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Your data matches 147 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
[1,1] => 1
[2] => 2
[1,1,1] => 1
[1,2] => 1
[2,1] => 1
[3] => 3
[1,1,1,1] => 1
[1,1,2] => 1
[1,2,1] => 1
[1,3] => 1
[2,1,1] => 1
[2,2] => 2
[3,1] => 1
[4] => 4
[1,1,1,1,1] => 1
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,1,3] => 1
[1,2,1,1] => 1
[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] => 2
[3,1,1] => 1
[3,2] => 2
[4,1] => 1
[5] => 5
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 1
[1,1,1,2,1] => 1
[1,1,1,3] => 1
[1,1,2,1,1] => 1
[1,1,2,2] => 1
[1,1,3,1] => 1
[1,1,4] => 1
[1,2,1,1,1] => 1
[1,2,1,2] => 1
[1,2,2,1] => 1
[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 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
[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: St000655
(load all 13 compositions to match this statistic)
(load all 13 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
[1,1] => [1,0,1,0]
=> 1
[2] => [1,1,0,0]
=> 2
[1,1,1] => [1,0,1,0,1,0]
=> 1
[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]
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[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]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,1,1] => [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
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[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]
=> 1
[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]
=> 2
[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]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,1,1,1] => [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
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[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]
=> 1
[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]
=> 1
[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]
=> 1
[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 length of the minimal rise of a Dyck path.
For the length of a maximal rise, see [[St000444]].
Matching statistic: St000685
(load all 7 compositions to match this statistic)
(load all 7 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: St001481
(load all 5 compositions to match this statistic)
(load all 5 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
[1,1] => [1,0,1,0]
=> 1
[2] => [1,1,0,0]
=> 2
[1,1,1] => [1,0,1,0,1,0]
=> 1
[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]
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[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]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,1,1] => [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
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[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]
=> 1
[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]
=> 2
[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]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,1,1,1] => [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
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[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]
=> 1
[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]
=> 1
[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]
=> 1
[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 minimal height of a peak of a Dyck path.
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
[1,1] => [1,0,1,0]
=> [[],[]]
=> 1
[2] => [1,1,0,0]
=> [[[]]]
=> 2
[1,1,1] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1
[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]
=> [[[[]]]]
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 1
[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]
=> [[[]],[[]]]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 4
[1,1,1,1,1] => [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
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 1
[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]
=> [[],[[]],[],[]]
=> 1
[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]
=> [[[]],[[[]]]]
=> 2
[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]
=> [[[[]]],[[]]]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> 5
[1,1,1,1,1,1] => [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
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[],[],[],[[]],[]]
=> 1
[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]
=> [[],[],[[]],[],[]]
=> 1
[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]
=> [[],[[]],[],[],[]]
=> 1
[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]
=> [[],[[]],[[]],[]]
=> 1
[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 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
[1,1] => [1,0,1,0]
=> ([(0,1)],2)
=> 1
[2] => [1,1,0,0]
=> ([],2)
=> 2
[1,1,1] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[1,2] => [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 1
[2,1] => [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 1
[3] => [1,1,1,0,0,0]
=> ([],3)
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4
[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
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 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
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 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
[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
[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
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 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
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> 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
[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
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> 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
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 5
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
[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
Description
The length of the shortest maximal antichain in a poset.
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
[1,1] => [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[2] => [1,1,0,0]
=> [2,1] => 1 = 2 - 1
[1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[3] => [1,1,1,0,0,0]
=> [3,1,2] => 2 = 3 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0 = 1 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1 = 2 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0 = 1 - 1
[4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 3 = 4 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0 = 1 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 0 = 1 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0 = 1 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0 = 1 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 0 = 1 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 0 = 1 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => 0 = 1 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0 = 1 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0 = 1 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0 = 1 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0 = 1 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => 1 = 2 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 0 = 1 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => 1 = 2 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 0 = 1 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 4 = 5 - 1
[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
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 0 = 1 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => 0 = 1 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,4,5] => 0 = 1 - 1
[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
[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
[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
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,3,4,5] => 0 = 1 - 1
[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 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,3,2,4,6,5] => 0 = 1 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4,6] => 0 = 1 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,4,5] => 0 = 1 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,2,3,5,6] => 0 = 1 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,2,3,6,5] => 0 = 1 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,2,3,4,6] => 0 = 1 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,2,3,4,5] => 0 = 1 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 0 = 1 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 0 = 1 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 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: St000025
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000025: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,1] => [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2] => [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,2,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
Description
The number of initial rises of a Dyck path.
In other words, this is the height of the first peak of $D$.
Matching statistic: St000026
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
St000026: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 1
[1,1] => [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2] => [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,2] => [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1
[3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,2] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,3] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[2,1,1,1] => [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[3,1,1] => [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[4,1] => [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[5] => [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,3] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,2,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,4] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,3] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,2] => [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,4,1] => [4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,5] => [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,2] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
Description
The position of the first return of a Dyck path.
The following 137 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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. St000310The minimal degree of a vertex 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. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001119The length of a shortest maximal path in a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. 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. St000654The first descent of a permutation. St000906The length of the shortest maximal chain in a poset. St000989The number of final rises of a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St000990The first ascent of a permutation. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St000667The greatest common divisor of the parts of the partition. St001571The Cartan determinant of the integer partition. St000120The number of left tunnels of a Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001933The largest multiplicity of a part in an integer partition. St000706The product of the factorials of the multiplicities of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000090The variation of a composition. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000003The number of standard Young tableaux of the partition. St000183The side length of the Durfee square of an integer partition. 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. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000517The Kreweras number of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000627The exponent of a binary word. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000847The number of standard Young tableaux whose descent set is the binary word. St000897The number of different multiplicities of parts of an integer partition. St000913The number of ways to refine the partition into singletons. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001256Number of simple reflexive modules that are 2-stable reflexive. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St001780The order of promotion on the set of standard tableaux of given shape. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. 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. 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. St001128The exponens consonantiae of a partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St000273The domination number of a graph. St000544The cop number of a graph. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001363The Euler characteristic of a graph according to Knill. St001339The irredundance number of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000284The Plancherel distribution on integer partitions. St000668The least common multiple of the parts of the partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. 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. St000933The number of multipartitions of sizes given by an integer partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000456The monochromatic index of a connected graph. St000460The hook length of the last cell along the main diagonal of an integer partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000929The constant term of the character polynomial of an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001884The number of borders of a binary word. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001877Number of indecomposable injective modules with projective dimension 2. 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. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000937The number of positive values of the symmetric group character corresponding to the partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000650The number of 3-rises of a permutation. 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. 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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