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Your data matches 137 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St000032
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St000032: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> 2
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 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]
=> 3
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 4
[1,1,0,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> 6
[1,1,0,1,0,1,0,0,1,0]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> 6
[1,0,1,1,0,1,0,1,0,0,1,0]
=> 4
[1,0,1,1,0,1,0,1,0,1,0,0]
=> 5
[1,1,0,0,1,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> 4
[1,1,0,0,1,0,1,1,0,0,1,0]
=> 4
[1,1,0,0,1,0,1,1,0,1,0,0]
=> 6
[1,1,0,0,1,1,0,0,1,0,1,0]
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> 6
[1,1,0,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> 6
[1,1,0,1,0,0,1,1,0,0,1,0]
=> 6
Description
The number of elements smaller than the given Dyck path in the Tamari Order.
Matching statistic: St000082
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00029: Dyck paths —to binary tree: left tree, up step, right tree, down step⟶ Binary trees
St000082: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000082: Binary trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [.,[.,.]]
=> 2
[1,0,1,1,0,0]
=> [[.,.],[.,.]]
=> 2
[1,1,0,0,1,0]
=> [[.,[.,.]],.]
=> 2
[1,1,0,1,0,0]
=> [.,[[.,.],.]]
=> 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]
=> [[.,.],[[.,.],.]]
=> 3
[1,1,0,0,1,0,1,0]
=> [[[.,[.,.]],.],.]
=> 2
[1,1,0,0,1,1,0,0]
=> [[.,[.,.]],[.,.]]
=> 4
[1,1,0,1,0,0,1,0]
=> [[.,[[.,.],.]],.]
=> 3
[1,1,0,1,0,1,0,0]
=> [.,[[[.,.],.],.]]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [[[[.,.],.],.],[.,.]]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [[[[.,.],.],[.,.]],.]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [[[.,.],.],[[.,.],.]]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [[[[.,.],[.,.]],.],.]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [[[.,.],[.,.]],[.,.]]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [[[.,.],[[.,.],.]],.]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [[.,.],[[[.,.],.],.]]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [[[[.,[.,.]],.],.],.]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [[[.,[.,.]],.],[.,.]]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [[[.,[.,.]],[.,.]],.]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [[.,[.,.]],[[.,.],.]]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [[[.,[[.,.],.]],.],.]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [[.,[[.,.],.]],[.,.]]
=> 6
[1,1,0,1,0,1,0,0,1,0]
=> [[.,[[[.,.],.],.]],.]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [.,[[[[.,.],.],.],.]]
=> 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [[[[[.,.],.],.],.],[.,.]]
=> 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [[[[[.,.],.],.],[.,.]],.]
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [[[[.,.],.],.],[[.,.],.]]
=> 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [[[[[.,.],.],[.,.]],.],.]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [[[[.,.],.],[.,.]],[.,.]]
=> 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [[[[.,.],.],[[.,.],.]],.]
=> 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [[[.,.],.],[[[.,.],.],.]]
=> 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [[[[[.,.],[.,.]],.],.],.]
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [[[[.,.],[.,.]],.],[.,.]]
=> 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [[[[.,.],[.,.]],[.,.]],.]
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [[[.,.],[.,.]],[[.,.],.]]
=> 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [[[[.,.],[[.,.],.]],.],.]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [[[.,.],[[.,.],.]],[.,.]]
=> 6
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [[[.,.],[[[.,.],.],.]],.]
=> 4
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [[.,.],[[[[.,.],.],.],.]]
=> 5
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[[[.,[.,.]],.],.],.],.]
=> 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [[[[.,[.,.]],.],.],[.,.]]
=> 4
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [[[[.,[.,.]],.],[.,.]],.]
=> 4
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [[[.,[.,.]],.],[[.,.],.]]
=> 6
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [[[[.,[.,.]],[.,.]],.],.]
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [[[.,[.,.]],[[.,.],.]],.]
=> 6
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [[[[.,[[.,.],.]],.],.],.]
=> 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [[[.,[[.,.],.]],.],[.,.]]
=> 6
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [[[.,[[.,.],.]],[.,.]],.]
=> 6
Description
The number of elements smaller than a binary tree in Tamari order.
Matching statistic: St000110
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000110: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [2,1] => 2
[1,0,1,1,0,0]
=> [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 6
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 6
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => 6
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => 4
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 5
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 4
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 4
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => 6
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => 6
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,1,4,6,5] => 6
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => 6
Description
The number of permutations less than or equal to a permutation in left weak order.
