Your data matches 101 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000701
(load all 16 compositions to match this statistic)
Mapping
St000701: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => 1
[.,[.,.]]
 => 1
[[.,.],.]
 => 1
[.,[.,[.,.]]]
 => 1
[.,[[.,.],.]]
 => 1
[[.,.],[.,.]]
 => 2
[[.,[.,.]],.]
 => 1
[[[.,.],.],.]
 => 1
[.,[.,[.,[.,.]]]]
 => 1
[.,[.,[[.,.],.]]]
 => 1
[.,[[.,.],[.,.]]]
 => 1
[.,[[.,[.,.]],.]]
 => 1
[.,[[[.,.],.],.]]
 => 1
[[.,.],[.,[.,.]]]
 => 2
[[.,.],[[.,.],.]]
 => 2
[[.,[.,.]],[.,.]]
 => 2
[[[.,.],.],[.,.]]
 => 2
[[.,[.,[.,.]]],.]
 => 1
[[.,[[.,.],.]],.]
 => 1
[[[.,.],[.,.]],.]
 => 1
[[[.,[.,.]],.],.]
 => 1
[[[[.,.],.],.],.]
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => 1
[.,[.,[[[.,.],.],.]]]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => 1
[.,[[.,.],[[.,.],.]]]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => 1
[.,[[[.,.],.],[.,.]]]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => 1
[.,[[.,[[.,.],.]],.]]
 => 1
[.,[[[.,.],[.,.]],.]]
 => 1
[.,[[[.,[.,.]],.],.]]
 => 1
[.,[[[[.,.],.],.],.]]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => 2
[[.,.],[.,[[.,.],.]]]
 => 2
[[.,.],[[.,.],[.,.]]]
 => 2
[[.,.],[[.,[.,.]],.]]
 => 2
[[.,.],[[[.,.],.],.]]
 => 2
[[.,[.,.]],[.,[.,.]]]
 => 2
[[.,[.,.]],[[.,.],.]]
 => 2
[[[.,.],.],[.,[.,.]]]
 => 2
[[[.,.],.],[[.,.],.]]
 => 2
[[.,[.,[.,.]]],[.,.]]
 => 2
[[.,[[.,.],.]],[.,.]]
 => 2
[[[.,.],[.,.]],[.,.]]
 => 2
[[[.,[.,.]],.],[.,.]]
 => 2
[[[[.,.],.],.],[.,.]]
 => 2
Description
The protection number of a binary tree. This is the minimal distance from the root to a leaf.
Matching statistic: St000700
Mapping
Mp00008: Binary trees —to complete tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [[],[]]
 => 1
[.,[.,.]]
 => [[],[[],[]]]
 => 1
[[.,.],.]
 => [[[],[]],[]]
 => 1
[.,[.,[.,.]]]
 => [[],[[],[[],[]]]]
 => 1
[.,[[.,.],.]]
 => [[],[[[],[]],[]]]
 => 1
[[.,.],[.,.]]
 => [[[],[]],[[],[]]]
 => 2
[[.,[.,.]],.]
 => [[[],[[],[]]],[]]
 => 1
[[[.,.],.],.]
 => [[[[],[]],[]],[]]
 => 1
[.,[.,[.,[.,.]]]]
 => [[],[[],[[],[[],[]]]]]
 => 1
[.,[.,[[.,.],.]]]
 => [[],[[],[[[],[]],[]]]]
 => 1
[.,[[.,.],[.,.]]]
 => [[],[[[],[]],[[],[]]]]
 => 1
[.,[[.,[.,.]],.]]
 => [[],[[[],[[],[]]],[]]]
 => 1
[.,[[[.,.],.],.]]
 => [[],[[[[],[]],[]],[]]]
 => 1
[[.,.],[.,[.,.]]]
 => [[[],[]],[[],[[],[]]]]
 => 2
[[.,.],[[.,.],.]]
 => [[[],[]],[[[],[]],[]]]
 => 2
[[.,[.,.]],[.,.]]
