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Your data matches 114 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St000701
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
St000701: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,[.,.]]]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> 1 = 0 + 1
[[[.,.],.],.]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> 2 = 1 + 1
[[.,[.,[.,[.,.]]]],.]
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> 1 = 0 + 1
Description
The protection number of a binary tree.
This is the minimal distance from the root to a leaf.
Matching statistic: St000700
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%
St000700: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,[.,.]]]
=> [[],[[],[[],[]]]]
=> 1 = 0 + 1
[.,[[.,.],.]]
=> [[],[[[],[]],[]]]
=> 1 = 0 + 1
[[.,.],[.,.]]
=> [[[],[]],[[],[]]]
=> 2 = 1 + 1
[[.,[.,.]],.]
=> [[[],[[],[]]],[]]
=> 1 = 0 + 1
[[[.,.],.],.]
=> [[[[],[]],[]],[]]
=> 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [[],[[],[[],[[],[]]]]]
=> 1 = 0 + 1
[.,[.,[[.,.],.]]]
=> [[],[[],[[[],[]],[]]]]
=> 1 = 0 + 1
[.,[[.,.],[.,.]]]
=> [[],[[[],[]],[[],[]]]]
=> 1 = 0 + 1
[.,[[.,[.,.]],.]]
=> [[],[[[],[[],[]]],[]]]
=> 1 = 0 + 1
[.,[[[.,.],.],.]]
=> [[],[[[[],[]],[]],[]]]
=> 1 = 0 + 1
[[.,.],[.,[.,.]]]
=> [[[],[]],[[],[[],[]]]]
=> 2 = 1 + 1
[[.,.],[[.,.],.]]
=> [[[],[]],[[[],[]],[]]]
=> 2 = 1 + 1
[[.,[.,.]],[.,.]]
=> [[[],[[],[]]],[[],[]]]
=> 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [[[[],[]],[]],[[],[]]]
=> 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [[[],[[],[[],[]]]],[]]
=> 1 = 0 + 1
[[.,[[.,.],.]],.]
=> [[[],[[[],[]],[]]],[]]
=> 1 = 0 + 1
[[[.,.],[.,.]],.]
=> [[[[],[]],[[],[]]],[]]
=> 1 = 0 + 1
[[[.,[.,.]],.],.]
=> [[[[],[[],[]]],[]],[]]
=> 1 = 0 + 1
[[[[.,.],.],.],.]
=> [[[[[],[]],[]],[]],[]]
=> 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [[],[[],[[],[[],[[],[]]]]]]
=> 1 = 0 + 1
[.,[.,[.,[[.,.],.]]]]
=> [[],[[],[[],[[[],[]],[]]]]]
=> 1 = 0 + 1
[.,[.,[[.,.],[.,.]]]]
=> [[],[[],[[[],[]],[[],[]]]]]
=> 1 = 0 + 1
[.,[.,[[.,[.,.]],.]]]
=> [[],[[],[[[],[[],[]]],[]]]]
=> 1 = 0 + 1
[.,[.,[[[.,.],.],.]]]
=> [[],[[],[[[[],[]],[]],[]]]]
=> 1 = 0 + 1
[.,[[.,.],[.,[.,.]]]]
=> [[],[[[],[]],[[],[[],[]]]]]
=> 1 = 0 + 1
[.,[[.,.],[[.,.],.]]]
=> [[],[[[],[]],[[[],[]],[]]]]
=> 1 = 0 + 1
[.,[[.,[.,.]],[.,.]]]
=> [[],[[[],[[],[]]],[[],[]]]]
=> 1 = 0 + 1
[.,[[[.,.],.],[.,.]]]
=> [[],[[[[],[]],[]],[[],[]]]]
=> 1 = 0 + 1
[.,[[.,[.,[.,.]]],.]]
=> [[],[[[],[[],[[],[]]]],[]]]
=> 1 = 0 + 1
[.,[[.,[[.,.],.]],.]]
=> [[],[[[],[[[],[]],[]]],[]]]
=> 1 = 0 + 1
[.,[[[.,.],[.,.]],.]]
