2,3,6] => ? = 1
[.,[[[[.,.],.],.],[.,.]]]
 => [2,3,4,6,5,1] => [6,2,3,4,1,5] => [5,1,4,3,2,6] => ? = 2
[.,[[.,[.,[.,[.,.]]]],.]]
 => [5,4,3,2,6,1] => [6,5,2,3,4,1] => [1,4,3,2,5,6] => ? = 2
[.,[[.,[.,[[.,.],.]]],.]]
 => [4,5,3,2,6,1] => [6,5,2,4,3,1] => [1,3,4,2,5,6] => ? = 1
[.,[[.,[[.,.],[.,.]]],.]]
 => [3,5,4,2,6,1] => [6,5,3,2,4,1] => [1,4,2,3,5,6] => ? = 1
[.,[[.,[[.,[.,.]],.]],.]]
 => [4,3,5,2,6,1] => [6,5,4,3,2,1] => [1,2,3,4,5,6] => ? = 0
[.,[[.,[[[.,.],.],.]],.]]
 => [3,4,5,2,6,1] => [6,5,3,4,2,1] => [1,2,4,3,5,6] => ? = 2
[.,[[[.,.],[.,[.,.]]],.]]
 => [2,5,4,3,6,1] => [6,2,5,3,4,1] => [1,4,3,5,2,6] => ? = 2
[.,[[[.,.],[[.,.],.]],.]]
 => [2,4,5,3,6,1] => [6,2,5,4,3,1] => [1,3,4,5,2,6] => ? = 1
[.,[[[.,[.,.]],[.,.]],.]]
 => [3,2,5,4,6,1] => [6,3,2,5,4,1] => [1,4,5,2,3,6] => ? = 1
[.,[[[[.,.],.],[.,.]],.]]
 => [2,3,5,4,6,1] => [6,2,3,5,4,1] => [1,4,5,3,2,6] => ? = 2
[.,[[[.,[.,[.,.]]],.],.]]
 => [4,3,2,5,6,1] => [6,4,2,3,5,1] => [1,5,3,2,4,6] => ? = 2
[.,[[[.,[[.,.],.]],.],.]]
 => [3,4,2,5,6,1] => [6,4,3,2,5,1] => [1,5,2,3,4,6] => ? = 1
[.,[[[[.,.],[.,.]],.],.]]
 => [2,4,3,5,6,1] => [6,2,4,3,5,1] => [1,5,3,4,2,6] => ? = 2
[.,[[[[.,[.,.]],.],.],.]]
 => [3,2,4,5,6,1] => [6,3,2,4,5,1] => [1,5,4,2,3,6] => ? = 2
[.,[[[[[.,.],.],.],.],.]]
 => [2,3,4,5,6,1] => [6,2,3,4,5,1] => [1,5,4,3,2,6] => ? = 3
[[.,.],[.,[.,[.,[.,.]]]]]
 => [1,6,5,4,3,2] => [1,6,2,3,4,5] => [5,4,3,2,6,1] => ? = 0
[[.,.],[.,[.,[[.,.],.]]]]
 => [1,5,6,4,3,2] => [1,6,2,3,5,4] => [4,5,3,2,6,1] => ? = 0
[[.,.],[.,[[.,.],[.,.]]]]
 => [1,4,6,5,3,2] => [1,6,2,4,3,5] => [5,3,4,2,6,1] => ? = 0
[[.,.],[.,[[.,[.,.]],.]]]
 => [1,5,4,6,3,2] => [1,6,2,5,4,3] => [3,4,5,2,6,1] => ? = 0
[[.,.],[.,[[[.,.],.],.]]]
 => [1,4,5,6,3,2] => [1,6,2,4,5,3] => [3,5,4,2,6,1] => ? = 1
[[.,.],[[.,.],[.,[.,.]]]]
 => [1,3,6,5,4,2] => [1,6,3,2,4,5] => [5,4,2,3,6,1] => ? = 0
[[.,.],[[.,.],[[.,.],.]]]
 => [1,3,5,6,4,2] => [1,6,3,2,5,4] => [4,5,2,3,6,1] => ? = 0
Description
The Castelnuovo-Mumford regularity of a permutation. The ''Castelnuovo-Mumford regularity'' of a permutation $\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety'' $X_\sigma$. Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for $\sigma$. It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].
Matching statistic: St001719
(load all 6 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001719: Lattices ⟶ ℤResult quality: 9% ●values known / values provided: 9%●distinct values known / distinct values provided: 14%
Values
[.,.]
 => [1] => [1] => ([(0,1)],2)
 => 1 = 0 + 1
[.,[.,.]]
 => [2,1] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[.,.],.]
 => [1,2] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[.,[.,[.,.]]]
 => [3,2,1] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
[.,[[.,.],.]]
 => [2,3,1] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => 1 = 0 + 1
[[.,.],[.,.]]
 => [3,1,2] => [3,1,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
[[.,[.,.]],.]
 => [2,1,3] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
[[[.,.],.],.]