 => [[[],[[],[]]],[[],[]]]
 => 2
[[[.,.],.],[.,.]]
 => [[[[],[]],[]],[[],[]]]
 => 2
[[.,[.,[.,.]]],.]
 => [[[],[[],[[],[]]]],[]]
 => 1
[[.,[[.,.],.]],.]
 => [[[],[[[],[]],[]]],[]]
 => 1
[[[.,.],[.,.]],.]
 => [[[[],[]],[[],[]]],[]]
 => 1
[[[.,[.,.]],.],.]
 => [[[[],[[],[]]],[]],[]]
 => 1
[[[[.,.],.],.],.]
 => [[[[[],[]],[]],[]],[]]
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => [[],[[],[[],[[],[[],[]]]]]]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [[],[[],[[],[[[],[]],[]]]]]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [[],[[],[[[],[]],[[],[]]]]]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [[],[[],[[[],[[],[]]],[]]]]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [[],[[],[[[[],[]],[]],[]]]]
 => 1
[.,[[.,.],[.,[.,.]]]]
 => [[],[[[],[]],[[],[[],[]]]]]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [[],[[[],[]],[[[],[]],[]]]]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [[],[[[],[[],[]]],[[],[]]]]
 => 1
[.,[[[.,.],.],[.,.]]]
 => [[],[[[[],[]],[]],[[],[]]]]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [[],[[[],[[],[[],[]]]],[]]]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [[],[[[],[[[],[]],[]]],[]]]
 => 1
[.,[[[.,.],[.,.]],.]]
 => [[],[[[[],[]],[[],[]]],[]]]
 => 1
[.,[[[.,[.,.]],.],.]]
 => [[],[[[[],[[],[]]],[]],[]]]
 => 1
[.,[[[[.,.],.],.],.]]
 => [[],[[[[[],[]],[]],[]],[]]]
 => 1
[[.,.],[.,[.,[.,.]]]]
 => [[[],[]],[[],[[],[[],[]]]]]
 => 2
[[.,.],[.,[[.,.],.]]]
 => [[[],[]],[[],[[[],[]],[]]]]
 => 2
[[.,.],[[.,.],[.,.]]]
 => [[[],[]],[[[],[]],[[],[]]]]
 => 2
[[.,.],[[.,[.,.]],.]]
 => [[[],[]],[[[],[[],[]]],[]]]
 => 2
[[.,.],[[[.,.],.],.]]
 => [[[],[]],[[[[],[]],[]],[]]]
 => 2
[[.,[.,.]],[.,[.,.]]]
 => [[[],[[],[]]],[[],[[],[]]]]
 => 2
[[.,[.,.]],[[.,.],.]]
 => [[[],[[],[]]],[[[],[]],[]]]
 => 2
[[[.,.],.],[.,[.,.]]]
 => [[[[],[]],[]],[[],[[],[]]]]
 => 2
[[[.,.],.],[[.,.],.]]
 => [[[[],[]],[]],[[[],[]],[]]]
 => 2
[[.,[.,[.,.]]],[.,.]]
 => [[[],[[],[[],[]]]],[[],[]]]
 => 2
[[.,[[.,.],.]],[.,.]]
 => [[[],[[[],[]],[]]],[[],[]]]
 => 2
[[[.,.],[.,.]],[.,.]]
 => [[[[],[]],[[],[]]],[[],[]]]
 => 2
[[[.,[.,.]],.],[.,.]]
 => [[[[],[[],[]]],[]],[[],[]]]
 => 2
[[[[.,.],.],.],[.,.]]
 => [[[[[],[]],[]],[]],[[],[]]]
 => 2
Description
The protection number of an ordered tree. This is the minimal distance from the root to a leaf.
Matching statistic: St001083
(load all 2 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => 0 = 1 - 1
[.,[.,.]]
 => [2,1] => [2,1] => 0 = 1 - 1
[[.,.],.]
