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Your data matches 150 different statistics following compositions of up to 3 maps.
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Matching statistic: St000657
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
St000657: Integer compositions ⟶ 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
[2] => 2 = 1 + 1
[1,1,1] => 1 = 0 + 1
[1,2] => 1 = 0 + 1
[2,1] => 1 = 0 + 1
[3] => 3 = 2 + 1
[1,1,1,1] => 1 = 0 + 1
[1,1,2] => 1 = 0 + 1
[1,2,1] => 1 = 0 + 1
[1,3] => 1 = 0 + 1
[2,1,1] => 1 = 0 + 1
[2,2] => 2 = 1 + 1
[3,1] => 1 = 0 + 1
[4] => 4 = 3 + 1
[1,1,1,1,1] => 1 = 0 + 1
[1,1,1,2] => 1 = 0 + 1
[1,1,2,1] => 1 = 0 + 1
[1,1,3] => 1 = 0 + 1
[1,2,1,1] => 1 = 0 + 1
[1,2,2] => 1 = 0 + 1
[1,3,1] => 1 = 0 + 1
[1,4] => 1 = 0 + 1
[2,1,1,1] => 1 = 0 + 1
[2,1,2] => 1 = 0 + 1
[2,2,1] => 1 = 0 + 1
[2,3] => 2 = 1 + 1
[3,1,1] => 1 = 0 + 1
[3,2] => 2 = 1 + 1
[4,1] => 1 = 0 + 1
[5] => 5 = 4 + 1
[1,1,1,1,1,1] => 1 = 0 + 1
[1,1,1,1,2] => 1 = 0 + 1
[1,1,1,2,1] => 1 = 0 + 1
[1,1,1,3] => 1 = 0 + 1
[1,1,2,1,1] => 1 = 0 + 1
[1,1,2,2] => 1 = 0 + 1
[1,1,3,1] => 1 = 0 + 1
[1,1,4] => 1 = 0 + 1
[1,2,1,1,1] => 1 = 0 + 1
[1,2,1,2] => 1 = 0 + 1
[1,2,2,1] => 1 = 0 + 1
[1,2,3] => 1 = 0 + 1
[1,3,1,1] => 1 = 0 + 1
[1,3,2] => 1 = 0 + 1
[1,4,1] => 1 = 0 + 1
[1,5] => 1 = 0 + 1
[2,1,1,1,1] => 1 = 0 + 1
[2,1,1,2] => 1 = 0 + 1
[2,1,2,1] => 1 = 0 + 1
Description
The smallest part of an integer composition.
Matching statistic: St001236
(load all 2 compositions to match this statistic)
(load all 2 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 = 0 + 1
[1,1] => 2 = 1 + 1
[2] => 1 = 0 + 1
[1,1,1] => 3 = 2 + 1
[1,2] => 1 = 0 + 1
[2,1] => 1 = 0 + 1
[3] => 1 = 0 + 1
[1,1,1,1] => 4 = 3 + 1
[1,1,2] => 1 = 0 + 1
[1,2,1] => 2 = 1 + 1
[1,3] => 1 = 0 + 1
[2,1,1] => 1 = 0 + 1
[2,2] => 1 = 0 + 1
[3,1] => 1 = 0 + 1
[4] => 1 = 0 + 1
[1,1,1,1,1] => 5 = 4 + 1
[1,1,1,2] => 1 = 0 + 1
[1,1,2,1] => 2 = 1 + 1
[1,1,3] => 1 = 0 + 1
[1,2,1,1] => 2 = 1 + 1
[1,2,2] => 1 = 0 + 1
[1,3,1] => 1 = 0 + 1
[1,4] => 1 = 0 + 1
[2,1,1,1] => 1 = 0 + 1
[2,1,2] => 1 = 0 + 1
[2,2,1] => 1 = 0 + 1
[2,3] => 1 = 0 + 1
[3,1,1] => 1 = 0 + 1
[3,2] => 1 = 0 + 1
[4,1] => 1 = 0 + 1
[5] => 1 = 0 + 1
[1,1,1,1,1,1] => 6 = 5 + 1
[1,1,1,1,2] => 1 = 0 + 1
[1,1,1,2,1] => 2 = 1 + 1
[1,1,1,3] => 1 = 0 + 1
[1,1,2,1,1] => 3 = 2 + 1
[1,1,2,2] => 1 = 0 + 1
[1,1,3,1] => 1 = 0 + 1
[1,1,4] => 1 = 0 + 1
[1,2,1,1,1] => 2 = 1 + 1
[1,2,1,2] => 1 = 0 + 1
[1,2,2,1] => 2 = 1 + 1
[1,2,3] => 1 = 0 + 1
[1,3,1,1] => 1 = 0 + 1
[1,3,2] => 1 = 0 + 1
[1,4,1] => 1 = 0 + 1
[1,5] => 1 = 0 + 1
[2,1,1,1,1] => 1 = 0 + 1
[2,1,1,2] => 1 = 0 + 1
[2,1,2,1] => 1 = 0 + 1
Description
The dominant dimension of the corresponding Comp-Nakayama algebra.
