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Your data matches 37 different statistics following compositions of up to 3 maps.
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Matching statistic: St000155
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Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000155: 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] => 2
[[.,.],[.,.]]
=> [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] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => 3
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => 2
[[.,.],[[.,.],.]]
=> [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] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => 3
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => 3
[.,[[.,.],[.,[.,.]]]]
=> [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] => 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => 4
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => 3
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => 3
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [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 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]].
Matching statistic: St000021
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000021: 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] => [2,3,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => 2
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,4,1] => 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,2,3,1] => 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,4,1] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,4,3,1] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,3,2,1] => 3
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,4,2] => 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4,1,3,2] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,4,1,3] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,2,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,1,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [3,4,2,5,1] => 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [3,5,2,4,1] => 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,3,2,5,1] => 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,3,2,4,1] => 3
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,2,4,3,1] => 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,5,1] => 2
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,2,3,4,1] => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,5,1] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,5,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,5,4,1] => 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,2,3,1] => 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,3,5,4,1] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,5,4,3,1] => 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,4,3,2,1] => 4
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [3,4,1,5,2] => 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [3,5,1,4,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [4,3,1,5,2] => 3
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [5,3,1,4,2] => 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [5,1,4,3,2] => 3
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,5,1,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [5,2,4,1,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,5,2,3] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [5,1,4,2,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,5,1,4] => 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,5,1,4] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,5,2,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,5,1,3,4] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [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: St000120
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(load all 5 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00028: Dyck paths —reverse⟶ Dyck paths
St000120: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> 0
[.,[.,.]]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[[.,.],.]
=> [1,1,0,0]
=> [1,1,0,0]
=> 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 3
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 3
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 3
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
Description
The number of left tunnels of a Dyck path.
A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b
Matching statistic: St000211
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => {{1}}
=> 0
[.,[.,.]]
=> [2,1] => {{1,2}}
=> 1
[[.,.],.]
=> [1,2] => {{1},{2}}
=> 0
[.,[.,[.,.]]]
=> [3,2,1] => {{1,3},{2}}
=> 1
[.,[[.,.],.]]
=> [2,3,1] => {{1,2,3}}
=> 2
[[.,.],[.,.]]
=> [3,1,2] => {{1,3},{2}}
=> 1
[[.,[.,.]],.]
=> [2,1,3] => {{1,2},{3}}
=> 1
[[[.,.],.],.]
=> [1,2,3] => {{1},{2},{3}}
=> 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => {{1,4},{2,3}}
=> 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => {{1,3},{2,4}}
=> 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => {{1,4},{2},{3}}
=> 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => {{1,3,4},{2}}
=> 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => {{1,2,3,4}}
=> 3
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => {{1,4},{2,3}}
=> 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => {{1,3},{2,4}}
=> 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => {{1,4},{2},{3}}
=> 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => {{1,4},{2},{3}}
=> 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => {{1,3},{2},{4}}
=> 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => {{1,2,3},{4}}
=> 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => {{1,3},{2},{4}}
=> 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => {{1,2},{3},{4}}
=> 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => {{1},{2},{3},{4}}
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => {{1,5},{2,4},{3}}
=> 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => {{1,4},{2,5},{3}}
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => {{1,5},{2,3,4}}
=> 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => {{1,4},{2,3,5}}
=> 3
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => {{1,3,5},{2,4}}
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => {{1,5},{2,4},{3}}
=> 2
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => {{1,4},{2,5},{3}}
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => {{1,5},{2,3},{4}}
=> 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => {{1,5},{2},{3},{4}}
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => {{1,4,5},{2,3}}
=> 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => {{1,3},{2,4,5}}
=> 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => {{1,4,5},{2},{3}}
=> 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => {{1,3,4,5},{2}}
=> 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => {{1,2,3,4,5}}
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => {{1,5},{2,4},{3}}
=> 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => {{1,4},{2,5},{3}}
=> 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => {{1,5},{2,3,4}}
=> 3
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => {{1,4},{2,3,5}}
=> 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => {{1,3,5},{2,4}}
=> 3
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => {{1,5},{2,4},{3}}
=> 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => {{1,5},{2,4},{3}}
=> 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => {{1,4},{2,5},{3}}
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => {{1,5},{2,3},{4}}
=> 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => {{1,5},{2,3},{4}}
=> 2
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => {{1,5},{2},{3},{4}}
=> 1
Description
The rank of the set partition.