 => [1,2,3] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1 = 0 + 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [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)
 => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [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)
 => ? = 1 + 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [4,2,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1 = 0 + 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [4,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 1 = 0 + 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [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 = 0 + 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [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 = 0 + 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [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)
 => 1 = 0 + 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
 => 1 = 0 + 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => 1 = 0 + 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [4,3,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 0 + 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,3,1,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
 => ? = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
 => ? = 0 + 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ? = 0 + 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [4,1,2,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [3,4,2,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
 => ? = 0 + 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [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)
 => ? = 0 + 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [3,2,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ? = 0 + 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [3,2,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,9),(3,11),(4,9),(4,10),(5,8),(5,11),(7,8),(8,6),(9,7),(10,7),(11,6)],12)
 => ? = 1 + 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [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)
 => ? = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [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)
 => ? = 1 + 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [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)
 => ? = 1 + 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [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
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [5,3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 0 + 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [5,3,1,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ? = 0 + 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [5,2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
 => ? = 0 + 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [5,2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
 => ? = 0 + 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [5,1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 1 + 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [3,5,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [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 = 0 + 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [4,5,2,1,3] => ([(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)
 => ? = 0 + 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [4,5,1,2,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
 => ? = 0 + 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [3,2,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [2,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [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 = 0 + 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [3,4,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
 => ? = 0 + 1
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [3,2,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
 => ? = 0 + 1
[[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [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)
 => ? = 0 + 1
[[.,[[.,.],[.,.]]],.]
 => [4,2,3,1,5] => [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)
 => ? = 0 + 1
[[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [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)
 => ? = 0 + 1
[[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [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)
 => ? = 1 + 1
[[[.,.],[.,[.,.]]],.]
 => [4,3,1,2,5] => [4,2,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
 => [3,4,1,2,5] => [4,1,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[[[.,[.,.]],[.,.]],.]
 => [4,2,1,3,5] => [2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 1 = 0 + 1
[[[[.,.],.],[.,.]],.]
 => [4,1,2,3,5] => [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 = 0 + 1
[[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [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)
 => ? = 0 + 1
[[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [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)
 => ? = 0 + 1
[[[[.,.],[.,.]],.],.]
 => [3,1,2,4,5] => [3,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
 => 1 = 0 + 1
[[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 0 + 1
[[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => [5,4,3,2,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,14),(2,13),(3,7),(4,13),(4,16),(5,14),(5,17),(6,16),(6,17),(8,12),(9,12),(10,8),(11,9),(12,7),(13,10),(14,11),(15,8),(15,9),(16,10),(16,15),(17,11),(17,15)],18)
 => ? = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [5,6,4,3,2,1] => [5,4,3,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,7),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,11),(14,8),(14,9)],15)
 => ? = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [6,4,5,3,2,1] => [5,4,2,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,14),(3,14),(4,11),(5,7),(6,12),(6,13),(8,10),(9,10),(10,7),(11,9),(12,8),(13,8),(13,9),(14,11),(14,13)],15)
 => ? = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [5,4,6,3,2,1] => [5,4,2,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,10),(5,11),(6,7),(6,9),(7,12),(8,11),(9,12),(11,9),(12,10)],13)
 => ? = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [4,5,6,3,2,1] => [5,4,1,2,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,11),(3,13),(4,7),(5,10),(5,13),(6,11),(6,12),(8,10),(9,7),(10,9),(11,8),(12,8),(13,9)],14)
 => ? = 1 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [6,5,3,4,2,1] => [5,3,4,2,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(2,14),(3,14),(4,11),(5,7),(6,12),(6,13),(8,10),(9,10),(10,7),(11,9),(12,8),(13,8),(13,9),(14,11),(14,13)],15)
 => ? = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [5,6,3,4,2,1] => [5,3,4,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,11),(9,12),(11,12),(12,10)],13)
 => ? = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [6,4,3,5,2,1] => [5,3,2,4,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,8),(5,10),(6,11),(7,11),(8,12),(9,12),(11,8),(11,9),(12,10)],13)
 => ? = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
 => [6,3,4,5,2,1] => [5,2,3,4,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,12),(4,11),(5,7),(6,11),(6,12),(8,7),(9,8),(10,8),(11,13),(12,13),(13,9),(13,10)],14)
 => ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [5,4,3,6,2,1] => [5,3,2,1,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,7),(3,7),(3,8),(4,10),(5,11),(6,9),(7,12),(8,12),(10,9),(11,10),(12,11)],13)
 => ? = 1 + 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [4,5,3,6,2,1] => [5,3,1,2,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
 => [5,3,4,6,2,1] => [5,2,3,1,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,9),(4,10),(5,11),(6,8),(7,11),(9,10),(10,8),(11,9)],12)
 => ? = 1 + 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [4,3,5,6,2,1] => [5,2,1,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,9),(5,11),(6,7),(6,10),(7,12),(8,10),(10,12),(11,9),(12,11)],13)
 => ? = 1 + 1
[.,[.,[[[[.,.],.],.],.]]]
 => [3,4,5,6,2,1] => [5,1,2,3,4,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,11),(2,13),(3,12),(4,7),(5,12),(5,14),(6,13),(6,14),(8,11),(9,8),(10,8),(11,7),(12,9),(13,10),(14,9),(14,10)],15)
 => ? = 2 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
 => [6,5,4,2,3,1] => [4,5,3,2,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(2,13),(3,12),(4,7),(5,11),(5,14),(6,12),(6,14),(8,10),(9,10),(10,7),(11,8),(12,9),(13,11),(14,8),(14,9)],15)
 => ? = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
 => [5,6,4,2,3,1] => [4,5,3,1,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,7),(4,7),(5,9),(6,10),(6,11),(7,11),(8,10),(10,12),(11,12),(12,9)],13)
 => ? = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [6,4,5,2,3,1] => [4,5,2,3,1,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(8,11),(9,12),(11,12),(12,10)],13)
 => ? = 0 + 1
[.,[[.,.],[[.,[.,.]],.]]]
