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Your data matches 173 different statistics following compositions of up to 3 maps.
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Matching statistic: St000331
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
St000331: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 1
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> 3
[1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> 1
Description
The number of upper interactions of a Dyck path.
An ''upper interaction'' in a Dyck path is defined as the occurrence of a factor '''$A^{k}$$B^{k}$''' for any '''${k ≥ 1}$''', where '''${A}$''' is a down-step and '''${B}$''' is a up-step.
Matching statistic: St000021
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00066: Permutations —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
Mp00066: Permutations —inverse⟶ Permutations
St000021: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [3,2,1] => 2
[1,1,1,0,0,0]
=> [3,2,1] => [2,3,1] => [3,1,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,4,3,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,3,4,2] => [1,4,2,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [2,1,4,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 3
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,4,2,1] => [4,3,1,2] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,3,1,4] => [3,1,2,4] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,4,3,1] => [4,1,3,2] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4,2,3,1] => [4,2,3,1] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [3,2,4,1] => [4,2,1,3] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,2,5,4,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,4,5,3] => [1,2,5,3,4] => 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,4,3,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [1,5,4,3,2] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,4,5,3,2] => [1,5,4,2,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,3,4,2,5] => [1,4,2,3,5] => 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,3,5,4,2] => [1,5,2,4,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [1,5,3,4,2] => [1,5,3,4,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,4,3,5,2] => [1,5,3,2,4] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,1,5,4,3] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,4,5,3] => [2,1,5,3,4] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2,1,5,4] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => 4
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,5,3,2,1] => [5,4,3,1,2] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,4,2,1,5] => [4,3,1,2,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,5,4,2,1] => [5,4,1,3,2] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5,3,4,2,1] => [5,4,2,3,1] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [4,3,5,2,1] => [5,4,2,1,3] => 3
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
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000155: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,1,0,0]
=> [1,2] => [1,2] => [2,1] => 1
[1,0,1,0,1,0]
=> [2,3,1] => [2,3,1] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [1,3,2] => [3,1,2] => [3,1,2] => 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,3,2] => [3,2,1] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [2,3,1] => 2
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,4,1] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1,4] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,4,1,3] => [1,4,2,3] => 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => [2,4,1,3] => 2
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,4,1,2] => [4,1,2,3] => 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1,2,4] => [3,1,4,2] => 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,3,4,2] => [4,2,3,1] => 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => [4,1,3,2] => 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,3,2,4] => [3,2,4,1] => 2
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [4,1,2,3] => [3,4,1,2] => 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => [3,4,2,1] => 2
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,4,3] => [2,4,3,1] => 2
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => [1,2,5,3,4] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [2,3,5,1,4] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => [1,5,2,3,4] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => [1,4,2,5,3] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [2,5,1,3,4] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,5,2,1,3] => [3,5,1,2,4] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [2,4,1,5,3] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => [1,4,5,2,3] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [5,2,1,3,4] => [2,4,5,1,3] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,5,3,4] => [1,4,5,3,2] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,2] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => [5,1,2,3,4] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => [4,1,2,5,3] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => [4,1,5,2,3] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [5,3,1,2,4] => [4,2,5,1,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [3,1,4,5,2] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => [5,2,3,4,1] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,3,4,2,5] => [4,2,3,5,1] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,1,4,5,2] => [5,1,3,4,2] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,5,2] => [5,1,2,4,3] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => [4,1,3,5,2] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,3,5,2,4] => [4,2,5,3,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => [4,1,5,3,2] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [1,5,3,2,4] => [4,3,5,2,1] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => 3
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: St000157
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00115: Set partitions —Kasraoui-Zeng⟶ Set partitions
Mp00258: Set partitions —Standard tableau associated to a set partition⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> {{1}}
=> {{1}}
=> [[1]]
=> 0
[1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> [[1],[2]]
=> 1
[1,1,0,0]
=> {{1,2}}
=> {{1,2}}
=> [[1,2]]
=> 0
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> [[1],[2],[3]]
=> 2
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1},{2,3}}
=> [[1,3],[2]]
=> 1
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1,2},{3}}
=> [[1,2],[3]]
=> 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> [[1,3],[2]]
=> 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> [[1,2,3]]
=> 0
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [[1],[2],[3],[4]]
=> 3
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1},{2},{3,4}}
=> [[1,4],[2],[3]]
=> 2