This is defined as the number of elements in the set partition minus the number of blocks, or, equivalently, the number of arcs in the one-line diagram associated to the set partition.
Matching statistic: St000329
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1,0]
=> [1,0]
=> 0
[.,[.,.]]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[[.,.],.]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 3
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 3
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 3
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St000703
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00066: Permutations —inverse⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00066: Permutations —inverse⟶ Permutations
St000703: 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] => [3,2,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => [3,1,2] => 2
[[.,.],[.,.]]
=> [3,1,2] => [2,3,1] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,3,1,2] => 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,2,3,1] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [4,2,1,3] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,1,2,3] => 3
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,4,2,1] => 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [3,2,4,1] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [2,3,4,1] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,1,2,4] => 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [2,3,1,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [5,4,3,1,2] => 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,4,2,3,1] => 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,4,2,1,3] => 3
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,4,1,2,3] => 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,3,4,2,1] => 2
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,3,4,1,2] => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,2,4,1] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,2,3,4,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [5,3,2,1,4] => 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,3,1,2,4] => 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [5,2,3,1,4] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [5,2,1,3,4] => 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,1,2,3,4] => 4
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [4,5,3,2,1] => 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,5,3,1,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [4,5,2,3,1] => 3
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,5,2,1,3] => 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [4,5,1,2,3] => 3
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,3,5,2,1] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,3,5,1,2] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [3,4,5,2,1] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [3,4,5,1,2] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [4,3,2,5,1] => 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [4,2,3,5,1] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,4,2,5,1] => 2
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [3,2,4,5,1] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [2,3,4,5,1] => 1
Description
The number of deficiencies of a permutation.
This is defined as
$$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$
The number of exceedances is [[St000155]].
Matching statistic: St000996
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St000996: 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]
=> [2,1,3] => 1
[.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> [2,3,1] => 2
[[.,.],[.,.]]
=> [1,1,0,0,1,0]
=> [3,1,2] => 1
[[.,[.,.]],.]
=> [1,1,0,1,0,0]
=> [1,3,2] => 1
[[[.,.],.],.]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 2
[.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 2
[.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 1
[.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> [2,3,1,4] => 2
[.,[[[.,.],.],.]]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3
[[.,.],[.,[.,.]]]
=> [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 2
[[.,.],[[.,.],.]]
=> [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2
[[.,[.,.]],[.,.]]
=> [1,1,0,1,0,0,1,0]
=> [3,1,2,4] => 1
[[[.,.],.],[.,.]]
=> [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
[[.,[.,[.,.]]],.]
=> [1,1,0,1,0,1,0,0]
=> [1,3,2,4] => 1
[[.,[[.,.],.]],.]
=> [1,1,0,1,1,0,0,0]
=> [1,3,4,2] => 2
[[[.,.],[.,.]],.]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => 1
[[[.,[.,.]],.],.]
=> [1,1,1,0,1,0,0,0]
=> [1,2,4,3] => 1
[[[[.,.],.],.],.]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 2
[.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 2
[.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => 3
[.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => 3
[.,[.,[[[.,.],.],.]]]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 3
[.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 2
[.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 2
[.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => 2
[.,[[[.,.],.],[.,.]]]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 1
[.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => 3
[.,[[.,[[.,.],.]],.]]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => 3
[.,[[[.,.],[.,.]],.]]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => 2
[.,[[[.,[.,.]],.],.]]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => 3
[.,[[[[.,.],.],.],.]]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4
[[.,.],[.,[.,[.,.]]]]
=> [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => 2
[[.,.],[.,[[.,.],.]]]
=> [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 2
[[.,.],[[.,.],[.,.]]]
=> [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 3
[[.,.],[[.,[.,.]],.]]