 => [5,4,6,2,3,1] => [4,5,2,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(2,10),(3,7),(4,7),(5,8),(6,8),(7,11),(8,9),(9,11),(11,10)],12)
 => ? = 0 + 1
[[[.,.],[.,.]],[.,[.,.]]]
 => [6,5,3,1,2,4] => [5,3,6,2,1,4] => ([(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 = 0 + 1
[[[.,.],[.,.]],[[.,.],.]]
 => [5,6,3,1,2,4] => [5,3,6,1,2,4] => ([(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 = 0 + 1
[[[.,.],[.,[.,.]]],[.,.]]
 => [6,4,3,1,2,5] => [5,3,2,6,1,4] => ([(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 = 0 + 1
[[[.,.],[[.,.],.]],[.,.]]
 => [6,3,4,1,2,5] => [5,2,3,6,1,4] => ([(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 = 0 + 1
[[[.,[.,.]],[.,.]],[.,.]]
 => [6,4,2,1,3,5] => [3,5,2,6,1,4] => ([(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 = 0 + 1
[[[[.,.],.],[.,.]],[.,.]]
 => [6,4,1,2,3,5] => [4,5,2,6,1,3] => ([(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 = 0 + 1
[[[[.,.],[.,.]],.],[.,.]]
 => [6,3,1,2,4,5] => [4,2,5,6,1,3] => ([(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 = 0 + 1
[[[.,[.,.]],[.,[.,.]]],.]
 => [5,4,2,1,3,6] => [3,5,2,1,6,4] => ([(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 = 0 + 1
[[[.,[.,.]],[[.,.],.]],.]
 => [4,5,2,1,3,6] => [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 = 0 + 1
[[[.,[.,[.,.]]],[.,.]],.]
 => [5,3,2,1,4,6] => [3,2,5,1,6,4] => ([(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 = 0 + 1
[[[.,[[.,.],.]],[.,.]],.]
 => [5,2,3,1,4,6] => [2,3,5,1,6,4] => ([(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 = 0 + 1
[[[[.,.],[.,.]],[.,.]],.]
 => [5,3,1,2,4,6] => [4,2,5,1,6,3] => ([(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 = 0 + 1
[[[[.,[.,.]],.],[.,.]],.]
 => [5,2,1,3,4,6] => [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 = 0 + 1
[[[[.,[.,.]],[.,.]],.],.]
 => [4,2,1,3,5,6] => [2,4,1,5,6,3] => ([(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 = 0 + 1
Description
The number of shortest chains of small intervals from the bottom to the top in a lattice. An interval $[a, b]$ in a lattice is small if $b$ is a join of elements covering $a$.
Matching statistic: St001882
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001882: Signed permutations ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 29%
Values
[.,.]
 => [1] => [1] => [1] => 0
[.,[.,.]]
 => [2,1] => [2,1] => [2,1] => 0
[[.,.],.]
 => [1,2] => [1,2] => [1,2] => 0
[.,[.,[.,.]]]
 => [3,2,1] => [3,1,2] => [3,1,2] => 0
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => [3,2,1] => 0
[[.,.],[.,.]]
 => [1,3,2] => [1,3,2] => [1,3,2] => 0
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,1,2,3] => [4,1,2,3] => 0
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,3,2] => [4,1,3,2] => 0
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [4,2,1,3] => [4,2,1,3] => 0
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,2,1] => [4,3,2,1] => 0
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [4,2,3,1] => [4,2,3,1] => 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [1,4,2,3] => [1,4,2,3] => 0
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 0
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 0
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 0
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [3,1,2,4] => [3,1,2,4] => 0
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 0
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,1,2,3,4] => [5,1,2,3,4] => ? = 0
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 0
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [5,1,3,2,4] => [5,1,3,2,4] => ? = 0
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 0
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [5,1,3,4,2] => [5,1,3,4,2] => ? = 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [5,2,1,3,4] => [5,2,1,3,4] => ? = 0
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [5,2,1,4,3] => [5,2,1,4,3] => ? = 0
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [5,3,2,1,4] => [5,3,2,1,4] => ? = 0
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [5,2,3,1,4] => [5,2,3,1,4] => ? = 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,4,2,3,1] => [5,4,2,3,1] => ? = 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 0
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [5,2,4,3,1] => [5,2,4,3,1] => ? = 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [5,3,2,4,1] => [5,3,2,4,1] => ? = 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [5,2,3,4,1] => [5,2,3,4,1] => ? = 2
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [1,5,2,3,4] => [1,5,2,3,4] => 0
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [1,5,2,4,3] => [1,5,2,4,3] => 0
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [1,5,3,2,4] => [1,5,3,2,4] => 0
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 0
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [1,5,3,4,2] => [1,5,3,4,2] => 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [2,1,5,3,4] => [2,1,5,3,4] => ? = 0
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => ? = 0
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [1,2,5,3,4] => [1,2,5,3,4] => 0
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 0
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [3,1,2,5,4] => [3,1,2,5,4] => ? = 0
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => ? = 0
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 0
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => ? = 0
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 0
[[.,[.,[.,[.,.]]]],.]
 => [4,3,2,1,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 0
[[.,[.,[[.,.],.]]],.]
 => [3,4,2,1,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 0
[[.,[[.,.],[.,.]]],.]
 => [2,4,3,1,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 0
[[.,[[.,[.,.]],.]],.]
