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Your data matches 150 different statistics following compositions of up to 3 maps.
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Matching statistic: St000035
(load all 57 compositions to match this statistic)
(load all 57 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000035: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 0
[.,[.,.]]
=> [2,1] => 1
[[.,.],.]
=> [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => 1
[[.,.],[.,.]]
=> [3,1,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => 2
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 2
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 2
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1
Description
The number of left outer peaks of a permutation.
A left outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $1$ if $w_1 > w_2$.
In other words, it is a peak in the word $[0,w_1,..., w_n]$.
This appears in [1, def.3.1]. The joint distribution with [[St000366]] is studied in [3], where left outer peaks are called ''exterior peaks''.
Matching statistic: St000834
(load all 44 compositions to match this statistic)
(load all 44 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000834: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 0
[.,[.,.]]
=> [2,1] => 0
[[.,.],.]
=> [1,2] => 1
[.,[.,[.,.]]]
=> [3,2,1] => 0
[.,[[.,.],.]]
=> [2,3,1] => 1
[[.,.],[.,.]]
=> [3,1,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 0
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 0
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 2
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 2
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1
Description
The number of right outer peaks of a permutation.
A right outer peak in a permutation $w = [w_1,..., w_n]$ is either a position $i$ such that $w_{i-1} < w_i > w_{i+1}$ or $n$ if $w_n > w_{n-1}$.
In other words, it is a peak in the word $[w_1,..., w_n,0]$.
Matching statistic: St001188
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001188: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001188: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> 0
[[.,.],.]
=> [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 2
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
Description
The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path.
Also the number of simple modules that are isolated vertices in the sense of 4.5. (4) in the reference.
Matching statistic: St001212
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001212: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001212: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> 0
[[.,.],.]
=> [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 1
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 2
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
Description
The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module.
Matching statistic: St001244
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St001244: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001244: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> 0
[.,[.,.]]
=> [1,1,0,0]
=> 0
[[.,.],.]
=> [1,0,1,0]
=> 1
[.,[.,[.,.]]]
=> [1,1,1,0,0,0]
=> 0
[.,[[.,.],.]]
=> [1,1,0,1,0,0]
=> 1
[[.,.],[.,.]]
=> [1,0,1,1,0,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> 1
[[[.,.],.],.]
=> [1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,.]]]]
=> [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[[.,.],.]]]
=> [1,1,1,0,1,0,0,0]
=> 1
[.,[[.,.],[.,.]]]
=> [1,1,0,1,1,0,0,0]
=> 1
[.,[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0]
=> 1
[.,[[[.,.],.],.]]
=> [1,1,0,1,0,1,0,0]
=> 2
[[.,.],[.,[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,.],[[.,.],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[[.,[.,.]],[.,.]]
=> [1,1,0,0,1,1,0,0]
=> 1
[[[.,.],.],[.,.]]
=> [1,0,1,0,1,1,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,1,0,0,0,1,0]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,0,0,1,0]
=> 1
[[[.,.],[.,.]],.]
=> [1,0,1,1,0,0,1,0]
=> 2
[[[.,[.,.]],.],.]
=> [1,1,0,0,1,0,1,0]
=> 1
[[[[.,.],.],.],.]
=> [1,0,1,0,1,0,1,0]
=> 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
[.,[.,[.,[[.,.],.]]]]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[.,[.,[[.,.],[.,.]]]]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[.,[.,[[.,[.,.]],.]]]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[.,[.,[[[.,.],.],.]]]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2
[.,[[.,.],[.,[.,.]]]]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[.,[[.,.],[[.,.],.]]]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[.,[[[.,.],.],[.,.]]]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2
[.,[[.,[.,[.,.]]],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[.,[[.,[[.,.],.]],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[.,[[[.,.],[.,.]],.]]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[.,[[[.,[.,.]],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[.,[[[[.,.],.],.],.]]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[[.,.],[.,[.,[.,.]]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[[.,.],[.,[[.,.],.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[[.,.],[[.,.],[.,.]]]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[[.,.],[[.,[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2
[[.,.],[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[[[.,.],.],[.,[.,.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[[[.,.],.],[[.,.],.]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
Description
The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path.
For the projective dimension see [[St001007]] and for 1-regularity see [[St001126]]. After applying the inverse zeta map [[Mp00032]], this statistic matches the number of rises of length at least 2 [[St000659]].
Matching statistic: St001665
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St001665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001665: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 0
[.,[.,.]]
=> [2,1] => 1
[[.,.],.]
=> [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => 1
[[.,.],[.,.]]
=> [3,1,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 2
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 2
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 2
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1
Description
The number of pure excedances of a permutation.
A pure excedance of a permutation $\pi$ is a position $i < \pi_i$ such that there is no $j < i$ with $i\leq \pi_j < \pi_i$.