Matching statistic: St000655
(load all 12 compositions to match this statistic)
(load all 12 compositions to match this statistic)
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%
St000655: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1 = 0 + 1
[1,1] => [1,0,1,0]
=> 1 = 0 + 1
[2] => [1,1,0,0]
=> 2 = 1 + 1
[1,1,1] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
Description
The length of the minimal rise of a Dyck path.
For the length of a maximal rise, see [[St000444]].
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 = 0 + 1
[1,1] => [1,0,1,0]
=> 2 = 1 + 1
[2] => [1,1,0,0]
=> 1 = 0 + 1
[1,1,1] => [1,0,1,0,1,0]
=> 3 = 2 + 1
[1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 3 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4] => [1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5 = 4 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 3 = 2 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 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: St001481
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
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%
St001481: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1 = 0 + 1
[1,1] => [1,0,1,0]
=> 1 = 0 + 1
[2] => [1,1,0,0]
=> 2 = 1 + 1
[1,1,1] => [1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1] => [1,1,0,0,1,0]
=> 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[4] => [1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 4 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1 = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1 = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1 = 0 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1 = 0 + 1
Description
The minimal height of a peak of a Dyck path.
Matching statistic: St000210
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000210: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ 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] => [1,0,1,0]
=> [1,2] => 0
[2] => [1,1,0,0]
=> [2,1] => 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
[3] => [1,1,1,0,0,0]
=> [3,1,2] => 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0
[4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 3
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 4
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,3,4,6] => 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,2,3,6,5] => 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 0
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)
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%
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] => [1,0]
=> [[]]
=> 1 = 0 + 1
[1,1] => [1,0,1,0]
=> [[],[]]
=> 1 = 0 + 1
[2] => [1,1,0,0]
=> [[[]]]
=> 2 = 1 + 1
[1,1,1] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
=> [[],[[]]]
=> 1 = 0 + 1
[2,1] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 3 = 2 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 1 = 0 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 1 = 0 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 1 = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 1 = 0 + 1
[4] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 4 = 3 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 1 = 0 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 1 = 0 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 1 = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 1 = 0 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 1 = 0 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 1 = 0 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 1 = 0 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 1 = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 1 = 0 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 1 = 0 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> 1 = 0 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 2 = 1 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 1 = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> 5 = 4 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[]]
=> 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[[]]]
=> 1 = 0 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[],[],[],[[]],[]]
=> 1 = 0 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[],[],[],[[[]]]]
=> 1 = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[],[],[[]],[],[]]
=> 1 = 0 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[],[],[[]],[[]]]
=> 1 = 0 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[],[],[[[]]],[]]
=> 1 = 0 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[],[],[[[[]]]]]
=> 1 = 0 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[],[[]],[],[],[]]
=> 1 = 0 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[],[[]],[],[[]]]
=> 1 = 0 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[],[[]],[[]],[]]
=> 1 = 0 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[],[[]],[[[]]]]
=> 1 = 0 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[],[[[]]],[],[]]
=> 1 = 0 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[],[[[]]],[[]]]
=> 1 = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[],[[[[]]]],[]]
=> 1 = 0 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> 1 = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[]],[],[],[],[]]
=> 1 = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[[]],[],[],[[]]]
=> 1 = 0 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[[]],[],[[]],[]]
=> 1 = 0 + 1
Description
The protection number of an ordered tree.
This is the minimal distance from the root to a leaf.
Matching statistic: St000908
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> ([],1)
=> 1 = 0 + 1
[1,1] => [1,0,1,0]
=> ([(0,1)],2)
=> 1 = 0 + 1
[2] => [1,1,0,0]
=> ([],2)
=> 2 = 1 + 1
[1,1,1] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2] => [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[2,1] => [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 1 = 0 + 1
[3] => [1,1,1,0,0,0]
=> ([],3)
=> 3 = 2 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1 = 0 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 1 = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 1 = 0 + 1
[4] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4 = 3 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1 = 0 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1 = 0 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 1 = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1 = 0 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 1 = 0 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1 = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> 1 = 0 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1 = 0 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2 = 1 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> 1 = 0 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2 = 1 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> 1 = 0 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 5 = 4 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 0 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 1 = 0 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1 = 0 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> 1 = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1 = 0 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> 1 = 0 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> 1 = 0 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> 1 = 0 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1 = 0 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> 1 = 0 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1 = 0 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> 1 = 0 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 1 = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> 1 = 0 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> 1 = 0 + 1
Description
The length of the shortest maximal antichain in a poset.