 => [3,2,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => ? = 0
[[.,[[[.,.],.],.]],.]
 => [2,3,4,1,5] => [4,2,3,1,5] => [4,2,3,1,5] => ? = 1
[[[.,.],[.,[.,.]]],.]
 => [1,4,3,2,5] => [1,4,2,3,5] => [1,4,2,3,5] => 0
[[[.,.],[[.,.],.]],.]
 => [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 0
[[[.,[.,.]],[.,.]],.]
 => [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => ? = 0
[[[[.,.],.],[.,.]],.]
 => [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 0
[[[.,[.,[.,.]]],.],.]
 => [3,2,1,4,5] => [3,1,2,4,5] => [3,1,2,4,5] => ? = 0
[[[.,[[.,.],.]],.],.]
 => [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => ? = 0
[[[[.,.],[.,.]],.],.]
 => [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 0
[[[[.,[.,.]],.],.],.]
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => ? = 0
[[[[[.,.],.],.],.],.]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [6,5,4,3,2,1] => [6,1,2,3,4,5] => [6,1,2,3,4,5] => ? = 0
[.,[.,[.,[.,[[.,.],.]]]]]
 => [5,6,4,3,2,1] => [6,1,2,3,5,4] => [6,1,2,3,5,4] => ? = 0
[.,[.,[.,[[.,.],[.,.]]]]]
 => [4,6,5,3,2,1] => [6,1,2,4,3,5] => [6,1,2,4,3,5] => ? = 0
[.,[.,[.,[[.,[.,.]],.]]]]
 => [5,4,6,3,2,1] => [6,1,2,5,4,3] => [6,1,2,5,4,3] => ? = 0
[.,[.,[.,[[[.,.],.],.]]]]
 => [4,5,6,3,2,1] => [6,1,2,4,5,3] => [6,1,2,4,5,3] => ? = 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [3,6,5,4,2,1] => [6,1,3,2,4,5] => [6,1,3,2,4,5] => ? = 0
[.,[.,[[.,.],[[.,.],.]]]]
 => [3,5,6,4,2,1] => [6,1,3,2,5,4] => [6,1,3,2,5,4] => ? = 0
[.,[.,[[.,[.,.]],[.,.]]]]
 => [4,3,6,5,2,1] => [6,1,4,3,2,5] => [6,1,4,3,2,5] => ? = 0
[.,[.,[[[.,.],.],[.,.]]]]
 => [3,4,6,5,2,1] => [6,1,3,4,2,5] => [6,1,3,4,2,5] => ? = 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [5,4,3,6,2,1] => [6,1,5,3,4,2] => [6,1,5,3,4,2] => ? = 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [4,5,3,6,2,1] => [6,1,5,4,3,2] => [6,1,5,4,3,2] => ? = 0
[.,[.,[[[.,.],[.,.]],.]]]
 => [3,5,4,6,2,1] => [6,1,3,5,4,2] => [6,1,3,5,4,2] => ? = 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [4,3,5,6,2,1] => [6,1,4,3,5,2] => [6,1,4,3,5,2] => ? = 1
[.,[.,[[[[.,.],.],.],.]]]
 => [3,4,5,6,2,1] => [6,1,3,4,5,2] => [6,1,3,4,5,2] => ? = 2
[.,[[.,.],[.,[.,[.,.]]]]]
 => [2,6,5,4,3,1] => [6,2,1,3,4,5] => [6,2,1,3,4,5] => ? = 0
[.,[[.,.],[.,[[.,.],.]]]]
 => [2,5,6,4,3,1] => [6,2,1,3,5,4] => [6,2,1,3,5,4] => ? = 0
[.,[[.,.],[[.,.],[.,.]]]]
 => [2,4,6,5,3,1] => [6,2,1,4,3,5] => [6,2,1,4,3,5] => ? = 0
[.,[[.,.],[[.,[.,.]],.]]]
 => [2,5,4,6,3,1] => [6,2,1,5,4,3] => [6,2,1,5,4,3] => ? = 0
[.,[[.,.],[[[.,.],.],.]]]
 => [2,4,5,6,3,1] => [6,2,1,4,5,3] => [6,2,1,4,5,3] => ? = 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => [3,2,6,5,4,1] => [6,3,2,1,4,5] => [6,3,2,1,4,5] => ? = 0
[.,[[.,[.,.]],[[.,.],.]]]
 => [3,2,5,6,4,1] => [6,3,2,1,5,4] => [6,3,2,1,5,4] => ? = 0
[.,[[[.,.],.],[.,[.,.]]]]
 => [2,3,6,5,4,1] => [6,2,3,1,4,5] => [6,2,3,1,4,5] => ? = 1
Description
The number of occurrences of a type-B 231 pattern in a signed permutation. For a signed permutation $\pi\in\mathfrak H_n$, a triple $-n \leq i < j < k\leq n$ is an occurrence of the type-B $231$ pattern, if $1 \leq j < k$, $\pi(i) < \pi(j)$ and $\pi(i)$ is one larger than $\pi(k)$, i.e., $\pi(i) = \pi(k) + 1$ if $\pi(k) \neq -1$ and $\pi(i) = 1$ otherwise.
Matching statistic: St000181
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 14%
Values
[.,.]
 => [1,0]
 => [1,0]
 => ([],1)
 => 1 = 0 + 1
[.,[.,.]]
 => [1,1,0,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[.,.],.]