Matching statistic: St001729
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St001729: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001729: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 0
[.,[.,.]]
=> [2,1] => 1
[[.,.],.]
=> [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => 1
[[.,.],[.,.]]
=> [3,1,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 2
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 2
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => 2
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 2
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => 1
Description
The number of visible descents of a permutation.
A visible descent of a permutation $\pi$ is a position $i$ such that $\pi(i+1) \leq \min(i, \pi(i))$.
Matching statistic: St001737
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
St001737: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001737: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => 0
[.,[.,.]]
=> [2,1] => 1
[[.,.],.]
=> [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => 1
[[.,.],[.,.]]
=> [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => 2
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => 1
Description
The number of descents of type 2 in a permutation.
A position $i\in[1,n-1]$ is a descent of type 2 of a permutation $\pi$ of $n$ letters, if it is a descent and if $\pi(j) < \pi(i)$ for all $j < i$.
Matching statistic: St000021
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00141: Binary trees —pruning number to logarithmic height⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1] => 0
[.,[.,.]]
=> [1,0,1,0]
=> [2,1] => 1
[[.,.],.]
=> [1,1,0,0]
=> [1,2] => 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [2,1,3] => 1
[[.,.],[.,.]]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[[.,[.,.]],.]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1
[[[.,.],.],.]
=> [1,1,0,1,0,0]
=> [3,1,2] => 1
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[.,[[.,.],[.,.]]]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[.,[[[.,.],.],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 1
[[.,.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[[.,.],[[.,.],.]]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[[.,[.,.]],[.,.]]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => 1
[[[.,.],.],[.,.]]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[[.,[.,[.,.]]],.]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[[.,[[.,.],.]],.]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[[[.,.],[.,.]],.]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => 1
[[[.,[.,.]],.],.]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 2
[[[[.,.],.],.],.]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => 1
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 1
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 1
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => 1
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 2
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 1
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 2
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 2
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => 1
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => 2
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => 2
[[.,.],[[[.,.],.],.]]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => 1
[[.,[.,.]],[.,[.,.]]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[[[.,.],.],[[.,.],.]]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => 1
[[.,[.,[.,.]]],[.,.]]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => 1
[[.,[[.,.],.]],[.,.]]
=> [1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => 1
Description
The number of descents of a permutation.
This can be described as an occurrence of the vincular mesh pattern ([2,1], {(1,0),(1,1),(1,2)}), i.e., the middle column is shaded, see [3].
Matching statistic: St000155
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(load all 4 compositions to match this statistic)
Mp00017: Binary trees —to 312-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => 0
[.,[.,.]]
=> [2,1] => [2,1] => 1
[[.,.],.]
=> [1,2] => [1,2] => 0
[.,[.,[.,.]]]
=> [3,2,1] => [3,1,2] => 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => 1
[[.,.],[.,.]]
=> [1,3,2] => [1,3,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,1,2,3] => 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,1,3,2] => 1
[.,[[.,.],[.,.]]]
=> [2,4,3,1] => [4,2,1,3] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,3,2,1] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,2,3,1] => 1
[[.,.],[.,[.,.]]]
=> [1,4,3,2] => [1,4,2,3] => 1
[[.,.],[[.,.],.]]
=> [1,3,4,2] => [1,4,3,2] => 1
[[.,[.,.]],[.,.]]
=> [2,1,4,3] => [2,1,4,3] => 2
[[[.,.],.],[.,.]]
=> [1,2,4,3] => [1,2,4,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,1,2,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => 1
[[[.,.],[.,.]],.]
=> [1,3,2,4] => [1,3,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,1,2,3,4] => 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,1,2,4,3] => 1
[.,[.,[[.,.],[.,.]]]]
=> [3,5,4,2,1] => [5,1,3,2,4] => 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,1,4,3,2] => 2
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,1,3,4,2] => 1
[.,[[.,.],[.,[.,.]]]]
=> [2,5,4,3,1] => [5,2,1,3,4] => 1
[.,[[.,.],[[.,.],.]]]
=> [2,4,5,3,1] => [5,2,1,4,3] => 1
[.,[[.,[.,.]],[.,.]]]
=> [3,2,5,4,1] => [5,3,2,1,4] => 2
[.,[[[.,.],.],[.,.]]]
=> [2,3,5,4,1] => [5,2,3,1,4] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,4,2,3,1] => 2
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,3,2,1] => 2
[.,[[[.,.],[.,.]],.]]
=> [2,4,3,5,1] => [5,2,4,3,1] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,3,2,4,1] => 2
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,2,3,4,1] => 1
[[.,.],[.,[.,[.,.]]]]
=> [1,5,4,3,2] => [1,5,2,3,4] => 1
[[.,.],[.,[[.,.],.]]]