 => [1,2] => [1,2] => 0 = 1 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [2,3,1] => 0 = 1 - 1
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => 0 = 1 - 1
[[.,.],[.,.]]
 => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => 0 = 1 - 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [3,4,2,1] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,3,2,1] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [2,4,3,1] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [2,3,4,1] => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [3,2,4,1] => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [3,4,1,2] => 0 = 1 - 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [4,3,1,2] => 0 = 1 - 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,4,2,3] => 1 = 2 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,3,1,4] => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [1,3,2,4] => 1 = 2 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [4,5,3,2,1] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [3,5,4,2,1] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [3,4,5,2,1] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [4,3,5,2,1] => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [4,5,2,3,1] => 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [5,4,2,3,1] => 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [3,5,2,4,1] => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [2,5,3,4,1] => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [3,4,2,5,1] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [4,3,2,5,1] => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [2,4,3,5,1] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [2,3,4,5,1] => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [3,2,4,5,1] => 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [4,5,3,1,2] => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [5,4,3,1,2] => 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [3,5,4,1,2] => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [3,4,5,1,2] => 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [4,3,5,1,2] => 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [4,5,2,1,3] => 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [5,4,2,1,3] => 0 = 1 - 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [4,5,1,2,3] => 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [5,4,1,2,3] => 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [3,5,2,1,4] => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [2,5,3,1,4] => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [3,5,1,2,4] => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [2,5,1,3,4] => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,5,2,3,4] => 1 = 2 - 1
Description
The number of boxed occurrences of 132 in a permutation. This is the number of occurrences of the pattern $132$ such that any entry between the three matched entries is either larger than the largest matched entry or smaller than the smallest matched entry.
Matching statistic: St001085
(load all 39 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00325: Permutations —ones to leading⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => 0 = 1 - 1
[.,[.,.]]
 => [2,1] => [2,1] => 0 = 1 - 1
[[.,.],.]
 => [1,2] => [1,2] => 0 = 1 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => 0 = 1 - 1
[.,[[.,.],.]]
 => [2,3,1] => [2,1,3] => 1 = 2 - 1
[[.,.],[.,.]]
 => [3,1,2] => [3,1,2] => 0 = 1 - 1
[[.,[.,.]],.]
 => [2,1,3] => [1,3,2] => 0 = 1 - 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,1,2] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [3,2,4,1] => 1 = 2 - 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [4,3,2,1] => 0 = 1 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [2,1,3,4] => 1 = 2 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [2,1,4,3] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [4,2,1,3] => 0 = 1 - 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [3,1,4,2] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [4,2,3,1] => 0 = 1 - 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [4,1,3,2] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,4,3,2] => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,4,2,3] => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [1,3,4,2] => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,3,2,4] => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,1,2,3] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [4,3,5,1,2] => 1 = 2 - 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [5,4,2,1,3] => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [3,2,4,5,1] => 1 = 2 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [3,2,5,4,1] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [5,4,1,3,2] => 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [4,3,5,2,1] => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [5,4,3,1,2] => 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [5,4,3,2,1] => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [2,1,3,4,5] => 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [2,1,4,3,5] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [2,1,3,5,4] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [2,1,5,3,4] => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [2,1,5,4,3] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [5,3,1,2,4] => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [4,2,5,1,3] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [5,3,2,1,4] => 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [3,1,4,5,2] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [3,1,5,4,2] => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [5,3,1,4,2] => 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [4,2,5,3,1] => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [5,2,1,4,3] => 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [4,1,5,3,2] => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [5,3,4,2,1] => 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [5,3,4,1,2] => 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [5,2,4,3,1] => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [5,2,4,1,3] => 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [5,1,4,2,3] => 0 = 1 - 1
Description
The number of occurrences of the vincular pattern |21-3 in a permutation. This is the number of occurrences of the pattern $213$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive. In other words, this is the number of ascents whose bottom value is strictly smaller and the top value is strictly larger than the first entry of the permutation.