=> [1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => 3
[[.,.],[[[.,.],.],.]]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3
[[.,[.,.]],[.,[.,.]]]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => 2
[[.,[.,.]],[[.,.],.]]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => 2
[[[.,.],.],[.,[.,.]]]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => 2
[[[.,.],.],[[.,.],.]]
=> [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => 2
[[.,[[.,.],.]],[.,.]]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => 1
[[[.,.],[.,.]],[.,.]]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1,2,5,3] => 2
[[[.,[.,.]],.],[.,.]]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,3,5] => 1
[[[[.,.],.],.],[.,.]]
=> [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St001907
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001907: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001907: Signed 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] => [3,2,1] => 1
[.,[[.,.],.]]
=> [2,3,1] => [2,3,1] => 2
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 0
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [4,3,2,1] => 2
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [3,4,2,1] => 2
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [4,2,3,1] => 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [3,2,4,1] => 2
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [2,3,4,1] => 3
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [4,3,1,2] => 2
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [3,4,1,2] => 2
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [4,2,1,3] => 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,2,3] => 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [3,2,1,4] => 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [2,3,1,4] => 2
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2,4] => 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 0
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [5,4,3,2,1] => 2
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [4,5,3,2,1] => 2
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [5,3,4,2,1] => 3
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [4,3,5,2,1] => 3
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [3,4,5,2,1] => 3
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [5,4,2,3,1] => 2
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [4,5,2,3,1] => 2
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [5,3,2,4,1] => 2
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [5,2,3,4,1] => 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [4,3,2,5,1] => 3
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [3,4,2,5,1] => 3
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [4,2,3,5,1] => 2
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [3,2,4,5,1] => 3
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [2,3,4,5,1] => 4
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [5,4,3,1,2] => 2
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [4,5,3,1,2] => 2
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [5,3,4,1,2] => 3
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [4,3,5,1,2] => 3
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [3,4,5,1,2] => 3
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [5,4,2,1,3] => 2
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [4,5,2,1,3] => 2
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [5,4,1,2,3] => 2
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [4,5,1,2,3] => 2
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [5,3,2,1,4] => 2
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [5,2,3,1,4] => 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [5,3,1,2,4] => 2
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [5,2,1,3,4] => 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,2,3,4] => 1
Description
The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation.
For a signed permutation $\sigma$, this equals
$$ \left\lfloor \dfrac{fexc(\sigma)+1}{2} \right\rfloor = exc(\sigma) + \left\lfloor \dfrac{neg(\sigma)+1}{2} \right\rfloor, $$
where
$$fexc(\sigma) = 2exc(\sigma) + neg(\sigma),$$
$$exc(\sigma) = |\{i \in [n-1] \,:\, \sigma(i) > i\}|,$$
$$neg(\sigma) = |\{i \in [n] \,:\, \sigma(i) < 0\}|.$$
This statistic has the same distribution as the descent statistic [[St001427]].
Matching statistic: St000325
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000325: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => 1 = 0 + 1
[.,[.,.]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[.,.],.]
=> [1,2] => [1,2] => 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,3,1] => 2 = 1 + 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => 3 = 2 + 1
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,4,1] => 3 = 2 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,2,3,1] => 3 = 2 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,4,1] => 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,4,3,1] => 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,3,2,1] => 4 = 3 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,4,2] => 3 = 2 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4,1,3,2] => 3 = 2 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,4,1,3] => 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,2,3] => 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,1,4] => 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2,4] => 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [3,4,2,5,1] => 3 = 2 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [3,5,2,4,1] => 3 = 2 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,3,2,5,1] => 4 = 3 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,3,2,4,1] => 4 = 3 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,2,4,3,1] => 4 = 3 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,5,1] => 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,2,3,4,1] => 3 = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,5,1] => 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,5,1] => 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,5,4,1] => 4 = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,2,3,1] => 4 = 3 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,3,5,4,1] => 3 = 2 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,5,4,3,1] => 4 = 3 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,4,3,2,1] => 5 = 4 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [3,4,1,5,2] => 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [3,5,1,4,2] => 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [4,3,1,5,2] => 4 = 3 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [5,3,1,4,2] => 4 = 3 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [5,1,4,3,2] => 4 = 3 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,5,1,3] => 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [5,2,4,1,3] => 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,5,2,3] => 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [5,1,4,2,3] => 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,5,1,4] => 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,5,1,4] => 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,5,2,4] => 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,5,1,3,4] => 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,2,3,4] => 2 = 1 + 1
Description
The width of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The width of the tree is given by the number of leaves of this tree.
