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Your data matches 88 different statistics following compositions of up to 3 maps.
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Matching statistic: St001236
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
St001236: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 2
[2] => 1
[1,1,1] => 3
[1,2] => 1
[2,1] => 1
[3] => 1
[1,1,1,1] => 4
[1,1,2] => 1
[1,2,1] => 2
[1,3] => 1
[2,1,1] => 1
[2,2] => 1
[3,1] => 1
[4] => 1
[1,1,1,1,1] => 5
[1,1,1,2] => 1
[1,1,2,1] => 2
[1,1,3] => 1
[1,2,1,1] => 2
[1,2,2] => 1
[1,3,1] => 1
[1,4] => 1
[2,1,1,1] => 1
[2,1,2] => 1
[2,2,1] => 1
[2,3] => 1
[3,1,1] => 1
[3,2] => 1
[4,1] => 1
[5] => 1
[1,1,1,1,1,1] => 6
[1,1,1,1,2] => 1
[1,1,1,2,1] => 2
[1,1,1,3] => 1
[1,1,2,1,1] => 3
[1,1,2,2] => 1
[1,1,3,1] => 1
[1,1,4] => 1
[1,2,1,1,1] => 2
[1,2,1,2] => 1
[1,2,2,1] => 2
[1,2,3] => 1
[1,3,1,1] => 1
[1,3,2] => 1
[1,4,1] => 1
[1,5] => 1
[2,1,1,1,1] => 1
[2,1,1,2] => 1
[2,1,2,1] => 1
Description
The dominant dimension of the corresponding Comp-Nakayama algebra.
Matching statistic: St000657
(load all 42 compositions to match this statistic)
(load all 42 compositions to match this statistic)
Mp00041: Integer compositions —conjugate⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => 1
[1,1] => [2] => 2
[2] => [1,1] => 1
[1,1,1] => [3] => 3
[1,2] => [1,2] => 1
[2,1] => [2,1] => 1
[3] => [1,1,1] => 1
[1,1,1,1] => [4] => 4
[1,1,2] => [1,3] => 1
[1,2,1] => [2,2] => 2
[1,3] => [1,1,2] => 1
[2,1,1] => [3,1] => 1
[2,2] => [1,2,1] => 1
[3,1] => [2,1,1] => 1
[4] => [1,1,1,1] => 1
[1,1,1,1,1] => [5] => 5
[1,1,1,2] => [1,4] => 1
[1,1,2,1] => [2,3] => 2
[1,1,3] => [1,1,3] => 1
[1,2,1,1] => [3,2] => 2
[1,2,2] => [1,2,2] => 1
[1,3,1] => [2,1,2] => 1
[1,4] => [1,1,1,2] => 1
[2,1,1,1] => [4,1] => 1
[2,1,2] => [1,3,1] => 1
[2,2,1] => [2,2,1] => 1
[2,3] => [1,1,2,1] => 1
[3,1,1] => [3,1,1] => 1
[3,2] => [1,2,1,1] => 1
[4,1] => [2,1,1,1] => 1
[5] => [1,1,1,1,1] => 1
[1,1,1,1,1,1] => [6] => 6
[1,1,1,1,2] => [1,5] => 1
[1,1,1,2,1] => [2,4] => 2
[1,1,1,3] => [1,1,4] => 1
[1,1,2,1,1] => [3,3] => 3
[1,1,2,2] => [1,2,3] => 1
[1,1,3,1] => [2,1,3] => 1
[1,1,4] => [1,1,1,3] => 1
[1,2,1,1,1] => [4,2] => 2
[1,2,1,2] => [1,3,2] => 1
[1,2,2,1] => [2,2,2] => 2
[1,2,3] => [1,1,2,2] => 1
[1,3,1,1] => [3,1,2] => 1
[1,3,2] => [1,2,1,2] => 1
[1,4,1] => [2,1,1,2] => 1
[1,5] => [1,1,1,1,2] => 1
[2,1,1,1,1] => [5,1] => 1
[2,1,1,2] => [1,4,1] => 1
[2,1,2,1] => [2,3,1] => 1
Description
The smallest part of an integer composition.