This is the same as the number of permutations less than or equal to the given permutation in right weak order.
Matching statistic: St001346
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001346: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001346: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [1,2] => 2
[1,0,1,1,0,0]
=> [2,3,1] => 2
[1,1,0,0,1,0]
=> [3,1,2] => 2
[1,1,0,1,0,0]
=> [2,1,3] => 3
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 4
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 3
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 6
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 6
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [5,6,3,2,4,1] => 6
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => 4
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => 5
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,1,2] => 4
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,1,2] => 4
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,1,2] => 6
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,1,2] => 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,1,2] => 6
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,1,3] => 6
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,1,3] => 6
Description
The number of parking functions that give the same permutation.
A '''parking function''' $(a_1,\dots,a_n)$ is a list of preferred parking spots of $n$ cars entering a one-way street. Once the cars have parked, the order of the cars gives a permutation of $\{1,\dots,n\}$. This statistic records the number of parking functions that yield the same permutation of cars.
Matching statistic: St001464
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001464: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001464: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [2,1] => 2
[1,0,1,1,0,0]
=> [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => 6
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => 6
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => 6
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => 4
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 5
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 4
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 4
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => 6
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => 6
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,1,4,6,5] => 6
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => 6
Description
The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise.
Matching statistic: St000071
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00232: Dyck paths —parallelogram poset⟶ Posets
St000071: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [1,1,1,0,0,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 4
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,7),(2,6),(3,1),(3,6),(4,5),(5,2),(5,3),(6,7)],8)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,6),(4,6),(5,1),(5,2),(7,3),(7,4)],8)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,7),(3,6),(4,2),(4,6),(5,3),(5,4),(6,7),(7,1)],8)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,6),(2,7),(3,4),(3,6),(4,2),(4,8),(5,1),(5,3),(6,8),(8,7)],9)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> 6
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ([(0,5),(1,7),(2,7),(3,4),(4,6),(5,3),(6,1),(6,2)],8)
=> 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,1,0,0,1,0,0]
=> ([(0,5),(1,7),(2,7),(4,6),(5,4),(6,1),(6,2),(7,3)],8)
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ([(0,5),(1,8),(2,7),(3,6),(4,1),(4,7),(5,3),(6,2),(6,4),(7,8)],9)
=> 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,0,0]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,0]
=> ([(0,5),(1,8),(2,8),(3,7),(4,7),(5,6),(6,1),(6,2),(8,3),(8,4)],9)
=> 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,0]
=> ([(0,5),(2,8),(3,7),(4,2),(4,7),(5,6),(6,3),(6,4),(7,8),(8,1)],9)
=> 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ([(0,5),(1,7),(2,8),(3,4),(3,7),(4,2),(4,9),(5,6),(6,1),(6,3),(7,9),(9,8)],10)
=> 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,0,0]
=> ([(0,6),(2,7),(3,7),(4,1),(5,4),(6,2),(6,3),(7,5)],8)
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(5,3),(5,4),(6,1),(6,2),(8,5)],9)
=> 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ([(0,6),(1,8),(2,8),(3,7),(4,7),(6,1),(6,2),(7,5),(8,3),(8,4)],9)
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ([(0,6),(1,9),(2,9),(3,8),(4,7),(5,3),(5,7),(6,1),(6,2),(7,8),(9,4),(9,5)],10)
=> 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ([(0,6),(2,8),(3,7),(4,2),(4,7),(5,1),(6,3),(6,4),(7,8),(8,5)],9)
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ([(0,6),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,4),(6,5),(8,9),(9,2),(9,3)],10)
=> 6
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ([(0,6),(2,9),(3,7),(4,2),(4,8),(5,4),(5,7),(6,3),(6,5),(7,8),(8,9),(9,1)],10)
=> 4
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ([(0,6),(1,7),(2,8),(3,4),(3,7),(4,5),(4,10),(5,2),(5,9),(6,1),(6,3),(7,10),(9,8),(10,9)],11)
=> 5
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ([(0,2),(0,3),(2,7),(3,7),(4,5),(5,1),(6,4),(7,6)],8)
=> 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> ([(0,3),(0,4),(1,7),(2,7),(3,8),(4,8),(5,6),(6,1),(6,2),(8,5)],9)
=> 4
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ([(0,4),(0,5),(2,8),(3,8),(4,7),(5,7),(6,2),(6,3),(7,6),(8,1)],9)
=> 4
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ([(0,3),(0,4),(1,9),(2,8),(3,7),(4,7),(5,1),(5,8),(6,2),(6,5),(7,6),(8,9)],10)
=> 6
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ([(0,4),(0,5),(2,7),(3,7),(4,8),(5,8),(6,1),(7,6),(8,2),(8,3)],9)
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(0,5),(1,8),(3,7),(4,9),(5,9),(6,1),(6,7),(7,8),(8,2),(9,3),(9,6)],10)
=> 6
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,6),(1,8),(3,7),(4,2),(5,4),(6,1),(6,7),(7,8),(8,5)],9)
=> 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> ([(0,4),(0,5),(1,9),(2,7),(3,7),(4,8),(5,1),(5,8),(6,2),(6,3),(8,9),(9,6)],10)
=> 6
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ([(0,5),(0,6),(1,8),(2,8),(4,9),(5,7),(6,4),(6,7),(7,9),(8,3),(9,1),(9,2)],10)
=> 6
Description
The number of maximal chains in a poset.