=> [[],[[[[],[]],[[],[]]],[]]]
=> 1 = 0 + 1
[.,[[[.,[.,.]],.],.]]
=> [[],[[[[],[[],[]]],[]],[]]]
=> 1 = 0 + 1
[.,[[[[.,.],.],.],.]]
=> [[],[[[[[],[]],[]],[]],[]]]
=> 1 = 0 + 1
[[.,.],[.,[.,[.,.]]]]
=> [[[],[]],[[],[[],[[],[]]]]]
=> 2 = 1 + 1
[[.,.],[.,[[.,.],.]]]
=> [[[],[]],[[],[[[],[]],[]]]]
=> 2 = 1 + 1
[[.,.],[[.,.],[.,.]]]
=> [[[],[]],[[[],[]],[[],[]]]]
=> 2 = 1 + 1
[[.,.],[[.,[.,.]],.]]
=> [[[],[]],[[[],[[],[]]],[]]]
=> 2 = 1 + 1
[[.,.],[[[.,.],.],.]]
=> [[[],[]],[[[[],[]],[]],[]]]
=> 2 = 1 + 1
[[.,[.,.]],[.,[.,.]]]
=> [[[],[[],[]]],[[],[[],[]]]]
=> 2 = 1 + 1
[[.,[.,.]],[[.,.],.]]
=> [[[],[[],[]]],[[[],[]],[]]]
=> 2 = 1 + 1
[[[.,.],.],[.,[.,.]]]
=> [[[[],[]],[]],[[],[[],[]]]]
=> 2 = 1 + 1
[[[.,.],.],[[.,.],.]]
=> [[[[],[]],[]],[[[],[]],[]]]
=> 2 = 1 + 1
[[.,[.,[.,.]]],[.,.]]
=> [[[],[[],[[],[]]]],[[],[]]]
=> 2 = 1 + 1
[[.,[[.,.],.]],[.,.]]
=> [[[],[[[],[]],[]]],[[],[]]]
=> 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [[[[],[]],[[],[]]],[[],[]]]
=> 2 = 1 + 1
[[[.,[.,.]],.],[.,.]]
=> [[[[],[[],[]]],[]],[[],[]]]
=> 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [[[[[],[]],[]],[]],[[],[]]]
=> 2 = 1 + 1
[[.,[.,[.,[.,.]]]],.]
=> [[[],[[],[[],[[],[]]]]],[]]
=> 1 = 0 + 1
[[.,[.,[[.,.],.]]],.]
=> [[[],[[],[[[],[]],[]]]],[]]
=> 1 = 0 + 1
[[.,[[.,.],[.,.]]],.]
=> [[[],[[[],[]],[[],[]]]],[]]
=> 1 = 0 + 1
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)
(load all 2 compositions to match this statistic)
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%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St001083: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,[.,.]]]
=> [3,2,1] => [2,3,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => 0
[[.,.],[.,.]]