 => [1,0,1,0]
 => [1,1,0,0]
 => ([(0,1)],2)
 => 1 = 0 + 1
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 0 + 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 + 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 0 + 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 0 + 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 0 + 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 0 + 1
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1 + 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ? = 0 + 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 0 + 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ? = 0 + 1
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ? = 1 + 1
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ? = 1 + 1
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 1 + 1
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2 + 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 0 + 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ? = 0 + 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ? = 0 + 1
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 1 + 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ? = 0 + 1
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ? = 0 + 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ? = 0 + 1
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 0 + 1
[[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ? = 0 + 1
[[.,[[.,[.,.]],.]],.]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ? = 0 + 1
[[.,[[[.,.],.],.]],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 1 + 1
[[[.,.],[.,[.,.]]],.]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ? = 0 + 1
[[[.,[.,.]],[.,.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ? = 0 + 1
[[[[.,.],[.,.]],.],.]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[[.,[.,.]],.],.],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[[[.,.],.],.],.],.]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ? = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 1 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 1 + 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 1 + 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 1 + 1
[.,[.,[[[[.,.],.],.],.]]]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 2 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ? = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? = 0 + 1
[.,[[.,.],[[.,[.,.]],.]]]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? = 0 + 1
[.,[[.,.],[[[.,.],.],.]]]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 1 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ? = 0 + 1
[.,[[.,[.,.]],[[.,.],.]]]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? = 0 + 1
[.,[[[.,.],.],[.,[.,.]]]]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0,1,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 1 + 1
Description
The number of connected components of the Hasse diagram for the poset.
Matching statistic: St001490
(load all 4 compositions to match this statistic)
Mapping
Mp00019: Binary trees —right rotate⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 14%
Values
[.,.]
 => [.,.]
 => [1,0]
 => [[1],[]]
 => 1 = 0 + 1
[.,[.,.]]
 => [[.,.],.]
 => [1,1,0,0]
 => [[2],[]]
 => 1 = 0 + 1
[[.,.],.]
 => [.,[.,.]]
 => [1,0,1,0]
 => [[1,1],[]]
 => 1 = 0 + 1
[.,[.,[.,.]]]
 => [[.,[.,.]],.]
 => [1,1,0,1,0,0]
 => [[3],[]]
 => 1 = 0 + 1
[.,[[.,.],.]]
 => [[[.,.],.],.]
 => [1,1,1,0,0,0]
 => [[2,2],[]]
 => 1 = 0 + 1
[[.,.],[.,.]]
 => [.,[.,[.,.]]]
 => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => 1 = 0 + 1
[[.,[.,.]],.]
 => [.,[[.,.],.]]
 => [1,0,1,1,0,0]
 => [[2,1],[]]
 => 1 = 0 + 1
[[[.,.],.],.]
 => [[.,.],[.,.]]
 => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => [[.,[.,[.,.]]],.]
 => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => [[.,[[.,.],.]],.]
 => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => 1 = 0 + 1
[.,[[.,.],[.,.]]]
 => [[[.,.],[.,.]],.]
 => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => 1 = 0 + 1
[.,[[.,[.,.]],.]]
 => [[[.,[.,.]],.],.]
 => [1,1,1,0,1,0,0,0]
 => [[2,2,2],[]]
 => ? = 0 + 1
[.,[[[.,.],.],.]]
 => [[[[.,.],.],.],.]
 => [1,1,1,1,0,0,0,0]
 => [[3,3],[]]
 => ? = 1 + 1
[[.,.],[.,[.,.]]]
 => [.,[.,[.,[.,.]]]]
 => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => 1 = 0 + 1
[[.,.],[[.,.],.]]
 => [.,[.,[[.,.],.]]]
 => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => 1 = 0 + 1
[[.,[.,.]],[.,.]]
 => [.,[[.,.],[.,.]]]
 => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => 1 = 0 + 1
[[[.,.],.],[.,.]]
 => [[.,.],[.,[.,.]]]
 => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => 1 = 0 + 1
[[.,[.,[.,.]]],.]
 => [.,[[.,[.,.]],.]]
 => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => 1 = 0 + 1
[[.,[[.,.],.]],.]
 => [.,[[[.,.],.],.]]
 => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => 1 = 0 + 1
[[[.,.],[.,.]],.]
 => [[.,.],[[.,.],.]]
 => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => 1 = 0 + 1
[[[.,[.,.]],.],.]
 => [[.,[.,.]],[.,.]]
 => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => 1 = 0 + 1
[[[[.,.],.],.],.]
 => [[[.,.],.],[.,.]]
 => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [[.,[.,[.,[.,.]]]],.]
 => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
 => [[.,[.,[[.,.],.]]],.]
 => [1,1,0,1,0,1,1,0,0,0]
 => [[4,4],[2]]
 => ? = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => [[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,1,0,0]
 => [[4,3],[1]]
 => ? = 0 + 1
[.,[.,[[.,[.,.]],.]]]
 => [[.,[[.,[.,.]],.]],.]
 => [1,1,0,1,1,0,1,0,0,0]
 => [[3,3,3],[1,1]]
 => ? = 0 + 1
[.,[.,[[[.,.],.],.]]]
 => [[.,[[[.,.],.],.]],.]
 => [1,1,0,1,1,1,0,0,0,0]
 => [[4,4],[1]]
 => ? = 1 + 1
[.,[[.,.],[.,[.,.]]]]
 => [[[.,.],[.,[.,.]]],.]