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> [[1,3],[2],[4]]
=> 2
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1},{2,4},{3}}
=> [[1,4],[2],[3]]
=> 2
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1},{2,3,4}}
=> [[1,3,4],[2]]
=> 1
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1,2},{3},{4}}
=> [[1,2],[3],[4]]
=> 2
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> [[1,2],[3,4]]
=> 1
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1,3},{2},{4}}
=> [[1,3],[2],[4]]
=> 2
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> [[1,4],[2],[3]]
=> 2
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1,3,4},{2}}
=> [[1,3,4],[2]]
=> 1
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1,2,3},{4}}
=> [[1,2,3],[4]]
=> 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,3},{2,4}}
=> [[1,3],[2,4]]
=> 2
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,2,4},{3}}
=> [[1,2,4],[3]]
=> 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> [[1,2,3,4]]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [[1],[2],[3],[4],[5]]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1},{2},{3},{4,5}}
=> [[1,5],[2],[3],[4]]
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> {{1},{2},{3,4},{5}}
=> [[1,4],[2],[3],[5]]
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1},{2},{3,5},{4}}
=> [[1,5],[2],[3],[4]]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1},{2},{3,4,5}}
=> [[1,4,5],[2],[3]]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> {{1},{2,3},{4,5}}
=> [[1,3],[2,5],[4]]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> [[1,4],[2],[3],[5]]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1},{2,5},{3},{4}}
=> [[1,5],[2],[3],[4]]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1},{2,4,5},{3}}
=> [[1,4,5],[2],[3]]
=> 2
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> [[1,3,4],[2],[5]]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1},{2,4},{3,5}}
=> [[1,4],[2,5],[3]]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1},{2,3,5},{4}}
=> [[1,3,5],[2],[4]]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1},{2,3,4,5}}
=> [[1,3,4,5],[2]]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [[1,2],[3],[4],[5]]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> [[1,2],[3,5],[4]]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> {{1,2},{3,4},{5}}
=> [[1,2],[3,4],[5]]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,2},{3,5},{4}}
=> [[1,2],[3,5],[4]]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,2},{3,4,5}}
=> [[1,2,5],[3,4]]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> [[1,3],[2],[4],[5]]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> [[1,3],[2,5],[4]]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> {{1,4},{2},{3},{5}}
=> [[1,4],[2],[3],[5]]
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> [[1,5],[2],[3],[4]]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> {{1,4,5},{2},{3}}
=> [[1,4,5],[2],[3]]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> {{1,3,4},{2},{5}}
=> [[1,3,4],[2],[5]]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> {{1,4},{2},{3,5}}
=> [[1,4],[2,5],[3]]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> {{1,3,5},{2},{4}}
=> [[1,3,5],[2],[4]]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> {{1,3,4,5},{2}}
=> [[1,3,4,5],[2]]
=> 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000211
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00216: Set partitions —inverse Wachs-White⟶ Set partitions
Mp00221: Set partitions —conjugate⟶ Set partitions
St000211: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> {{1}}
=> {{1}}
=> {{1}}
=> 0
[1,0,1,0]
=> {{1},{2}}
=> {{1},{2}}
=> {{1,2}}
=> 1
[1,1,0,0]
=> {{1,2}}
=> {{1,2}}
=> {{1},{2}}
=> 0
[1,0,1,0,1,0]
=> {{1},{2},{3}}
=> {{1},{2},{3}}
=> {{1,2,3}}
=> 2
[1,0,1,1,0,0]
=> {{1},{2,3}}
=> {{1,2},{3}}
=> {{1,2},{3}}
=> 1
[1,1,0,0,1,0]
=> {{1,2},{3}}
=> {{1},{2,3}}
=> {{1,3},{2}}
=> 1
[1,1,0,1,0,0]
=> {{1,3},{2}}
=> {{1,3},{2}}
=> {{1},{2,3}}
=> 1
[1,1,1,0,0,0]
=> {{1,2,3}}
=> {{1,2,3}}
=> {{1},{2},{3}}
=> 0
[1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> 3
[1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> {{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> 2
[1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> {{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> 2
[1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> {{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> 2
[1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> {{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> 1
[1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> {{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> 2
[1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> {{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> 1
[1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> {{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> 2
[1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> {{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> 2
[1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> {{1,3},{2,4}}
=> {{1,3},{2,4}}
=> 2
[1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> {{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> 1
[1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> {{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> 1
[1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> {{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> 1
[1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> {{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> {{1,2,3,4,5}}
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> {{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> 3
[1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> {{1},{2,3},{4},{5}}
=> {{1,2,3,5},{4}}
=> 3
[1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> {{1,3},{2},{4},{5}}
=> {{1,2,3},{4,5}}
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> {{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> {{1},{2},{3,4},{5}}
=> {{1,2,4,5},{3}}
=> 3
[1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> {{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> {{1,2,5},{3,4}}
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> {{1,4},{2},{3},{5}}
=> {{1,2},{3,4,5}}
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> {{1,3},{2,4},{5}}
=> {{1,2,4},{3,5}}
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> {{1},{2,3,4},{5}}
=> {{1,2,5},{3},{4}}
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> {{1},{2,5},{3,4}}
=> {{1,4},{2,3},{5}}
=> {{1,2},{3,5},{4}}
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> {{1},{2,3,5},{4}}
=> {{1,3,4},{2},{5}}
=> {{1,2},{3},{4,5}}
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> {{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> {{1},{2},{3},{4,5}}
=> {{1,3,4,5},{2}}
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> {{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> {{1},{2,3},{4,5}}
=> {{1,3,5},{2},{4}}