Matching statistic: St000685
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000685: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000685: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,1] => [1,0,1,0]
=> 2
[2] => [1,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2] => [1,0,1,1,0,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> 1
[3] => [1,1,1,0,0,0]
=> 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1
Description
The dominant dimension of the LNakayama algebra associated to a Dyck path.
To every Dyck path there is an LNakayama algebra associated as described in [[St000684]].
Matching statistic: St000655
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[1,1] => [2] => [1,1,0,0]
=> 2
[2] => [1,1] => [1,0,1,0]
=> 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 3
[1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[3] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,1,1,2,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[1,1,1,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,2,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 1
[1,1,3,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 1
[1,1,4] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1
[1,2,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,2,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1
[1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2
[1,2,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1
[1,5] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[2,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
[2,1,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
Description
The length of the minimal rise of a Dyck path.
For the length of a maximal rise, see [[St000444]].
Matching statistic: St001481
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001481: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001481: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[1,1] => [2] => [1,1,0,0]
=> 2
[2] => [1,1] => [1,0,1,0]
=> 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 3
[1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[3] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,1,1,2,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 2
[1,1,1,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[1,1,2,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 1
[1,1,3,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 1
[1,1,4] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1
[1,2,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[1,2,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1
[1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 2
[1,2,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1
[1,5] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[2,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
[2,1,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
Description
The minimal height of a peak of a Dyck path.
Matching statistic: St000210
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000210: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000210: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => 0 = 1 - 1
[1,1] => [1,0,1,0]
=> [2,1] => 1 = 2 - 1
[2] => [1,1,0,0]
=> [1,2] => 0 = 1 - 1
[1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 2 = 3 - 1
[1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 0 = 1 - 1
[2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 0 = 1 - 1
[3] => [1,1,1,0,0,0]
=> [1,2,3] => 0 = 1 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 3 = 4 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0 = 1 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 1 = 2 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0 = 1 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0 = 1 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0 = 1 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0 = 1 - 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 4 = 5 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 0 = 1 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 1 = 2 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 0 = 1 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 1 = 2 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 0 = 1 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 0 = 1 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 0 = 1 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 0 = 1 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 0 = 1 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 0 = 1 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0 = 1 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0 = 1 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0 = 1 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => 5 = 6 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => 0 = 1 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => 1 = 2 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => 0 = 1 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => 2 = 3 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => 0 = 1 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => 0 = 1 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => 0 = 1 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => 1 = 2 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => 0 = 1 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => 1 = 2 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => 0 = 1 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => 0 = 1 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,5,4,6] => 0 = 1 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => 0 = 1 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 0 = 1 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => 0 = 1 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => 0 = 1 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => 0 = 1 - 1
Description
Minimum over maximum difference of elements in cycles.
Given a cycle $C$ in a permutation, we can compute the maximum distance between elements in the cycle, that is $\max \{ a_i-a_j | a_i, a_j \in C \}$.
The statistic is then the minimum of this value over all cycles in the permutation.
For example, all permutations with a fixed-point has statistic value 0,
and all permutations of $[n]$ with only one cycle, has statistic value $n-1$.