Matching statistic: St001086
(load all 5 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => 0 = 1 - 1
[.,[.,.]]
 => [2,1] => [2,1] => 0 = 1 - 1
[[.,.],.]
 => [1,2] => [1,2] => 0 = 1 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => 0 = 1 - 1
[.,[[.,.],.]]
 => [2,3,1] => [3,1,2] => 0 = 1 - 1
[[.,.],[.,.]]
 => [3,1,2] => [2,3,1] => 0 = 1 - 1
[[.,[.,.]],.]
 => [2,1,3] => [1,3,2] => 1 = 2 - 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,3,2] => 1 = 2 - 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [3,4,2,1] => 0 = 1 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,1,2] => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [3,4,1,2] => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [4,2,3,1] => 0 = 1 - 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [4,1,2,3] => 0 = 1 - 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [3,2,4,1] => 0 = 1 - 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [2,3,4,1] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,4,3,2] => 1 = 2 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,3,4,2] => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [1,4,2,3] => 1 = 2 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,3,2,4] => 1 = 2 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,1,4,3,2] => 1 = 2 - 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [4,5,3,2,1] => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,4,1,3,2] => 1 = 2 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [4,5,1,3,2] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [5,3,4,2,1] => 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [5,1,3,4,2] => 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [4,3,5,2,1] => 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [3,4,5,2,1] => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,4,3,1,2] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [4,5,3,1,2] => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [5,3,4,1,2] => 0 = 1 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [4,3,5,1,2] => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [3,4,5,1,2] => 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [5,4,2,3,1] => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [5,1,4,2,3] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [4,5,2,3,1] => 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [5,4,1,2,3] => 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [4,5,1,2,3] => 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [5,3,2,4,1] => 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [5,1,3,2,4] => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [5,2,3,4,1] => 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [5,1,2,3,4] => 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [4,3,2,5,1] => 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [3,4,2,5,1] => 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [4,2,3,5,1] => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [3,2,4,5,1] => 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [2,3,4,5,1] => 0 = 1 - 1
Description
The number of occurrences of the consecutive pattern 132 in a permutation. This is the number of occurrences of the pattern $132$, where the matched entries are all adjacent.
Matching statistic: St001683
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => 0 = 1 - 1
[.,[.,.]]
 => [2,1] => [2,1] => 0 = 1 - 1
[[.,.],.]
 => [1,2] => [1,2] => 0 = 1 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [2,3,1] => 0 = 1 - 1
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => 0 = 1 - 1
[[.,.],[.,.]]
 => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => 0 = 1 - 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [3,4,2,1] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,3,2,1] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [2,4,3,1] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [2,3,4,1] => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [3,2,4,1] => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [3,4,1,2] => 0 = 1 - 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [4,3,1,2] => 0 = 1 - 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,4,2,3] => 1 = 2 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,3,1,4] => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [1,3,2,4] => 1 = 2 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [4,5,3,2,1] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [3,5,4,2,1] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [3,4,5,2,1] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [4,3,5,2,1] => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [4,5,2,3,1] => 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [5,4,2,3,1] => 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [3,5,2,4,1] => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [2,5,3,4,1] => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [3,4,2,5,1] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [4,3,2,5,1] => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [2,4,3,5,1] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [2,3,4,5,1] => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [3,2,4,5,1] => 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [4,5,3,1,2] => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [5,4,3,1,2] => 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [3,5,4,1,2] => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [3,4,5,1,2] => 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [4,3,5,1,2] => 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [4,5,2,1,3] => 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [5,4,2,1,3] => 0 = 1 - 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [4,5,1,2,3] => 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [5,4,1,2,3] => 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [3,5,2,1,4] => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [2,5,3,1,4] => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [3,5,1,2,4] => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [2,5,1,3,4] => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,5,2,3,4] => 1 = 2 - 1
Description
The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation.
Matching statistic: St001685
(load all 2 compositions to match this statistic)
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => 0 = 1 - 1
[.,[.,.]]