Matching statistic: St000100
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00065: Permutations —permutation poset⟶ Posets
St000100: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [2,1] => ([],2)
=> 2
[1,0,1,1,0,0]
=> [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => ([(1,2)],3)
=> 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(0,4),(3,2),(4,1),(4,3)],5)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 6
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => ([(0,4),(3,2),(4,5),(5,1),(5,3)],6)
=> 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => ([(0,4),(1,5),(2,5),(3,2),(4,1),(4,3)],6)
=> 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => ([(0,5),(3,4),(4,2),(5,1),(5,3)],6)
=> 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => ([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => ([(0,3),(0,4),(1,5),(3,5),(4,1),(5,2)],6)
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => ([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> 6
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => ([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> 4
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => ([(0,2),(0,5),(3,4),(4,1),(5,3)],6)
=> 5
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> 4
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> 4
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => ([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> 6
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> 6
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,1,4,6,5] => ([(0,5),(1,2),(2,5),(5,3),(5,4)],6)
=> 6
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => ([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 6
Description
The number of linear extensions of a poset.
Matching statistic: St000468
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000468: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00160: Permutations —graph of inversions⟶ Graphs
St000468: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 2
[1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
=> 6
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ([(4,5)],6)
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => ([(4,5)],6)
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
=> 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => ([(4,5)],6)
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => ([(3,5),(4,5)],6)
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => ([(1,2),(3,5),(4,5)],6)
=> 6
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => ([(2,5),(3,5),(4,5)],6)
=> 4
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 5
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => ([(4,5)],6)
=> 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 4
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 4
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => ([(1,2),(3,5),(4,5)],6)
=> 6
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => ([(2,5),(3,4)],6)
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => ([(1,2),(3,5),(4,5)],6)
=> 6
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => ([(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,1,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> 6
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => ([(1,2),(3,5),(4,5)],6)
=> 6
Description
The Hosoya index of a graph.
This is the total number of matchings in the graph.
Matching statistic: St000708
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00108: Permutations —cycle type⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [2,1] => [2]
=> 2
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [2,3,1] => [3]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> 3
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> 4
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,1,1]
=> 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,1]
=> 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> 6
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> 6
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5]
=> 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [2,1,1,1,1]
=> 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [2,1,1,1,1]
=> 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [3,1,1,1]
=> 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [2,1,1,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [2,2,1,1]
=> 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [3,1,1,1]
=> 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [4,1,1]
=> 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [2,1,1,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [2,2,1,1]
=> 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [2,2,1,1]
=> 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => [3,2,1]
=> 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => [3,1,1,1]
=> 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => [3,2,1]
=> 6
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => [4,1,1]
=> 4
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [5,1]
=> 5
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [2,1,1,1,1]
=> 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => [2,2,1,1]
=> 4
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => [2,2,1,1]
=> 4
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => [3,2,1]
=> 6
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,2,1,1]
=> 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [3,2,1]
=> 6
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => [3,1,1,1]
=> 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,1,4,6,5] => [3,2,1]
=> 6
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => [3,2,1]
=> 6
Description
The product of the parts of an integer partition.