Note that, due to the construction of this tree, the width of the tree is always one more than the number of descents [[St000021]]. This also matches the number of runs in a permutation [[St000470]].
See also [[St000308]] for the height of this tree.
Matching statistic: St000470
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00014: Binary trees —to 132-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000470: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[.,.]
=> [1] => [1] => 1 = 0 + 1
[.,[.,.]]
=> [2,1] => [2,1] => 2 = 1 + 1
[[.,.],.]
=> [1,2] => [1,2] => 1 = 0 + 1
[.,[.,[.,.]]]
=> [3,2,1] => [2,3,1] => 2 = 1 + 1
[.,[[.,.],.]]
=> [2,3,1] => [3,2,1] => 3 = 2 + 1
[[.,.],[.,.]]
=> [3,1,2] => [3,1,2] => 2 = 1 + 1
[[.,[.,.]],.]
=> [2,1,3] => [2,1,3] => 2 = 1 + 1
[[[.,.],.],.]
=> [1,2,3] => [1,2,3] => 1 = 0 + 1
[.,[.,[.,[.,.]]]]
=> [4,3,2,1] => [3,2,4,1] => 3 = 2 + 1
[.,[.,[[.,.],.]]]
=> [3,4,2,1] => [4,2,3,1] => 3 = 2 + 1
[.,[[.,.],[.,.]]]
=> [4,2,3,1] => [2,3,4,1] => 2 = 1 + 1
[.,[[.,[.,.]],.]]
=> [3,2,4,1] => [2,4,3,1] => 3 = 2 + 1
[.,[[[.,.],.],.]]
=> [2,3,4,1] => [4,3,2,1] => 4 = 3 + 1
[[.,.],[.,[.,.]]]
=> [4,3,1,2] => [3,1,4,2] => 3 = 2 + 1
[[.,.],[[.,.],.]]
=> [3,4,1,2] => [4,1,3,2] => 3 = 2 + 1
[[.,[.,.]],[.,.]]
=> [4,2,1,3] => [2,4,1,3] => 2 = 1 + 1
[[[.,.],.],[.,.]]
=> [4,1,2,3] => [4,1,2,3] => 2 = 1 + 1
[[.,[.,[.,.]]],.]
=> [3,2,1,4] => [2,3,1,4] => 2 = 1 + 1
[[.,[[.,.],.]],.]
=> [2,3,1,4] => [3,2,1,4] => 3 = 2 + 1
[[[.,.],[.,.]],.]
=> [3,1,2,4] => [3,1,2,4] => 2 = 1 + 1
[[[.,[.,.]],.],.]
=> [2,1,3,4] => [2,1,3,4] => 2 = 1 + 1
[[[[.,.],.],.],.]
=> [1,2,3,4] => [1,2,3,4] => 1 = 0 + 1
[.,[.,[.,[.,[.,.]]]]]
=> [5,4,3,2,1] => [3,4,2,5,1] => 3 = 2 + 1
[.,[.,[.,[[.,.],.]]]]
=> [4,5,3,2,1] => [3,5,2,4,1] => 3 = 2 + 1
[.,[.,[[.,.],[.,.]]]]
=> [5,3,4,2,1] => [4,3,2,5,1] => 4 = 3 + 1
[.,[.,[[.,[.,.]],.]]]
=> [4,3,5,2,1] => [5,3,2,4,1] => 4 = 3 + 1
[.,[.,[[[.,.],.],.]]]
=> [3,4,5,2,1] => [5,2,4,3,1] => 4 = 3 + 1
[.,[[.,.],[.,[.,.]]]]