 => [2,1] => [2,1] => 0 = 1 - 1
[[.,.],.]
 => [1,2] => [1,2] => 0 = 1 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [2,3,1] => 0 = 1 - 1
[.,[[.,.],.]]
 => [2,3,1] => [3,2,1] => 0 = 1 - 1
[[.,.],[.,.]]
 => [3,1,2] => [1,3,2] => 1 = 2 - 1
[[.,[.,.]],.]
 => [2,1,3] => [2,1,3] => 0 = 1 - 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [3,4,2,1] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,3,2,1] => 0 = 1 - 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [2,4,3,1] => 1 = 2 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [2,3,4,1] => 0 = 1 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [3,2,4,1] => 0 = 1 - 1
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [3,4,1,2] => 0 = 1 - 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [4,3,1,2] => 0 = 1 - 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [2,4,1,3] => 1 = 2 - 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,4,2,3] => 1 = 2 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [2,3,1,4] => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [3,2,1,4] => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [1,3,2,4] => 1 = 2 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [2,1,3,4] => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [4,5,3,2,1] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [3,5,4,2,1] => 1 = 2 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [3,4,5,2,1] => 0 = 1 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [4,3,5,2,1] => 0 = 1 - 1
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [4,5,2,3,1] => 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [5,4,2,3,1] => 0 = 1 - 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [3,5,2,4,1] => 1 = 2 - 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [2,5,3,4,1] => 1 = 2 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [3,4,2,5,1] => 0 = 1 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [4,3,2,5,1] => 0 = 1 - 1
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [2,4,3,5,1] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [2,3,4,5,1] => 0 = 1 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [3,2,4,5,1] => 0 = 1 - 1
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [4,5,3,1,2] => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [5,4,3,1,2] => 0 = 1 - 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [3,5,4,1,2] => 1 = 2 - 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [3,4,5,1,2] => 0 = 1 - 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [4,3,5,1,2] => 0 = 1 - 1
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [4,5,2,1,3] => 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [5,4,2,1,3] => 0 = 1 - 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [4,5,1,2,3] => 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [5,4,1,2,3] => 0 = 1 - 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [3,5,2,1,4] => 1 = 2 - 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [2,5,3,1,4] => 1 = 2 - 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [3,5,1,2,4] => 1 = 2 - 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [2,5,1,3,4] => 1 = 2 - 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,5,2,3,4] => 1 = 2 - 1
Description
The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.
Matching statistic: St001744
(load all 5 compositions to match this statistic)
Mapping
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1] => 0 = 1 - 1
[.,[.,.]]
 => [2,1] => [2,1] => 0 = 1 - 1
[[.,.],.]
 => [1,2] => [1,2] => 0 = 1 - 1
[.,[.,[.,.]]]
 => [3,2,1] => [3,2,1] => 0 = 1 - 1
[.,[[.,.],.]]
 => [2,3,1] => [3,1,2] => 1 = 2 - 1
[[.,.],[.,.]]
 => [1,3,2] => [2,1,3] => 0 = 1 - 1
[[.,[.,.]],.]
 => [2,1,3] => [1,3,2] => 0 = 1 - 1
[[[.,.],.],.]
 => [1,2,3] => [1,2,3] => 0 = 1 - 1
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [4,1,3,2] => 1 = 2 - 1
[.,[[.,.],[.,.]]]
 => [2,4,3,1] => [3,1,4,2] => 0 = 1 - 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [4,3,1,2] => 1 = 2 - 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [3,4,1,2] => 1 = 2 - 1
[[.,.],[.,[.,.]]]
 => [1,4,3,2] => [2,1,4,3] => 0 = 1 - 1
[[.,.],[[.,.],.]]
 => [1,3,4,2] => [2,4,1,3] => 1 = 2 - 1
[[.,[.,.]],[.,.]]
 => [2,1,4,3] => [3,2,1,4] => 0 = 1 - 1
[[[.,.],.],[.,.]]