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> {{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> {{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> {{1},{2},{3,5},{4}}
=> {{1,4,5},{2,3}}
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> {{1,2},{3,5},{4}}
=> {{1,4},{2,3},{5}}
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> {{1},{2,5},{3},{4}}
=> {{1,5},{2,3,4}}
=> 3
[1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> {{1},{2,3,4,5}}
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> {{1,4},{2,5},{3}}
=> {{1,3,4},{2,5}}
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> {{1},{2,4},{3,5}}
=> {{1,3,5},{2,4}}
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> {{1,5},{2,3},{4}}
=> {{1},{2,3,5},{4}}
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> {{1,4},{2},{3,5}}
=> {{1,3},{2,4,5}}
=> 3
[1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> {{1,3,5},{2,4}}
=> {{1},{2,4},{3,5}}
=> 2
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: St000245
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [3,1,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [3,2,1] => 0
[1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => [1,3,2] => 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [4,1,2,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [3,1,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [4,3,1,2] => 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,4,2,3] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [4,2,1,3] => 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [3,2,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,3,2,1] => [4,3,2,1] => 0
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,2,1,3] => [2,4,1,3] => 2
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => [1,3,2,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4,3,1,2] => [1,4,3,2] => 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [1,2,4,3] => 2
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [5,1,2,3,4] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [4,1,2,3,5] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [5,4,1,2,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => [1,5,2,3,4] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [3,1,2,4,5] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [5,3,1,2,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [4,3,1,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [5,4,3,1,2] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [3,5,1,2,4] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => [1,4,2,3,5] => 3
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,5,4,2,3] => [1,5,4,2,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,5,2,4,3] => [5,1,4,2,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => [1,2,5,3,4] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [5,2,1,3,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [4,2,1,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [5,4,2,1,3] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [2,5,1,3,4] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,2,1,4,5] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [5,3,2,1,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => [4,3,2,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => [5,4,3,2,1] => 0
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,3,2,1,4] => [3,5,2,1,4] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [2,4,1,3,5] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5,4,2,1,3] => [2,5,4,1,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => [5,2,4,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [2,1,5,3,4] => 2
Description
The number of ascents of a permutation.
Matching statistic: St000662
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
Mp00069: Permutations —complement⟶ Permutations
St000662: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [2,1] => 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,2] => 0
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [3,1,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,3,1] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [1,3,2] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,2,3] => 0
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [4,3,1,2] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [4,2,3,1] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,2,3] => [4,1,3,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [4,1,2,3] => 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [3,4,2,1] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [3,4,1,2] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [2,4,3,1] => 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [1,4,3,2] => 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,3,1,2] => [1,2,4,3] => 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [2,3,4,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,1,3,2] => [1,4,2,3] => 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [3,1,4,2] => [2,4,1,3] => 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [5,4,3,2,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [5,4,3,1,2] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [5,4,2,3,1] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => [5,4,1,3,2] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,2,5,4,3] => [5,4,1,2,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [5,3,4,2,1] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [5,3,4,1,2] => 3
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => [5,2,4,3,1] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => [5,1,4,3,2] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,2,3] => [5,1,2,4,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,4,3,2,5] => [5,2,3,4,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,2,4,3] => [5,1,4,2,3] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,4,2,5,3] => [5,2,4,1,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [5,1,2,3,4] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [4,5,3,2,1] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [4,5,3,1,2] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [4,5,2,3,1] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => [4,5,1,3,2] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [4,5,1,2,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [3,5,4,2,1] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => [3,5,4,1,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [2,5,4,3,1] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [1,5,4,3,2] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,4,1,2,3] => [1,2,5,4,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,3,1,2,5] => [2,3,5,4,1] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,1,2,4,3] => [1,5,4,2,3] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [4,1,2,5,3] => [2,5,4,1,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,3,1,2] => [1,2,3,5,4] => 1
Description
The staircase size of the code of a permutation.