 => [1,1,1,0,0,1,0,1,0,0]
 => [[4,2],[]]
 => ? = 0 + 1
[.,[[.,.],[[.,.],.]]]
 => [[[.,.],[[.,.],.]],.]
 => [1,1,1,0,0,1,1,0,0,0]
 => [[3,3,2],[1]]
 => ? = 0 + 1
[.,[[.,[.,.]],[.,.]]]
 => [[[.,[.,.]],[.,.]],.]
 => [1,1,1,0,1,0,0,1,0,0]
 => [[3,2,2],[]]
 => ? = 0 + 1
[.,[[[.,.],.],[.,.]]]
 => [[[[.,.],.],[.,.]],.]
 => [1,1,1,1,0,0,0,1,0,0]
 => [[4,3],[]]
 => ? = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => [[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,1,0,1,0,0,0]
 => [[2,2,2,2],[]]
 => ? = 1 + 1
[.,[[.,[[.,.],.]],.]]
 => [[[.,[[.,.],.]],.],.]
 => [1,1,1,0,1,1,0,0,0,0]
 => [[3,3,2],[]]
 => ? = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => [[[[.,.],[.,.]],.],.]
 => [1,1,1,1,0,0,1,0,0,0]
 => [[3,3,3],[1]]
 => ? = 1 + 1
[.,[[[.,[.,.]],.],.]]
 => [[[[.,[.,.]],.],.],.]
 => [1,1,1,1,0,1,0,0,0,0]
 => [[4,4],[]]
 => ? = 1 + 1
[.,[[[[.,.],.],.],.]]
 => [[[[[.,.],.],.],.],.]
 => [1,1,1,1,1,0,0,0,0,0]
 => [[3,3,3],[]]
 => ? = 2 + 1
[[.,.],[.,[.,[.,.]]]]
 => [.,[.,[.,[.,[.,.]]]]]
 => [1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
 => [.,[.,[.,[[.,.],.]]]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => 1 = 0 + 1
[[.,.],[[.,.],[.,.]]]
 => [.,[.,[[.,.],[.,.]]]]
 => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => 1 = 0 + 1
[[.,.],[[.,[.,.]],.]]
 => [.,[.,[[.,[.,.]],.]]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => 1 = 0 + 1
[[.,.],[[[.,.],.],.]]
 => [.,[.,[[[.,.],.],.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [[2,2,1,1],[]]
 => ? = 1 + 1
[[.,[.,.]],[.,[.,.]]]
 => [.,[[.,.],[.,[.,.]]]]
 => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
 => [.,[[.,.],[[.,.],.]]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => 1 = 0 + 1
[[[.,.],.],[.,[.,.]]]
 => [[.,.],[.,[.,[.,.]]]]
 => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
 => [[.,.],[.,[[.,.],.]]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => 1 = 0 + 1
[[.,[.,[.,.]]],[.,.]]
 => [.,[[.,[.,.]],[.,.]]]
 => [1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
 => [.,[[[.,.],.],[.,.]]]
 => [1,0,1,1,1,0,0,0,1,0]
 => [[2,2,2,1],[1]]
 => ? = 0 + 1
[[[.,.],[.,.]],[.,.]]
 => [[.,.],[[.,.],[.,.]]]
 => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
 => [[.,[.,.]],[.,[.,.]]]
 => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
 => [[[.,.],.],[.,[.,.]]]
 => [1,1,1,0,0,0,1,0,1,0]
 => [[2,2,2,2],[1,1]]
 => ? = 0 + 1
[[.,[.,[.,[.,.]]]],.]
 => [.,[[.,[.,[.,.]]],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
 => [.,[[.,[[.,.],.]],.]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [[3,3,1],[1]]
 => ? = 0 + 1
[[.,[[.,.],[.,.]]],.]
 => [.,[[[.,.],[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [[3,2,1],[]]
 => ? = 0 + 1
[[.,[[.,[.,.]],.]],.]
 => [.,[[[.,[.,.]],.],.]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [[2,2,2,1],[]]
 => ? = 0 + 1
[[.,[[[.,.],.],.]],.]
 => [.,[[[[.,.],.],.],.]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [[3,3,1],[]]
 => ? = 1 + 1
[[[.,.],[.,[.,.]]],.]
 => [[.,.],[[.,[.,.]],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
 => [[.,.],[[[.,.],.],.]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [[3,3,2],[1,1]]
 => ? = 0 + 1
[[[.,[.,.]],[.,.]],.]
 => [[.,[.,.]],[[.,.],.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => 1 = 0 + 1
[[[[.,.],.],[.,.]],.]
 => [[[.,.],.],[[.,.],.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [[3,2,2],[1]]
 => ? = 0 + 1
[[[.,[.,[.,.]]],.],.]
 => [[.,[.,[.,.]]],[.,.]]
 => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
 => [[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,1,0,0,0,1,0]
 => [[3,3,3],[2,1]]
 => ? = 0 + 1
[[[[.,.],[.,.]],.],.]
 => [[[.,.],[.,.]],[.,.]]
 => [1,1,1,0,0,1,0,0,1,0]
 => [[3,3,2],[2]]
 => ? = 0 + 1
[[[[.,[.,.]],.],.],.]
 => [[[.,[.,.]],.],[.,.]]
 => [1,1,1,0,1,0,0,0,1,0]
 => [[2,2,2,2],[1]]
 => ? = 0 + 1
[[[[[.,.],.],.],.],.]
 => [[[[.,.],.],.],[.,.]]