=> [5,4,2,3,1] => [4,2,3,5,1] => 3 = 2 + 1
[.,[[.,.],[[.,.],.]]]
=> [4,5,2,3,1] => [5,2,3,4,1] => 3 = 2 + 1
[.,[[.,[.,.]],[.,.]]]
=> [5,3,2,4,1] => [3,2,4,5,1] => 3 = 2 + 1
[.,[[[.,.],.],[.,.]]]
=> [5,2,3,4,1] => [2,3,4,5,1] => 2 = 1 + 1
[.,[[.,[.,[.,.]]],.]]
=> [4,3,2,5,1] => [3,2,5,4,1] => 4 = 3 + 1
[.,[[.,[[.,.],.]],.]]
=> [3,4,2,5,1] => [5,4,2,3,1] => 4 = 3 + 1
[.,[[[.,.],[.,.]],.]]
=> [4,2,3,5,1] => [2,3,5,4,1] => 3 = 2 + 1
[.,[[[.,[.,.]],.],.]]
=> [3,2,4,5,1] => [2,5,4,3,1] => 4 = 3 + 1
[.,[[[[.,.],.],.],.]]
=> [2,3,4,5,1] => [5,4,3,2,1] => 5 = 4 + 1
[[.,.],[.,[.,[.,.]]]]
=> [5,4,3,1,2] => [3,4,1,5,2] => 3 = 2 + 1
[[.,.],[.,[[.,.],.]]]
=> [4,5,3,1,2] => [3,5,1,4,2] => 3 = 2 + 1
[[.,.],[[.,.],[.,.]]]
=> [5,3,4,1,2] => [4,3,1,5,2] => 4 = 3 + 1
[[.,.],[[.,[.,.]],.]]
=> [4,3,5,1,2] => [5,3,1,4,2] => 4 = 3 + 1
[[.,.],[[[.,.],.],.]]
=> [3,4,5,1,2] => [5,1,4,3,2] => 4 = 3 + 1
[[.,[.,.]],[.,[.,.]]]
=> [5,4,2,1,3] => [4,2,5,1,3] => 3 = 2 + 1
[[.,[.,.]],[[.,.],.]]
=> [4,5,2,1,3] => [5,2,4,1,3] => 3 = 2 + 1
[[[.,.],.],[.,[.,.]]]
=> [5,4,1,2,3] => [4,1,5,2,3] => 3 = 2 + 1
[[[.,.],.],[[.,.],.]]
=> [4,5,1,2,3] => [5,1,4,2,3] => 3 = 2 + 1
[[.,[.,[.,.]]],[.,.]]
=> [5,3,2,1,4] => [3,2,5,1,4] => 3 = 2 + 1
[[.,[[.,.],.]],[.,.]]
=> [5,2,3,1,4] => [2,3,5,1,4] => 2 = 1 + 1
[[[.,.],[.,.]],[.,.]]
=> [5,3,1,2,4] => [3,1,5,2,4] => 3 = 2 + 1
[[[.,[.,.]],.],[.,.]]
=> [5,2,1,3,4] => [2,5,1,3,4] => 2 = 1 + 1
[[[[.,.],.],.],[.,.]]
=> [5,1,2,3,4] => [5,1,2,3,4] => 2 = 1 + 1
Description
The number of runs in a permutation.
A run in a permutation is an inclusion-wise maximal increasing substring, i.e., a contiguous subsequence.
This is the same as the number of descents plus 1.
The following 27 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000053The number of valleys of the Dyck path. St000157The number of descents of a standard tableau. St000245The number of ascents of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000662The staircase size of the code of a permutation. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St000015The number of peaks of a Dyck path. St000105The number of blocks in the set partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000314The number of left-to-right-maxima of a permutation. St000676The number of odd rises of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000702The number of weak deficiencies of a permutation. St000216The absolute length of a permutation. St000288The number of ones in a binary word. St000354The number of recoils of a permutation. St000809The reduced reflection length of the permutation. St000829The Ulam distance of a permutation to the identity permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001427The number of descents of a signed permutation. St001152The number of pairs with even minimum in a perfect matching. St001864The number of excedances of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001946The number of descents in a parking function.
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