Matching statistic: St000883
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000883: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,0,0]
=> [2,1] => [2,1] => 2
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => 2
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => 2
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => 2
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => 2
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => 3
[1,1,0,0,1,0,1,0]
=> [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
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => 3
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [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] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 3
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 4
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 3
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => 4
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 4
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 4
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 6
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 6
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,3,2,1,5] => 4
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,4,3,2,1] => 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [1,2,3,4,6,5] => 2
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [1,2,3,5,4,6] => 2
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [1,2,3,6,5,4] => 3
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => [1,2,4,3,5,6] => 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => [1,2,4,3,6,5] => 4
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,2,4,5,3,6] => [1,2,5,4,3,6] => 3
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => [1,2,6,5,4,3] => 4
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => [1,3,2,4,5,6] => 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => [1,3,2,4,6,5] => 4
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => [1,3,2,5,4,6] => 4
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,3,2,5,6,4] => [1,3,2,6,5,4] => 6
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,3,4,2,5,6] => [1,4,3,2,5,6] => 3
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,3,4,2,6,5] => [1,4,3,2,6,5] => 6
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,3,4,5,2,6] => [1,5,4,3,2,6] => 4
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => [1,6,5,4,3,2] => 5
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => [2,1,3,4,5,6] => 2
[1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => [2,1,3,4,6,5] => 4
[1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => [2,1,3,5,4,6] => 4
[1,1,0,0,1,0,1,1,0,1,0,0]
=> [2,1,3,5,6,4] => [2,1,3,6,5,4] => 6
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => [2,1,4,3,5,6] => 4
[1,1,0,0,1,1,0,1,0,0,1,0]
=> [2,1,4,5,3,6] => [2,1,5,4,3,6] => 6
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [2,3,1,4,5,6] => [3,2,1,4,5,6] => 3
[1,1,0,1,0,0,1,0,1,1,0,0]
=> [2,3,1,4,6,5] => [3,2,1,4,6,5] => 6
[1,1,0,1,0,0,1,1,0,0,1,0]
=> [2,3,1,5,4,6] => [3,2,1,5,4,6] => 6
Description
The number of longest increasing subsequences of a permutation.
The following 127 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001814The number of partitions interlacing the given partition. St001959The product of the heights of the peaks of a Dyck path. St000548The number of different non-empty partial sums of an integer partition. St000189The number of elements in the poset. St000363The number of minimal vertex covers of a graph. St000656The number of cuts of a poset. St000909The number of maximal chains of maximal size in a poset. St000910The number of maximal chains of minimal length in a poset. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001717The largest size of an interval in a poset. St001813The product of the sizes of the principal order filters in a poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001377The major index minus the number of inversions of a permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000255The number of reduced Kogan faces with the permutation as type. St001684The reduced word complexity of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001220The width of a permutation. St000422The energy of a graph, if it is integral. St001330The hat guessing number of a graph. St000456The monochromatic index of a connected graph. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001889The size of the connectivity set of a signed permutation. St001118The acyclic chromatic index of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000117The number of centered tunnels of a Dyck path. St000742The number of big ascents of a permutation after prepending zero. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001727The number of invisible inversions of a permutation. St000141The maximum drop size of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000653The last descent of a permutation. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001209The pmaj statistic of a parking function. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001497The position of the largest weak excedence of a permutation. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001811The Castelnuovo-Mumford regularity of a permutation. St000673The number of non-fixed points of a permutation. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001267The length of the Lyndon factorization of the binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001822The number of alignments of a signed permutation. St001866The nesting alignments of a signed permutation. St001903The number of fixed points of a parking function. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001060The distinguishing index of a graph. St000264The girth of a graph, which is not a tree. St001875The number of simple modules with projective dimension at most 1. 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 $eA$. 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. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001902The number of potential covers of a poset. St000454The largest eigenvalue of a graph if it is integral. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. 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. St000871The number of very big ascents of a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000647The number of big descents of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001863The number of weak excedances of a signed permutation. St000678The number of up steps after the last double rise of a Dyck path. St000834The number of right outer peaks of a permutation. St000906The length of the shortest maximal chain in a poset. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St000455The second largest eigenvalue of a graph if it is integral. St000359The number of occurrences of the pattern 23-1. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000259The diameter of a connected graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000245The number of ascents of a permutation. St000389The number of runs of ones of odd length in a binary word. St000444The length of the maximal rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000842The breadth of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St001645The pebbling number of a connected graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000260The radius of a connected graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000884The number of isolated descents of a permutation. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001394The genus of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St001820The size of the image of the pop stack sorting operator. St000247The number of singleton blocks of a set partition. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000534The number of 2-rises of a permutation. St000546The number of global descents of a permutation. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000731The number of double exceedences of a permutation. St000931The number of occurrences of the pattern UUU in a Dyck path. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001846The number of elements which do not have a complement in the lattice. St001868The number of alignments of type NE of a signed permutation.
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