Matching statistic: St000700
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [[]]
=> 1
[1,1] => [2] => [1,1,0,0]
=> [[[]]]
=> 2
[2] => [1,1] => [1,0,1,0]
=> [[],[]]
=> 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 3
[1,2] => [2,1] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 1
[2,1] => [1,2] => [1,0,1,1,0,0]
=> [[],[[]]]
=> 1
[3] => [1,1,1] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 4
[1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 1
[1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2
[1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 1
[2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 1
[2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 1
[3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 1
[4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> 5
[1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 1
[1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 2
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> 1
[1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 2
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 1
[1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 1
[1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 1
[2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 1
[2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 1
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 1
[2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 1
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 1
[3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 1
[4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 1
[5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[[[[[[]]]]]]]
=> 6
[1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [[[[[[]]]]],[]]
=> 1
[1,1,1,2,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [[[[[]]]],[[]]]
=> 2
[1,1,1,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [[[[[]]]],[],[]]
=> 1
[1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [[[[]]],[[[]]]]
=> 3
[1,1,2,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [[[[]]],[[]],[]]
=> 1
[1,1,3,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [[[[]]],[],[[]]]
=> 1
[1,1,4] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [[[[]]],[],[],[]]
=> 1
[1,2,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[[]],[[[[]]]]]
=> 2
[1,2,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [[[]],[[[]]],[]]
=> 1
[1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [[[]],[[]],[[]]]
=> 2
[1,2,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [[[]],[[]],[],[]]
=> 1
[1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [[[]],[],[[[]]]]
=> 1
[1,3,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[[]],[],[[]],[]]
=> 1
[1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[[]],[],[],[[]]]
=> 1
[1,5] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[]],[],[],[],[]]
=> 1
[2,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> 1
[2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[],[[[[]]]],[]]
=> 1
[2,1,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[],[[[]]],[[]]]
=> 1
Description
The protection number of an ordered tree.
This is the minimal distance from the root to a leaf.
Matching statistic: St001107
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
[2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3 = 4 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1 = 2 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 0 = 1 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,1,1,2] => [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]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,2,1] => [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]
=> [1,1,1,1,0,1,1,0,0,0,0,1,0,0]
=> 1 = 2 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,2,1,1] => [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]
=> [1,1,1,0,1,1,1,0,0,0,1,0,0,0]
=> 2 = 3 - 1
[1,1,2,2] => [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]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[1,2,1,1,1] => [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]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> 1 = 2 - 1
[1,2,1,2] => [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]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,2,2,1] => [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]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> 0 = 1 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,1,1,1,1] => [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]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 0 = 1 - 1
[2,1,1,2] => [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]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[2,1,2,1] => [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]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.
In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Matching statistic: St000487
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000487: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000487: Permutations ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => ? = 1
[1,1] => [1,0,1,0]
=> [2,1] => 2
[2] => [1,1,0,0]
=> [1,2] => 1
[1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 3
[1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 1
[2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 1
[3] => [1,1,1,0,0,0]
=> [1,2,3] => 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 4
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 2
[1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 1
[4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => 5
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => 2
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => 2
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => 6
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => 2
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => 3
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,1,4,6,5] => 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,3,1,4,5,6] => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [2,1,4,5,6,3] => 2
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [2,1,4,5,3,6] => 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,6,5] => 2
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,5,6] => 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,1,3,5,6,4] => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [2,1,3,5,4,6] => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [2,1,3,4,6,5] => 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,3,4,5,6,2] => 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,3,4,5,2,6] => 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,3,4,2,6,5] => 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,3,4,2,5,6] => 1
Description
The length of the shortest cycle of a permutation.