 => [1,2,4,3] => [2,3,1,4] => 0 = 1 - 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,4,3,2] => 0 = 1 - 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,3,4,2] => 0 = 1 - 1
[[[.,.],[.,.]],.]
 => [1,3,2,4] => [1,2,4,3] => 0 = 1 - 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,3,2,4] => 0 = 1 - 1
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [5,4,3,2,1] => 0 = 1 - 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [5,1,4,3,2] => 1 = 2 - 1
[.,[.,[[.,.],[.,.]]]]
 => [3,5,4,2,1] => [4,1,5,3,2] => 0 = 1 - 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [5,4,1,3,2] => 1 = 2 - 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [4,5,1,3,2] => 1 = 2 - 1
[.,[[.,.],[.,[.,.]]]]
 => [2,5,4,3,1] => [3,1,5,4,2] => 0 = 1 - 1
[.,[[.,.],[[.,.],.]]]
 => [2,4,5,3,1] => [3,5,1,4,2] => 1 = 2 - 1
[.,[[.,[.,.]],[.,.]]]
 => [3,2,5,4,1] => [4,3,1,5,2] => 0 = 1 - 1
[.,[[[.,.],.],[.,.]]]
 => [2,3,5,4,1] => [3,4,1,5,2] => 0 = 1 - 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [5,4,3,1,2] => 1 = 2 - 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [4,5,3,1,2] => 1 = 2 - 1
[.,[[[.,.],[.,.]],.]]
 => [2,4,3,5,1] => [3,5,4,1,2] => 1 = 2 - 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [4,3,5,1,2] => 1 = 2 - 1
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [3,4,5,1,2] => 1 = 2 - 1
[[.,.],[.,[.,[.,.]]]]
 => [1,5,4,3,2] => [2,1,5,4,3] => 0 = 1 - 1
[[.,.],[.,[[.,.],.]]]
 => [1,4,5,3,2] => [2,5,1,4,3] => 1 = 2 - 1
[[.,.],[[.,.],[.,.]]]
 => [1,3,5,4,2] => [2,4,1,5,3] => 0 = 1 - 1
[[.,.],[[.,[.,.]],.]]
 => [1,4,3,5,2] => [2,5,4,1,3] => 1 = 2 - 1
[[.,.],[[[.,.],.],.]]
 => [1,3,4,5,2] => [2,4,5,1,3] => 1 = 2 - 1
[[.,[.,.]],[.,[.,.]]]
 => [2,1,5,4,3] => [3,2,1,5,4] => 0 = 1 - 1
[[.,[.,.]],[[.,.],.]]
 => [2,1,4,5,3] => [3,2,5,1,4] => 1 = 2 - 1
[[[.,.],.],[.,[.,.]]]
 => [1,2,5,4,3] => [2,3,1,5,4] => 0 = 1 - 1
[[[.,.],.],[[.,.],.]]
 => [1,2,4,5,3] => [2,3,5,1,4] => 1 = 2 - 1
[[.,[.,[.,.]]],[.,.]]
 => [3,2,1,5,4] => [4,3,2,1,5] => 0 = 1 - 1
[[.,[[.,.],.]],[.,.]]
 => [2,3,1,5,4] => [3,4,2,1,5] => 0 = 1 - 1
[[[.,.],[.,.]],[.,.]]
 => [1,3,2,5,4] => [2,4,3,1,5] => 0 = 1 - 1
[[[.,[.,.]],.],[.,.]]
 => [2,1,3,5,4] => [3,2,4,1,5] => 0 = 1 - 1
[[[[.,.],.],.],[.,.]]