=> [3,1,2] => [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,4,2,1] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,2,1] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,4,3,1] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,3,4,1] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,2,4,1] => 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,1,2] => 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4,3,1,2] => 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,4,1,3] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,4,2,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,5,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,2,1] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,5,4,2,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [3,4,5,2,1] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,3,5,2,1] => 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,5,2,3,1] => 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,4,2,3,1] => 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,5,2,4,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,5,3,4,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,4,2,5,1] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [4,3,2,5,1] => 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,4,3,5,1] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,3,4,5,1] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [3,2,4,5,1] => 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,5,3,1,2] => 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [5,4,3,1,2] => 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,5,4,1,2] => 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [3,4,5,1,2] => 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [4,3,5,1,2] => 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,5,2,1,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [5,4,2,1,3] => 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,5,1,2,3] => 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [5,4,1,2,3] => 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,5,2,1,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,5,3,1,4] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,5,1,2,4] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,5,1,3,4] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,5,2,3,4] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [3,4,2,1,5] => 0
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,3,2,1,5] => 0
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [2,4,3,1,5] => 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)
(load all 39 compositions to match this statistic)
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%
Mp00325: Permutations —ones to leading⟶ Permutations
St001085: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [2,1,3] => 1
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => 0
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,1,2] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,2,4,1] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,3,2,1] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,1,3,4] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,1,4,3] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,2,1,3] => 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,1,4,2] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [4,2,3,1] => 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,3,2] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,3,2] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,4,2,3] => 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3,4,2] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,2,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,1,2,3] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,3,5,1,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,4,2,1,3] => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [3,2,4,5,1] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,2,5,4,1] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,4,1,3,2] => 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,3,5,2,1] => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,4,3,1,2] => 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,4,3,2,1] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [2,1,3,4,5] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [2,1,4,3,5] => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,1,3,5,4] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,1,5,3,4] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,1,5,4,3] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,3,1,2,4] => 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,2,5,1,3] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [5,3,2,1,4] => 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [3,1,4,5,2] => 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,1,5,4,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [5,3,1,4,2] => 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,2,5,3,1] => 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [5,2,1,4,3] => 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,1,5,3,2] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [5,3,4,2,1] => 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [5,3,4,1,2] => 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [5,2,4,3,1] => 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [5,2,4,1,3] => 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,4,2,3] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,5,4,3,2] => 0
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [1,5,4,2,3] => 0
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [1,5,3,4,2] => 0
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)
(load all 5 compositions to match this statistic)
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%
Mp00329: Permutations —Tanimoto⟶ Permutations
St001086: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => 0
[[.,.],[.,.]]
=> [3,1,2] => [2,3,1] => 0
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [3,4,2,1] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,1,2] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,4,1,2] => 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,2,3,1] => 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4,1,2,3] => 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,2,4,1] => 0
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,3,4,1] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,3,2] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,4,2] => 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,4,2,3] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,2,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,4,3,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,5,3,2,1] => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,1,3,2] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,5,1,3,2] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,3,4,2,1] => 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,1,3,4,2] => 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [4,3,5,2,1] => 0
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [3,4,5,2,1] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,3,1,2] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [4,5,3,1,2] => 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [5,3,4,1,2] => 0
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [4,3,5,1,2] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [3,4,5,1,2] => 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,4,2,3,1] => 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [5,1,4,2,3] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [4,5,2,3,1] => 0
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [5,4,1,2,3] => 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [4,5,1,2,3] => 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [5,3,2,4,1] => 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [5,1,3,2,4] => 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [5,2,3,4,1] => 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [5,1,2,3,4] => 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,3,2,5,1] => 0
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [3,4,2,5,1] => 0
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [4,2,3,5,1] => 0
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [3,2,4,5,1] => 0
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [2,3,4,5,1] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,5,4,3,2] => 1
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [1,4,5,3,2] => 0
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [1,5,3,4,2] => 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: St001625
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 0
[.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 1
[[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 0
[[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 0
[.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 0
[.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 0
[.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 0
[.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
[.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 1
[[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 1
[[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 1
[[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 1
[[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 1
[[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 1
[[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 1
[[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> 1
[[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 1
[[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 1
[[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
[[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 1
[[[[.,.],.],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 1
[[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 0
Description
The Möbius invariant of a lattice.
The '''Möbius invariant''' of a lattice $L$ is the value of the Möbius function applied to least and greatest element, that is $\mu(L)=\mu_L(\hat{0},\hat{1})$, where $\hat{0}$ is the least element of $L$ and $\hat{1}$ is the greatest element of $L$.
For the definition of the Möbius function, see [[St000914]].
Matching statistic: St001683
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%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St001683: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,[.,.]]]
=> [3,2,1] => [2,3,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => 0
[[.,.],[.,.]]