Matching statistic: St000310
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000310: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000310: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [1] => ([],1)
=> 0
[1,1] => [1,0,1,0]
=> [1,2] => ([],2)
=> 0
[2] => [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> 0
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => ([(1,2)],3)
=> 0
[3] => [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => ([(2,3)],4)
=> 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => ([(2,3)],4)
=> 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> 0
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> 0
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> 0
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => ([(2,3),(2,4),(3,4)],5)
=> 0
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => ([(2,3),(2,4),(3,4)],5)
=> 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => ([(0,1),(2,3),(2,4),(3,4)],5)
=> 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> 0
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> 0
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => ([(4,5)],6)
=> 0
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => ([(4,5)],6)
=> 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
=> 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,2,5,4,3,6] => ([(3,4),(3,5),(4,5)],6)
=> 0
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,3,2,4,5,6] => ([(4,5)],6)
=> 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => ([(2,5),(3,4)],6)
=> 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => ([(2,5),(3,4)],6)
=> 0
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => ([(3,4),(3,5),(4,5)],6)
=> 0
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => ([(1,2),(3,4),(3,5),(4,5)],6)
=> 0
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 0
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => ([(4,5)],6)
=> 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => ([(2,5),(3,4)],6)
=> 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => ([(2,5),(3,4)],6)
=> 0
Description
The minimal degree of a vertex of a graph.
Matching statistic: St001107
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ 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
Mp00222: Dyck paths —peaks-to-valleys⟶ 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] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 0
[3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 0
[4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[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]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 0
[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]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 0
[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]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 0
[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]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 0
[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]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 0
[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]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 0
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 0
[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]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 0
[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]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 0
[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]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 0
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 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]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 0
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 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]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 0
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[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]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[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]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 0
[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]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> 0
[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]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 0
[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]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> 0
[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]
=> [1,0,1,0,1,1,0,1,1,0,0,1,0,0]
=> 0
[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]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> 0
[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]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 0
[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]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> 0
[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]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> 0
[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]
=> [1,0,1,1,0,1,1,0,0,1,0,0,1,0]
=> 0
[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]
=> [1,0,1,1,0,1,1,1,0,0,1,0,0,0]
=> 0
[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]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> 0
[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]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> 0
[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]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> 0
[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]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 0
[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]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> 0
[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]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> 0
[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]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> 0
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.
The following 140 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001119The length of a shortest maximal path in a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000297The number of leading ones in a binary word. St000314The number of left-to-right-maxima of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000382The first part of an integer composition. St000383The last part of an integer composition. St000617The number of global maxima of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001316The domatic number of a graph. St001322The size of a minimal independent dominating set in a graph. St001829The common independence number of a graph. St000439The position of the first down step of a Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000487The length of the shortest cycle of a permutation. St000993The multiplicity of the largest part of an integer partition. St001075The minimal size of a block of a set partition. St000989The number of final rises of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000654The first descent of a permutation. St000906The length of the shortest maximal chain in a poset. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000990The first ascent of a permutation. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001252Half the sum of the even parts of a partition. St001587Half of the largest even part of an integer partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000929The constant term of the character polynomial of an integer partition. St000090The variation of a composition. St000379The number of Hamiltonian cycles in a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001586The number of odd parts smaller than the largest even part in an integer partition. 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. St000219The number of occurrences of the pattern 231 in a permutation. St000225Difference between largest and smallest parts in a partition. St000290The major index of a binary word. St000291The number of descents of a binary word. St000293The number of inversions of a binary word. St000296The length of the symmetric border of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000347The inversion sum of a binary word. St000629The defect of a binary word. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. 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. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000944The 3-degree of an integer partition. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. 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. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001214The aft of an integer partition. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St001541The Gini index of an integer partition. St001584The area statistic between a Dyck path and its bounce path. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001657The number of twos in an integer partition. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001730The number of times the path corresponding to a binary word crosses the base line. St001918The degree of the cyclic sieving polynomial corresponding to an integer 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. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001071The beta invariant of the graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001271The competition number of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000274The number of perfect matchings of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000699The toughness times the least common multiple of 1,. St001095The number of non-isomorphic posets with precisely one further covering relation. St001141The number of occurrences of hills of size 3 in a Dyck path. St001651The Frankl number of a lattice. St000260The radius of a connected graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000478Another weight of a partition according to Alladi. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. 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. St000941The number of characters of the symmetric group whose value on the partition is even. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001176The size of a partition minus its first part. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000627The exponent of a binary word. St001884The number of borders of a binary word. 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. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000259The diameter 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph.
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