The code $c(\pi)$ of a permutation $\pi$ of length $n$ is given by the sequence $(c_1,\ldots,c_{n})$ with $c_i = |\{j > i : \pi(j) < \pi(i)\}|$. This is a bijection between permutations and all sequences $(c_1,\ldots,c_n)$ with $0 \leq c_i \leq n-i$.
The staircase size of the code is the maximal $k$ such that there exists a subsequence $(c_{i_k},\ldots,c_{i_1})$ of $c(\pi)$ with $c_{i_j} \geq j$.
This statistic is mapped through [[Mp00062]] to the number of descents, showing that together with the number of inversions [[St000018]] it is Euler-Mahonian.
Matching statistic: St001298
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00326: Permutations —weak order rowmotion⟶ Permutations
St001298: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [2,1] => [2,1] => [1,2] => 1
[1,1,0,0]
=> [1,2] => [1,2] => [2,1] => 0
[1,0,1,0,1,0]
=> [3,2,1] => [3,2,1] => [1,2,3] => 2
[1,0,1,1,0,0]
=> [2,3,1] => [2,3,1] => [2,1,3] => 1
[1,1,0,0,1,0]
=> [3,1,2] => [1,3,2] => [2,3,1] => 1
[1,1,0,1,0,0]
=> [2,1,3] => [2,1,3] => [3,1,2] => 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [3,2,1] => 0
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 3
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,4,2,1] => [3,1,2,4] => 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [2,4,3,1] => [2,1,3,4] => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [2,3,1,4] => 2
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,3,4,1] => [3,2,1,4] => 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [1,4,3,2] => [2,3,4,1] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => [1,3,2,4] => 2
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [2,1,4,3] => [3,4,1,2] => 2
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,2,1,4] => [4,1,2,3] => 2
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,3,1,4] => [4,2,1,3] => 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [1,2,4,3] => [3,4,2,1] => 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [1,3,2,4] => [4,2,3,1] => 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [4,3,1,2] => 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [4,3,2,1] => 0
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,2,3,4,5] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => [4,1,2,3,5] => 3
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [3,5,4,2,1] => [3,1,2,4,5] => 3
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => [3,4,1,2,5] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,4,5,2,1] => [4,3,1,2,5] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [2,5,4,3,1] => [2,1,3,4,5] => 3
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [4,2,5,3,1] => [2,4,1,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [3,2,5,4,1] => [2,3,1,4,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => [2,3,4,1,5] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,4,2,5,1] => [3,2,4,1,5] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [2,3,5,4,1] => [3,2,1,4,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [2,4,3,5,1] => [3,4,2,1,5] => 2
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => [4,2,3,1,5] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [4,3,2,1,5] => 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [1,5,4,3,2] => [2,3,4,5,1] => 3
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [4,1,5,3,2] => [1,4,2,3,5] => 3
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,5,4,2] => [1,3,2,4,5] => 3
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [4,3,1,5,2] => [1,3,4,2,5] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [3,4,1,5,2] => [3,1,4,2,5] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [2,1,5,4,3] => [3,4,5,1,2] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [4,2,1,5,3] => [1,2,4,3,5] => 3
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [3,2,1,5,4] => [4,5,1,2,3] => 3
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [5,1,2,3,4] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,4,2,1,5] => [5,3,1,2,4] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [2,3,1,5,4] => [4,5,2,1,3] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [2,4,3,1,5] => [5,2,1,3,4] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,2,4,1,5] => [5,2,3,1,4] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [5,3,2,1,4] => 1
Description
The number of repeated entries in the Lehmer code of a permutation.
The Lehmer code of a permutation $\pi$ is the sequence $(v_1,\dots,v_n)$, with $v_i=|\{j > i: \pi(j) < \pi(i)\}$. This statistic counts the number of distinct elements in this sequence.