=> [3,1,2] => [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,4,2,1] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,2,1] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,4,3,1] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,3,4,1] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,2,4,1] => 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,1,2] => 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4,3,1,2] => 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,4,1,3] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,4,2,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,5,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,2,1] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,5,4,2,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [3,4,5,2,1] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,3,5,2,1] => 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,5,2,3,1] => 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,4,2,3,1] => 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,5,2,4,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,5,3,4,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,4,2,5,1] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [4,3,2,5,1] => 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,4,3,5,1] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,3,4,5,1] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [3,2,4,5,1] => 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,5,3,1,2] => 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [5,4,3,1,2] => 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,5,4,1,2] => 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [3,4,5,1,2] => 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [4,3,5,1,2] => 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,5,2,1,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [5,4,2,1,3] => 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,5,1,2,3] => 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [5,4,1,2,3] => 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,5,2,1,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,5,3,1,4] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,5,1,2,4] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,5,1,3,4] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,5,2,3,4] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [3,4,2,1,5] => 0
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,3,2,1,5] => 0
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [2,4,3,1,5] => 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)
(load all 2 compositions to match this statistic)
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%
Mp00257: Permutations —Alexandersson Kebede⟶ Permutations
St001685: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,[.,.]]]
=> [3,2,1] => [2,3,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => 0
[[.,.],[.,.]]
=> [3,1,2] => [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,4,2,1] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,2,1] => 0
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,4,3,1] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,3,4,1] => 0
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,2,4,1] => 0
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,1,2] => 0
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4,3,1,2] => 0
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,4,1,3] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [1,4,2,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,1,4] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => 0
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [1,3,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [4,5,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,2,1] => 0
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [3,5,4,2,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [3,4,5,2,1] => 0
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,3,5,2,1] => 0
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,5,2,3,1] => 0
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,4,2,3,1] => 0
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,5,2,4,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,5,3,4,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,4,2,5,1] => 0
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [4,3,2,5,1] => 0
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,4,3,5,1] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,3,4,5,1] => 0
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [3,2,4,5,1] => 0
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,5,3,1,2] => 0
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [5,4,3,1,2] => 0
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [3,5,4,1,2] => 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [3,4,5,1,2] => 0
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [4,3,5,1,2] => 0
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,5,2,1,3] => 0
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [5,4,2,1,3] => 0
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,5,1,2,3] => 0
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [5,4,1,2,3] => 0
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,5,2,1,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,5,3,1,4] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,5,1,2,4] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,5,1,3,4] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [1,5,2,3,4] => 1
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [3,4,2,1,5] => 0
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [4,3,2,1,5] => 0
[[.,[[.,.],[.,.]]],.]
=> [4,2,3,1,5] => [2,4,3,1,5] => 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)
(load all 5 compositions to match this statistic)
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%
Mp00329: Permutations —Tanimoto⟶ Permutations
St001744: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => 1
[[.,.],[.,.]]
=> [1,3,2] => [2,1,3] => 0
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,1,4,2] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,1,2] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,4,1,2] => 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,4,3] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,1,3] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,2,1,4] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,3,1,4] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,3,2] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,4,2] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,2,4,3] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,2,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,4,3,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,1,5,3,2] => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,1,3,2] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,5,1,3,2] => 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [3,1,5,4,2] => 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,5,1,4,2] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [4,3,1,5,2] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,4,1,5,2] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,3,1,2] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [4,5,3,1,2] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [3,5,4,1,2] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [4,3,5,1,2] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [3,4,5,1,2] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,1,5,4,3] => 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [2,5,1,4,3] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [2,4,1,5,3] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,5,4,1,3] => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,4,5,1,3] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [3,2,1,5,4] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,2,5,1,4] => 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [2,3,1,5,4] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [2,3,5,1,4] => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [4,3,2,1,5] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,4,2,1,5] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [2,4,3,1,5] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [3,2,4,1,5] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [2,3,4,1,5] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,5,4,3,2] => 0
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [1,4,5,3,2] => 0
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [1,3,5,4,2] => 0
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: St000317
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00329: Permutations —Tanimoto⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00329: Permutations —Tanimoto⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000317: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,[.,[.,.]]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => [3,1,2] => 1
[[.,.],[.,.]]
=> [1,3,2] => [2,1,3] => [2,1,3] => 0
[[.,[.,.]],.]