=> [1,4,5,3,2] => [1,5,2,4,3] => 1
[[.,.],[[.,.],[.,.]]]
=> [1,3,5,4,2] => [1,5,3,2,4] => 1
[[.,.],[[.,[.,.]],.]]
=> [1,4,3,5,2] => [1,5,4,3,2] => 2
[[.,.],[[[.,.],.],.]]
=> [1,3,4,5,2] => [1,5,3,4,2] => 1
[[.,[.,.]],[.,[.,.]]]
=> [2,1,5,4,3] => [2,1,5,3,4] => 2
[[.,[.,.]],[[.,.],.]]
=> [2,1,4,5,3] => [2,1,5,4,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,2,5,4,3] => [1,2,5,3,4] => 1
[[[.,.],.],[[.,.],.]]
=> [1,2,4,5,3] => [1,2,5,4,3] => 1
[[.,[.,[.,.]]],[.,.]]
=> [3,2,1,5,4] => [3,1,2,5,4] => 2
[[.,[[.,.],.]],[.,.]]
=> [2,3,1,5,4] => [3,2,1,5,4] => 2
[[[.,.],[.,.]],[.,.]]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[[[[.,.],.],.],[.,.]]
=> [1,2,3,5,4] => [1,2,3,5,4] => 1
Description
The number of exceedances (also excedences) of a permutation.
This is defined as $exc(\sigma) = \#\{ i : \sigma(i) > i \}$.
It is known that the number of exceedances is equidistributed with the number of descents, and that the bistatistic $(exc,den)$ is [[Permutations/Descents-Major#Euler-Mahonian_statistics|Euler-Mahonian]]. Here, $den$ is the Denert index of a permutation, see [[St000156]].
The following 140 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000245The number of ascents of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000662The staircase size of the code of a permutation. St000670The reversal length of a permutation. St000703The number of deficiencies of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000884The number of isolated descents of a permutation. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001354The number of series nodes in the modular decomposition of a graph. St001928The number of non-overlapping descents in a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000142The number of even parts of a partition. St000157The number of descents of a standard tableau. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000211The rank of the set partition. St000291The number of descents of a binary word. St000340The number of non-final maximal constant sub-paths of length greater than one. St000386The number of factors DDU in a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000665The number of rafts of a permutation. St000919The number of maximal left branches of a binary tree. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001280The number of parts of an integer partition that are at least two. St001427The number of descents of a signed permutation. St001489The maximum of the number of descents and the number of inverse descents. St001657The number of twos in an integer partition. St001726The number of visible inversions of a permutation. St001840The number of descents of a set partition. St001874Lusztig's a-function for the symmetric group. St000010The length of the partition. St000201The number of leaf nodes in a binary tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000507The number of ascents of a standard tableau. St000935The number of ordered refinements of an integer partition. St001389The number of partitions of the same length below the given integer partition. St000251The number of nonsingleton blocks of a set partition. St000254The nesting number of a set partition. St000354The number of recoils of a permutation. St000390The number of runs of ones in a binary word. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000253The crossing number of a set partition. St000288The number of ones in a binary word. St000389The number of runs of ones of odd length in a binary word. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000829The Ulam distance of a permutation to the identity permutation. St000702The number of weak deficiencies of a permutation. St000619The number of cyclic descents of a permutation. St000306The bounce count of a Dyck path. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000668The least common multiple of the parts of the partition. St000259The diameter of a connected graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St000260The radius of a connected graph. St000647The number of big descents of a permutation. St001330The hat guessing number of a graph. St001568The smallest positive integer that does not appear twice in the partition. St000353The number of inner valleys of a permutation. St001469The holeyness of a permutation. St001394The genus of a permutation. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000646The number of big ascents of a permutation. St000711The number of big exceedences of a permutation. St000023The number of inner peaks of a permutation. St000317The cycle descent number of a permutation. St000456The monochromatic index of a connected graph. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001274The number of indecomposable injective modules with projective dimension equal to two. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001728The number of invisible descents of a permutation. St000015The number of peaks of a Dyck path. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000455The second largest eigenvalue of a graph if it is integral. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000815The number of semistandard Young tableaux of partition weight of given shape. 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. St000993The multiplicity of the largest part of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St000454The largest eigenvalue of a graph if it is integral. St001877Number of indecomposable injective modules with projective dimension 2. St001864The number of excedances of a signed permutation. St001896The number of right descents of a signed permutations. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000534The number of 2-rises of a permutation. St001597The Frobenius rank of a skew partition. St001060The distinguishing index of a graph. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St000392The length of the longest run of ones in a binary word. St000758The length of the longest staircase fitting into an integer composition. St000982The length of the longest constant subword. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St000920The logarithmic height of a Dyck path. St001624The breadth of a lattice.
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