Matching statistic: St001489
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
St001489: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 0
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [1,3,2] => 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [2,3,1] => 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,4,1,2] => [3,1,4,2] => 2
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [1,4,3,2] => 2
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,4,1,3] => [4,2,1,3] => 2
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => [2,3,1,4] => 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => [2,3,4,1] => 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [2,4,3,1] => 2
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [3,2,4,1] => 2
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [3,4,2,1] => 2
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,5,1,2,3] => [1,4,2,5,3] => 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => [1,5,3,2,4] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => [3,1,4,2,5] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => [3,4,1,5,2] => 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,4,1,2] => [3,1,5,4,2] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,5,1,2] => [4,3,1,5,2] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,5,3,1,2] => [4,1,5,3,2] => 3
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [1,5,4,3,2] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => [2,1,5,3,4] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => [4,2,1,3,5] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => [4,5,2,1,3] => 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,2,4,1,3] => [5,2,1,4,3] => 3
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [2,3,1,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => [5,2,3,1,4] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [2,3,4,1,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [2,3,4,5,1] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [2,3,5,4,1] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [2,4,3,1,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => [2,4,3,5,1] => 2
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,5,2,3,1] => [4,2,5,3,1] => 3
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => [2,5,4,3,1] => 3
Description
The maximum of the number of descents and the number of inverse descents.
This is, the maximum of [[St000021]] and [[St000354]].
Matching statistic: St000105
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00240: Permutations —weak exceedance partition⟶ Set partitions
St000105: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => {{1}}
=> 1 = 0 + 1
[1,0,1,0]
=> [2,1] => [2,1] => {{1,2}}
=> 1 = 0 + 1
[1,1,0,0]
=> [1,2] => [1,2] => {{1},{2}}
=> 2 = 1 + 1
[1,0,1,0,1,0]
=> [2,3,1] => [2,3,1] => {{1,2,3}}
=> 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => {{1,2},{3}}
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [1,3,2] => [3,1,2] => {{1,3},{2}}
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [3,1,2] => [1,3,2] => {{1},{2,3}}
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => {{1},{2},{3}}
=> 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [2,3,4,1] => {{1,2,3,4}}
=> 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [2,3,1,4] => {{1,2,3},{4}}
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,4,1,3] => {{1,2,4},{3}}
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => {{1,4},{2},{3}}
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => {{1,2},{3},{4}}
=> 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [3,4,1,2] => {{1,3},{2,4}}
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [3,1,2,4] => {{1,3},{2},{4}}
=> 3 = 2 + 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,3,4,2] => {{1},{2,3,4}}
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => {{1,3,4},{2}}
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,3,2,4] => {{1},{2,3},{4}}
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [4,1,2,3] => {{1,4},{2},{3}}
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => {{1},{2,4},{3}}
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,2,4,3] => {{1},{2},{3,4}}
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => {{1},{2},{3},{4}}
=> 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => {{1,2,3,4,5}}
=> 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => {{1,2,3,4},{5}}
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => {{1,2,3,5},{4}}
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => {{1,5},{2},{3},{4}}
=> 4 = 3 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => {{1,2,3},{4},{5}}
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => {{1,2,4},{3,5}}
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => {{1,2,4},{3},{5}}
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => {{1,4},{2},{3,5}}
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,5,2,1,3] => {{1,4},{2,5},{3}}
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => {{1,4},{2},{3},{5}}
=> 4 = 3 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => {{1,2,5},{3},{4}}
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [5,2,1,3,4] => {{1,5},{2},{3},{4}}
=> 4 = 3 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,1,5,3,4] => {{1,2},{3,5},{4}}
=> 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => {{1,2},{3},{4},{5}}
=> 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => {{1,3,5},{2,4}}
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => {{1,3},{2,4},{5}}
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => {{1,3},{2,5},{4}}
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [5,3,1,2,4] => {{1,5},{2,3},{4}}
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => {{1,3},{2},{4},{5}}
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,3,4,5,2] => {{1},{2,3,4,5}}
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,3,4,2,5] => {{1},{2,3,4},{5}}
=> 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,1,4,5,2] => {{1,3,4,5},{2}}
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [3,4,1,5,2] => {{1,3},{2,4,5}}
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => {{1,3,4},{2},{5}}
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,3,5,2,4] => {{1},{2,3,5},{4}}
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => {{1,3,5},{2},{4}}
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [1,5,3,2,4] => {{1},{2,5},{3},{4}}
=> 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,3,2,4,5] => {{1},{2,3},{4},{5}}
=> 4 = 3 + 1
Description
The number of blocks in the set partition.