 => [1,2,3,5,4] => [2,3,4,1,5] => 0 = 1 - 1
Description
The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. Let $\nu$ be a (partial) permutation of $[k]$ with $m$ letters together with dashes between some of its letters. An occurrence of $\nu$ in a permutation $\tau$ is a subsequence $\tau_{a_1},\dots,\tau_{a_m}$ such that $a_i + 1 = a_{i+1}$ whenever there is a dash between the $i$-th and the $(i+1)$-st letter of $\nu$, which is order isomorphic to $\nu$. Thus, $\nu$ is a vincular pattern, except that it is not required to be a permutation. An arrow pattern of size $k$ consists of such a generalized vincular pattern $\nu$ and arrows $b_1\to c_1, b_2\to c_2,\dots$, such that precisely the numbers $1,\dots,k$ appear in the vincular pattern and the arrows. Let $\Phi$ be the map [[Mp00087]]. Let $\tau$ be a permutation and $\sigma = \Phi(\tau)$. Then a subsequence $w = (x_{a_1},\dots,x_{a_m})$ of $\tau$ is an occurrence of the arrow pattern if $w$ is an occurrence of $\nu$, for each arrow $b\to c$ we have $\sigma(x_b) = x_c$ and $x_1 < x_2 < \dots < x_k$.
Matching statistic: St000553
Mapping
Mp00013: Binary trees —to poset⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
Mp00111: Graphs —complement⟶ Graphs
St000553: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => ([],1)
 => ([],1)
 => ([],1)
 => 1
[.,[.,.]]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[[.,.],.]
 => ([(0,1)],2)
 => ([],2)
 => ([(0,1)],2)
 => 1
[.,[.,[.,.]]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[.,[[.,.],.]]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[[.,.],[.,.]]
 => ([(0,2),(1,2)],3)
 => ([(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
[[.,[.,.]],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[[[.,.],.],.]
 => ([(0,2),(2,1)],3)
 => ([],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
[.,[.,[.,[.,.]]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[.,[[.,.],.]]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,.],[.,.]]]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[.,[.,.]],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[[[.,.],.],.]]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[.,.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[.,.],[[.,.],.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[.,[.,.]],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[[.,.],.],[.,.]]
 => ([(0,3),(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[[.,[.,[.,.]]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[.,[[.,.],.]],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[.,.],[.,.]],.]
 => ([(0,3),(1,3),(3,2)],4)
 => ([(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[.,[.,.]],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[[[[.,.],.],.],.]
 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[.,[.,[.,[.,[.,.]]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[.,[.,[[.,.],.]]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[.,[[.,.],[.,.]]]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[.,[[.,[.,.]],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[.,[[[.,.],.],.]]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[[.,.],[.,[.,.]]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[[.,.],[[.,.],.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[[.,[.,.]],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[[[.,.],.],[.,.]]]
 => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[[.,[.,[.,.]]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[[.,[[.,.],.]],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[[[.,.],[.,.]],.]]
 => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[[[.,[.,.]],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[.,[[[[.,.],.],.],.]]
 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[[.,.],[.,[.,[.,.]]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[.,.],[.,[[.,.],.]]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[.,.],[[.,.],[.,.]]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[.,.],[[.,[.,.]],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[.,.],[[[.,.],.],.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[.,[.,.]],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
[[.,[.,.]],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
[[[.,.],.],[.,[.,.]]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
[[[.,.],.],[[.,.],.]]
 => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
 => 2
[[.,[.,[.,.]]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[.,[[.,.],.]],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[.,.],[.,.]],[.,.]]
 => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[.,[.,.]],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[[[[.,.],.],.],[.,.]]
 => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
Description
The number of blocks of a graph. A cut vertex is a vertex whose deletion increases the number of connected components. A block is a maximal connected subgraph which itself has no cut vertices. Two distinct blocks cannot overlap in more than a single cut vertex.
Matching statistic: St000920
Mapping
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St000920: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
 => [1] => [1,0]
 => [1,0]
 => 1
[.,[.,.]]
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 1
[[.,.],.]
 => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 1
[.,[.,[.,.]]]
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1
[.,[[.,.],.]]
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[[.,.],[.,.]]
 => [3,1,2] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 1
[[.,[.,.]],.]
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1
[[[.,.],.],.]
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[.,[.,[.,[.,.]]]]
 => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[.,[.,[[.,.],.]]]
 => [3,4,2,1] => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[.,[[.,.],[.,.]]]
 => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[.,[[.,[.,.]],.]]
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[.,[[[.,.],.],.]]
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[[.,.],[.,[.,.]]]
 => [4,3,1,2] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[[.,.],[[.,.],.]]
 => [3,4,1,2] => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[[.,[.,.]],[.,.]]
 => [4,2,1,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[[[.,.],.],[.,.]]
 => [4,1,2,3] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[[.,[.,[.,.]]],.]
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[[.,[[.,.],.]],.]
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[[[.,.],[.,.]],.]
 => [3,1,2,4] => [1,1,1,0,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1
[[[.,[.,.]],.],.]
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[[[[.,.],.],.],.]
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[.,[.,[.,[.,[.,.]]]]]
 => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[.,[.,[.,[[.,.],.]]]]
 => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[.,[.,[[.,.],[.,.]]]]
 => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[.,[.,[[.,[.,.]],.]]]
 => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[.,[.,[[[.,.],.],.]]]
 => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[.,[[.,.],[.,[.,.]]]]
 => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[.,[[.,.],[[.,.],.]]]
 => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[.,[[.,[.,.]],[.,.]]]
 => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[.,[[[.,.],.],[.,.]]]
 => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[.,[[.,[.,[.,.]]],.]]
 => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[.,[[.,[[.,.],.]],.]]
 => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[.,[[[.,.],[.,.]],.]]
 => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
[.,[[[.,[.,.]],.],.]]
 => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[.,[[[[.,.],.],.],.]]
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2
[[.,.],[.,[.,[.,.]]]]
 => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[[.,.],[.,[[.,.],.]]]
 => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[[.,.],[[.,.],[.,.]]]
 => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[[.,.],[[.,[.,.]],.]]
 => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[[.,.],[[[.,.],.],.]]
 => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 2
[[.,[.,.]],[.,[.,.]]]
 => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[[.,[.,.]],[[.,.],.]]
 => [4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[[[.,.],.],[.,[.,.]]]
 => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[[[.,.],.],[[.,.],.]]
 => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 1
[[.,[.,[.,.]]],[.,.]]
 => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[[.,[[.,.],.]],[.,.]]
 => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[[[.,.],[.,.]],[.,.]]
 => [5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[[[.,[.,.]],.],[.,.]]
 => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
[[[[.,.],.],.],[.,.]]
 => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 1
Description
The logarithmic height of a Dyck path. This is the floor of the binary logarithm of the usual height increased by one: $$ \lfloor\log_2(1+height(D))\rfloor $$
The following 91 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001399The distinguishing number of a poset. St000317The cycle descent number of a permutation. St000356The number of occurrences of the pattern 13-2. St000366The number of double descents of a permutation. St000386The number of factors DDU in a Dyck path. St000552The number of cut vertices of a graph. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001271The competition number of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000779The tier of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001570The minimal number of edges to add to make a graph Hamiltonian. St001432The order dimension of the partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001128The exponens consonantiae of a partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001281The normalized isoperimetric number of a graph. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St000668The least common multiple of the parts of the partition. St001568The smallest positive integer that does not appear twice in the partition. St001597The Frobenius rank of a skew partition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000260The radius of a connected graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000100The number of linear extensions of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000632The jump number of the poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000302The determinant of the distance matrix of a connected graph. St000456The monochromatic index of a connected graph. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001737The number of descents of type 2 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001330The hat guessing number of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000768The number of peaks in an integer composition. St000455The second largest eigenvalue of a graph if it is integral. St000648The number of 2-excedences of a permutation. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001779The order of promotion on the set of linear extensions of a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001964The interval resolution global dimension of a poset. St001875The number of simple modules with projective dimension at most 1. St001868The number of alignments of type NE of a signed permutation. St001060The distinguishing index of a graph. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001487The number of inner corners of a skew partition.