Matching statistic: St000993
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1]
=> [1]
=> ? = 1
[1,1] => [2] => [2]
=> [1,1]
=> 2
[2] => [1,1] => [1,1]
=> [2]
=> 1
[1,1,1] => [3] => [3]
=> [1,1,1]
=> 3
[1,2] => [2,1] => [2,1]
=> [2,1]
=> 1
[2,1] => [1,2] => [2,1]
=> [2,1]
=> 1
[3] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[1,1,1,1] => [4] => [4]
=> [1,1,1,1]
=> 4
[1,1,2] => [3,1] => [3,1]
=> [2,1,1]
=> 1
[1,2,1] => [2,2] => [2,2]
=> [2,2]
=> 2
[1,3] => [2,1,1] => [2,1,1]
=> [3,1]
=> 1
[2,1,1] => [1,3] => [3,1]
=> [2,1,1]
=> 1
[2,2] => [1,2,1] => [2,1,1]
=> [3,1]
=> 1
[3,1] => [1,1,2] => [2,1,1]
=> [3,1]
=> 1
[4] => [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1
[1,1,1,1,1] => [5] => [5]
=> [1,1,1,1,1]
=> 5
[1,1,1,2] => [4,1] => [4,1]
=> [2,1,1,1]
=> 1
[1,1,2,1] => [3,2] => [3,2]
=> [2,2,1]
=> 2
[1,1,3] => [3,1,1] => [3,1,1]
=> [3,1,1]
=> 1
[1,2,1,1] => [2,3] => [3,2]
=> [2,2,1]
=> 2
[1,2,2] => [2,2,1] => [2,2,1]
=> [3,2]
=> 1
[1,3,1] => [2,1,2] => [2,2,1]
=> [3,2]
=> 1
[1,4] => [2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 1
[2,1,1,1] => [1,4] => [4,1]
=> [2,1,1,1]
=> 1
[2,1,2] => [1,3,1] => [3,1,1]
=> [3,1,1]
=> 1
[2,2,1] => [1,2,2] => [2,2,1]
=> [3,2]
=> 1
[2,3] => [1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 1
[3,1,1] => [1,1,3] => [3,1,1]
=> [3,1,1]
=> 1
[3,2] => [1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 1
[4,1] => [1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 1
[5] => [1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 1
[1,1,1,1,1,1] => [6] => [6]
=> [1,1,1,1,1,1]
=> 6
[1,1,1,1,2] => [5,1] => [5,1]
=> [2,1,1,1,1]
=> 1
[1,1,1,2,1] => [4,2] => [4,2]
=> [2,2,1,1]
=> 2
[1,1,1,3] => [4,1,1] => [4,1,1]
=> [3,1,1,1]
=> 1
[1,1,2,1,1] => [3,3] => [3,3]
=> [2,2,2]
=> 3
[1,1,2,2] => [3,2,1] => [3,2,1]
=> [3,2,1]
=> 1
[1,1,3,1] => [3,1,2] => [3,2,1]
=> [3,2,1]
=> 1
[1,1,4] => [3,1,1,1] => [3,1,1,1]
=> [4,1,1]
=> 1
[1,2,1,1,1] => [2,4] => [4,2]
=> [2,2,1,1]
=> 2
[1,2,1,2] => [2,3,1] => [3,2,1]
=> [3,2,1]
=> 1
[1,2,2,1] => [2,2,2] => [2,2,2]
=> [3,3]
=> 2
[1,2,3] => [2,2,1,1] => [2,2,1,1]
=> [4,2]
=> 1
[1,3,1,1] => [2,1,3] => [3,2,1]
=> [3,2,1]
=> 1
[1,3,2] => [2,1,2,1] => [2,2,1,1]
=> [4,2]
=> 1
[1,4,1] => [2,1,1,2] => [2,2,1,1]
=> [4,2]
=> 1
[1,5] => [2,1,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1
[2,1,1,1,1] => [1,5] => [5,1]
=> [2,1,1,1,1]
=> 1
[2,1,1,2] => [1,4,1] => [4,1,1]
=> [3,1,1,1]
=> 1
[2,1,2,1] => [1,3,2] => [3,2,1]
=> [3,2,1]
=> 1
[2,1,3] => [1,3,1,1] => [3,1,1,1]
=> [4,1,1]
=> 1
Description
The multiplicity of the largest part of an integer partition.
The following 78 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001075The minimal size of a block of a set partition. St000667The greatest common divisor of the parts of the partition. St001571The Cartan determinant of the integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000379The number of Hamiltonian cycles in a graph. St000699The toughness times the least common multiple of 1,. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000264The girth of a graph, which is not a tree. St000455The second largest eigenvalue of a graph if it is integral. St000284The Plancherel distribution on integer partitions. St000456The monochromatic index of a connected graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. 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. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001568The smallest positive integer that does not appear twice in the partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001330The hat guessing number of a graph. St000907The number of maximal antichains of minimal length in a poset. St000850The number of 1/2-balanced pairs in a poset. St000650The number of 3-rises of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph.
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