The generating function of this statistic yields the famous [[wiki:Stirling numbers of the second kind|Stirling numbers of the second kind]] $S_2(n,k)$ given by the number of [[SetPartitions|set partitions]] of $\{ 1,\ldots,n\}$ into $k$ blocks, see [1].
The following 163 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000213The number of weak exceedances (also weak excedences) of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000507The number of ascents of a standard tableau. St000354The number of recoils of a permutation. St000702The number of weak deficiencies of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000829The Ulam distance of a permutation to the identity permutation. St001427The number of descents of a signed permutation. St001812The biclique partition number of a graph. St000742The number of big ascents of a permutation after prepending zero. St000291The number of descents of a binary word. St000646The number of big ascents of a permutation. St000703The number of deficiencies of a permutation. St000336The leg major index of a standard tableau. St000670The reversal length of a permutation. St000619The number of cyclic descents of a permutation. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000837The number of ascents of distance 2 of a permutation. St000711The number of big exceedences of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001432The order dimension of the partition. St000454The largest eigenvalue of a graph if it is integral. St000259The diameter of a connected graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000260The radius of a connected graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001060The distinguishing index of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000836The number of descents of distance 2 of a permutation. St001875The number of simple modules with projective dimension at most 1. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000144The pyramid weight of the Dyck path. St000316The number of non-left-to-right-maxima of a permutation. St000710The number of big deficiencies of a permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001388The number of non-attacking neighbors of a permutation. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000035The number of left outer peaks of a permutation. St000153The number of adjacent cycles of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000374The number of exclusive right-to-left minima of a permutation. St000443The number of long tunnels of a Dyck path. St000996The number of exclusive left-to-right maxima 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. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001668The number of points of the poset minus the width of the poset. St000628The balance of a binary word. St000080The rank of the poset. St000528The height of a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. 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 $eA$. 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 $eA$. St000264The girth of a graph, which is not a tree. St000460The hook length of the last cell along the main diagonal of an integer partition. St000474Dyson's crank of a partition. St000667The greatest common divisor of the parts of the partition. St000993The multiplicity of the largest part of an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001498The normalised height of a Nakayama algebra with magnitude 1. St001571The Cartan determinant of the integer partition. St001933The largest multiplicity of a part in an integer partition. St001330The hat guessing number of a graph. St000292The number of ascents of a binary word. St001568The smallest positive integer that does not appear twice in the partition. St000390The number of runs of ones in a binary word. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000477The weight of a partition according to Alladi. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001279The sum of the parts of an integer partition that are at least two. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001527The cyclic permutation representation number of an integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001820The size of the image of the pop stack sorting operator. St001720The minimal length of a chain of small intervals in a lattice. St001569The maximal modular displacement of a permutation. St001769The reflection length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001894The depth of a signed permutation. St001896The number of right descents of a signed permutations. St000455The second largest eigenvalue of a graph if it is integral. St001637The number of (upper) dissectors of a poset. St001889The size of the connectivity set of a signed permutation. St000939The number of characters of the symmetric group whose value on the partition is positive. St000451The length of the longest pattern of the form k 1 2. St001864The number of excedances of a signed permutation. St001863The number of weak excedances of a signed permutation. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000381The largest part of an integer composition. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001596The number of two-by-two squares inside a skew partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001497The position of the largest weak excedence of a permutation. St000624The normalized sum of the minimal distances to a greater element. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000039The number of crossings of a permutation. St000120The number of left tunnels of a Dyck path. St000445The number of rises of length 1 of a Dyck path. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001712The number of natural descents of a standard Young tableau. St001811The Castelnuovo-Mumford regularity of a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). 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. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001462The number of factors of a standard tableaux under concatenation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001948The number of augmented double ascents of a permutation. St000808The number of up steps of the associated bargraph. St000907The number of maximal antichains of minimal length in a poset. St000956The maximal displacement of a permutation. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St000493The los statistic of a set partition. St000497The lcb statistic of a set partition. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000392The length of the longest run of ones in a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001626The number of maximal proper sublattices of a lattice. St000386The number of factors DDU in a Dyck path. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St001877Number of indecomposable injective modules with projective dimension 2. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. St000237The number of small exceedances.
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