=> [2,1,3] => [1,3,2] => [1,3,2] => 0
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => [2,3,4,1] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => [4,3,1,2] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [3,1,4,2] => [3,4,1,2] => 0
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,1,2] => [3,1,4,2] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [3,4,1,2] => [4,1,3,2] => 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [2,1,4,3] => [2,1,4,3] => 0
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [2,4,1,3] => [4,2,1,3] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [3,2,1,4] => [2,3,1,4] => 0
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [2,3,1,4] => [3,2,1,4] => 0
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [1,4,3,2] => [1,3,4,2] => 0
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [1,3,4,2] => [1,4,3,2] => 0
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,2,4,3] => [1,2,4,3] => 0
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [1,3,2,4] => [1,3,2,4] => 0
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [2,3,4,5,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,4,3,2] => [5,3,4,1,2] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [4,1,5,3,2] => [4,3,5,1,2] => 0
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,1,3,2] => [4,3,1,5,2] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [4,5,1,3,2] => [5,3,1,4,2] => 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [3,1,5,4,2] => [3,4,1,5,2] => 0
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [3,5,1,4,2] => [5,4,3,1,2] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [4,3,1,5,2] => [3,4,5,2,1] => 0
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [3,4,1,5,2] => [4,5,3,1,2] => 0
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,3,1,2] => [3,1,4,5,2] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [4,5,3,1,2] => [3,1,5,4,2] => 1
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [3,5,4,1,2] => [4,1,3,5,2] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [4,3,5,1,2] => [5,1,4,3,2] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [3,4,5,1,2] => [5,1,3,4,2] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [2,1,5,4,3] => [2,1,4,5,3] => 0
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [2,5,1,4,3] => [5,2,4,1,3] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [2,4,1,5,3] => [4,2,5,1,3] => 0
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [2,5,4,1,3] => [4,2,1,5,3] => 1
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [2,4,5,1,3] => [5,2,1,4,3] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [3,2,1,5,4] => [2,3,1,5,4] => 0
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [3,2,5,1,4] => [5,3,2,1,4] => 1
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [2,3,1,5,4] => [3,2,1,5,4] => 0
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [2,3,5,1,4] => [5,2,3,1,4] => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [4,3,2,1,5] => [2,3,4,1,5] => 0
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,4,2,1,5] => [2,4,3,1,5] => 0
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [2,4,3,1,5] => [3,2,4,1,5] => 0
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [3,2,4,1,5] => [4,3,2,1,5] => 0
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [2,3,4,1,5] => [4,2,3,1,5] => 0
[[.,[.,[.,[.,.]]]],.]
=> [4,3,2,1,5] => [1,5,4,3,2] => [1,3,4,5,2] => 0
[[.,[.,[[.,.],.]]],.]
=> [3,4,2,1,5] => [1,4,5,3,2] => [1,3,5,4,2] => 0
[[.,[[.,.],[.,.]]],.]
=> [2,4,3,1,5] => [1,3,5,4,2] => [1,4,3,5,2] => 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.
The following 104 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000779The tier of a permutation. 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. St001570The minimal number of edges to add to make a graph Hamiltonian. St001691The number of kings in a graph. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000374The number of exclusive right-to-left minima of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000553The number of blocks of a graph. St000920The logarithmic height of a Dyck path. St000996The number of exclusive left-to-right maxima of a permutation. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001399The distinguishing number of a poset. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000542The number of left-to-right-minima of a permutation. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000225Difference between largest and smallest parts in a partition. St000944The 3-degree of an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000379The number of Hamiltonian cycles in a graph. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001176The size of a partition minus its first part. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000478Another weight of a partition according to Alladi. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000929The constant term of the character polynomial of an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000632The jump number of the poset. 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. St001811The Castelnuovo-Mumford regularity of a permutation. 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. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001486The number of corners of the ribbon associated with an integer composition. St000259The diameter of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000934The 2-degree of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001820The size of the image of the pop stack sorting operator. St001846The number of elements which do not have a complement in the lattice. St000768The number of peaks in an integer composition. St001877Number of indecomposable injective modules with projective dimension 2. St000648The number of 2-excedences of a permutation. 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". 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. 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. St001964The interval resolution global dimension of a poset. St001868The number of alignments of type NE of a signed permutation. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra.
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