 => [1,1,1,1,0,0,0,0,1,0]
 => [[3,3,3],[2]]
 => ? = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [[.,[.,[.,[.,[.,.]]]]],.]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [[6],[]]
 => ? = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [[.,[.,[.,[[.,.],.]]]],.]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [[5,5],[3]]
 => ? = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [[.,[.,[[.,.],[.,.]]]],.]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [[5,4],[2]]
 => ? = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [[.,[.,[[.,[.,.]],.]]],.]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [[4,4,4],[2,2]]
 => ? = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [[.,[.,[[[.,.],.],.]]],.]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [[5,5],[2]]
 => ? = 1 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [[.,[[.,.],[.,[.,.]]]],.]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [[5,3],[1]]
 => ? = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [[.,[[.,.],[[.,.],.]]],.]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [[4,4,3],[2,1]]
 => ? = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [[.,[[.,[.,.]],[.,.]]],.]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [[4,3,3],[1,1]]
 => ? = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
 => [[.,[[[.,.],.],[.,.]]],.]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [[5,4],[1]]
 => ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [[.,[[.,[.,[.,.]]],.]],.]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [[3,3,3,3],[1,1,1]]
 => ? = 1 + 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [[.,[[.,[[.,.],.]],.]],.]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [[4,4,3],[1,1]]
 => ? = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
 => [[.,[[[.,.],[.,.]],.]],.]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [[4,4,4],[2,1]]
 => ? = 1 + 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [[.,[[[.,[.,.]],.],.]],.]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [[5,5],[1]]
 => ? = 1 + 1
[.,[.,[[[[.,.],.],.],.]]]
 => [[.,[[[[.,.],.],.],.]],.]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [[4,4,4],[1,1]]
 => ? = 2 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
 => [[[.,.],[.,[.,[.,.]]]],.]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [[5,2],[]]
 => ? = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
 => [[[.,.],[.,[[.,.],.]]],.]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [[4,4,2],[2]]
 => ? = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [[[.,.],[[.,.],[.,.]]],.]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [[4,3,2],[1]]
 => ? = 0 + 1
[.,[[.,.],[[.,[.,.]],.]]]
 => [[[.,.],[[.,[.,.]],.]],.]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [[3,3,3,2],[1,1]]
 => ? = 0 + 1
[.,[[.,.],[[[.,.],.],.]]]
 => [[[.,.],[[[.,.],.],.]],.]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [[4,4,2],[1]]
 => ? = 1 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => [[[.,[.,.]],[.,[.,.]]],.]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [[4,2,2],[]]
 => ? = 0 + 1
[.,[[.,[.,.]],[[.,.],.]]]
 => [[[.,[.,.]],[[.,.],.]],.]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [[3,3,2,2],[1]]
 => ? = 0 + 1
[.,[[[.,.],.],[.,[.,.]]]]
 => [[[[.,.],.],[.,[.,.]]],.]
 => [1,1,1,1,0,0,0,1,0,1,0,0]
 => [[5,3],[]]
 => ? = 1 + 1
Description
The number of connected components of a skew partition.
Matching statistic: St001890
Mapping
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St001890: Posets ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 14%
Values
[.,.]
 => [1,0]
 => [1,0]
 => ([],1)
 => ? = 0 + 1
[.,[.,.]]
 => [1,1,0,0]
 => [1,0,1,0]
 => ([(0,1)],2)
 => 1 = 0 + 1
[[.,.],.]
 => [1,0,1,0]
 => [1,1,0,0]
 => ([(0,1)],2)
 => 1 = 0 + 1
[.,[.,[.,.]]]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[.,[[.,.],.]]
 => [1,1,0,1,0,0]
 => [1,1,1,0,0,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 1 = 0 + 1
[[.,.],[.,.]]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[.,[.,.]],.]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[[[.,.],.],.]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,0]
 => ([(0,2),(2,1)],3)
 => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[.,[.,[[.,.],.]]]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 0 + 1
[.,[[.,.],[.,.]]]
 => [1,1,0,1,1,0,0,0]
 => [1,1,1,0,0,0,1,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 0 + 1
[.,[[.,[.,.]],.]]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 0 + 1
[.,[[[.,.],.],.]]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,1,0,0,0,0]
 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ? = 1 + 1
[[.,.],[.,[.,.]]]
 => [1,0,1,1,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[.,.],[[.,.],.]]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
 => 1 = 0 + 1
[[.,[.,.]],[.,.]]
 => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[.,.],.],[.,.]]
 => [1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[.,[.,[.,.]]],.]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[.,[[.,.],.]],.]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
 => 1 = 0 + 1
[[[.,.],[.,.]],.]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[.,[.,.]],.],.]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[[[[.,.],.],.],.]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 0 + 1
[.,[.,[[.,.],[.,.]]]]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 0 + 1
[.,[.,[[.,[.,.]],.]]]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 0 + 1
[.,[.,[[[.,.],.],.]]]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 1 + 1
[.,[[.,.],[.,[.,.]]]]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ? = 0 + 1
[.,[[.,.],[[.,.],.]]]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 0 + 1
[.,[[.,[.,.]],[.,.]]]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ? = 0 + 1
[.,[[[.,.],.],[.,.]]]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 1 + 1
[.,[[.,[.,[.,.]]],.]]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ? = 1 + 1
[.,[[.,[[.,.],.]],.]]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => ([(0,3),(0,4),(1,6),(2,5),(3,7),(4,2),(4,7),(5,6),(7,1),(7,5)],8)
 => ? = 0 + 1
[.,[[[.,.],[.,.]],.]]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
 => ? = 1 + 1
[.,[[[.,[.,.]],.],.]]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 1 + 1
[.,[[[[.,.],.],.],.]]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ? = 2 + 1
[[.,.],[.,[.,[.,.]]]]
 => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[.,.],[.,[[.,.],.]]]
 => [1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 0 + 1
[[.,.],[[.,.],[.,.]]]
 => [1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,0,1,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ? = 0 + 1
[[.,.],[[.,[.,.]],.]]
 => [1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ? = 0 + 1
[[.,.],[[[.,.],.],.]]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
 => ? = 1 + 1
[[.,[.,.]],[.,[.,.]]]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[.,[.,.]],[[.,.],.]]
 => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ? = 0 + 1
[[[.,.],.],[.,[.,.]]]
 => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[.,.],.],[[.,.],.]]
 => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
 => ? = 0 + 1
[[.,[.,[.,.]]],[.,.]]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[.,[[.,.],.]],[.,.]]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ? = 0 + 1
[[[.,.],[.,.]],[.,.]]
 => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,0,1,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[.,[.,.]],.],[.,.]]
 => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[[.,.],.],.],[.,.]]
 => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[.,[.,[.,[.,.]]]],.]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 0 + 1
[[.,[[.,.],[.,.]]],.]
 => [1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ? = 0 + 1
[[.,[[.,[.,.]],.]],.]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ? = 0 + 1
[[.,[[[.,.],.],.]],.]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
 => ? = 1 + 1
[[[.,.],[.,[.,.]]],.]
 => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[.,.],[[.,.],.]],.]
 => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
 => ? = 0 + 1
[[[.,[.,.]],[.,.]],.]
 => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[[.,.],.],[.,.]],.]
 => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,0,1,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[.,[.,[.,.]]],.],.]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[.,[[.,.],.]],.],.]
 => [1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
 => ? = 0 + 1
[[[[.,.],[.,.]],.],.]
 => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[[.,[.,.]],.],.],.]
 => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[[[[[.,.],.],.],.],.]
 => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => 1 = 0 + 1
[.,[.,[.,[.,[.,[.,.]]]]]]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0]
 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ? = 0 + 1
[.,[.,[.,[.,[[.,.],.]]]]]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
 => ? = 0 + 1
[.,[.,[.,[[.,.],[.,.]]]]]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
 => ? = 0 + 1
[.,[.,[.,[[.,[.,.]],.]]]]
 => [1,1,1,1,1,0,0,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
 => ? = 0 + 1
[.,[.,[.,[[[.,.],.],.]]]]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 1 + 1
[.,[.,[[.,.],[.,[.,.]]]]]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
 => ? = 0 + 1
[.,[.,[[.,.],[[.,.],.]]]]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,10),(2,7),(3,8),(4,3),(4,6),(5,1),(5,6),(6,8),(6,10),(8,9),(9,7),(10,2),(10,9)],11)
 => ? = 0 + 1
[.,[.,[[.,[.,.]],[.,.]]]]
 => [1,1,1,1,0,0,1,1,0,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
 => ? = 0 + 1
[.,[.,[[[.,.],.],[.,.]]]]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
 => ? = 1 + 1
[.,[.,[[.,[.,[.,.]]],.]]]
 => [1,1,1,1,1,0,0,0,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
 => ? = 1 + 1
[.,[.,[[.,[[.,.],.]],.]]]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 0 + 1
[.,[.,[[[.,.],[.,.]],.]]]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => ([(0,4),(0,5),(1,7),(2,9),(3,6),(4,8),(5,2),(5,8),(6,7),(8,3),(8,9),(9,1),(9,6)],10)
 => ? = 1 + 1
[.,[.,[[[.,[.,.]],.],.]]]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => ([(0,5),(1,8),(2,7),(3,2),(3,6),(4,1),(4,6),(5,3),(5,4),(6,7),(6,8),(7,9),(8,9)],10)
 => ? = 1 + 1
[.,[.,[[[[.,.],.],.],.]]]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
 => ? = 2 + 1
[.,[[.,.],[.,[.,[.,.]]]]]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0]
 => ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
 => ? = 0 + 1
[.,[[.,.],[.,[[.,.],.]]]]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,1,0,0,0]
 => ([(0,3),(0,5),(1,8),(2,7),(3,6),(4,2),(4,9),(5,1),(5,6),(6,4),(6,8),(8,9),(9,7)],10)
 => ? = 0 + 1
[.,[[.,.],[[.,.],[.,.]]]]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => ([(0,4),(0,5),(1,6),(3,7),(4,8),(5,1),(5,8),(6,7),(7,2),(8,3),(8,6)],9)
 => ? = 0 + 1
[.,[[.,.],[[.,[.,.]],.]]]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
 => ? = 0 + 1
[.,[[.,.],[[[.,.],.],.]]]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
 => ? = 1 + 1
[.,[[.,[.,.]],[.,[.,.]]]]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,1,0,0,0,1,0,1,0]
 => ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
 => ? = 0 + 1
[.,[[.,[.,.]],[[.,.],.]]]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => ([(0,5),(1,7),(2,8),(3,6),(4,3),(4,8),(5,2),(5,4),(6,7),(8,1),(8,6)],9)
 => ? = 0 + 1
Description
The maximum magnitude of the Möbius function of a poset. The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices.