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Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000900
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000900: 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] => 1
[1,3] => 1
[2,1,1] => 1
[2,2] => 2
[3,1] => 1
[4] => 1
[1,1,1,1,1] => 5
[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] => 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] => 1
[1,1,1,3] => 1
[1,1,2,1,1] => 1
[1,1,2,2] => 2
[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 minimal number of repetitions of a part in an integer composition.
This is the smallest letter in the word obtained by applying the delta morphism.
Matching statistic: St000657
(load all 24 compositions to match this statistic)
(load all 24 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ 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] => [3] => 3
[1,2] => [1,1] => 1
[2,1] => [1,1] => 1
[3] => [1] => 1
[1,1,1,1] => [4] => 4
[1,1,2] => [2,1] => 1
[1,2,1] => [1,1,1] => 1
[1,3] => [1,1] => 1
[2,1,1] => [1,2] => 1
[2,2] => [2] => 2
[3,1] => [1,1] => 1
[4] => [1] => 1
[1,1,1,1,1] => [5] => 5
[1,1,1,2] => [3,1] => 1
[1,1,2,1] => [2,1,1] => 1
[1,1,3] => [2,1] => 1
[1,2,1,1] => [1,1,2] => 1
[1,2,2] => [1,2] => 1
[1,3,1] => [1,1,1] => 1
[1,4] => [1,1] => 1
[2,1,1,1] => [1,3] => 1
[2,1,2] => [1,1,1] => 1
[2,2,1] => [2,1] => 1
[2,3] => [1,1] => 1
[3,1,1] => [1,2] => 1
[3,2] => [1,1] => 1
[4,1] => [1,1] => 1
[5] => [1] => 1
[1,1,1,1,1,1] => [6] => 6
[1,1,1,1,2] => [4,1] => 1
[1,1,1,2,1] => [3,1,1] => 1
[1,1,1,3] => [3,1] => 1
[1,1,2,1,1] => [2,1,2] => 1
[1,1,2,2] => [2,2] => 2
[1,1,3,1] => [2,1,1] => 1
[1,1,4] => [2,1] => 1
[1,2,1,1,1] => [1,1,3] => 1
[1,2,1,2] => [1,1,1,1] => 1
[1,2,2,1] => [1,2,1] => 1
[1,2,3] => [1,1,1] => 1
[1,3,1,1] => [1,1,2] => 1
[1,3,2] => [1,1,1] => 1
[1,4,1] => [1,1,1] => 1
[1,5] => [1,1] => 1
[2,1,1,1,1] => [1,4] => 1
[2,1,1,2] => [1,2,1] => 1
[2,1,2,1] => [1,1,1,1] => 1
Description
The smallest part of an integer composition.
Matching statistic: St000655
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ 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%
Mp00173: Integer compositions —rotate front to back⟶ 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%
Values
[1] => [1] => [1] => [1,0]
=> 1
[1,1] => [2] => [2] => [1,1,0,0]
=> 2
[2] => [1] => [1] => [1,0]
=> 1
[1,1,1] => [3] => [3] => [1,1,1,0,0,0]
=> 3
[1,2] => [1,1] => [1,1] => [1,0,1,0]
=> 1
[2,1] => [1,1] => [1,1] => [1,0,1,0]
=> 1
[3] => [1] => [1] => [1,0]
=> 1
[1,1,1,1] => [4] => [4] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,2] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,2,1] => [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,3] => [1,1] => [1,1] => [1,0,1,0]
=> 1
[2,1,1] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,2] => [2] => [2] => [1,1,0,0]
=> 2
[3,1] => [1,1] => [1,1] => [1,0,1,0]
=> 1
[4] => [1] => [1] => [1,0]
=> 1
[1,1,1,1,1] => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,2] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1] => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,3] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,2,1,1] => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,2,2] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,3,1] => [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,4] => [1,1] => [1,1] => [1,0,1,0]
=> 1
[2,1,1,1] => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[2,1,2] => [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[2,2,1] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3] => [1,1] => [1,1] => [1,0,1,0]
=> 1
[3,1,1] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,2] => [1,1] => [1,1] => [1,0,1,0]
=> 1
[4,1] => [1,1] => [1,1] => [1,0,1,0]
=> 1
[5] => [1] => [1] => [1,0]
=> 1
[1,1,1,1,1,1] => [6] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[1,1,1,1,2] => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,1,2,1] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,3] => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1] => [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,2,2] => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> 2
[1,1,3,1] => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,4] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,2,1,1,1] => [1,1,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,2,1,2] => [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,2,2,1] => [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[1,2,3] => [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,3,1,1] => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3,2] => [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,4,1] => [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
=> 1
[1,5] => [1,1] => [1,1] => [1,0,1,0]
=> 1
[2,1,1,1,1] => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[2,1,1,2] => [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,2,1] => [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
Description
The length of the minimal rise of a Dyck path.
For the length of a maximal rise, see [[St000444]].
Matching statistic: St000700
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 78% ●values known / values provided: 99%●distinct values known / distinct values provided: 78%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 78% ●values known / values provided: 99%●distinct values known / distinct values provided: 78%
Values
[1] => [1] => [1,0]
=> [[]]
=> 1
[1,1] => [2] => [1,1,0,0]
=> [[[]]]
=> 2
[2] => [1] => [1,0]
=> [[]]
=> 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 3
[1,2] => [1,1] => [1,0,1,0]
=> [[],[]]
=> 1
[2,1] => [1,1] => [1,0,1,0]
=> [[],[]]
=> 1
[3] => [1] => [1,0]
=> [[]]
=> 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 4
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 1
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1
[1,3] => [1,1] => [1,0,1,0]
=> [[],[]]
=> 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> [[],[[]]]
=> 1
[2,2] => [2] => [1,1,0,0]
=> [[[]]]
=> 2
[3,1] => [1,1] => [1,0,1,0]
=> [[],[]]
=> 1
[4] => [1] => [1,0]
=> [[]]
=> 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> 5
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> [[],[[]]]
=> 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1
[1,4] => [1,1] => [1,0,1,0]
=> [[],[]]
=> 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 1
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 1
[2,3] => [1,1] => [1,0,1,0]
=> [[],[]]
=> 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> [[],[[]]]
=> 1
[3,2] => [1,1] => [1,0,1,0]
=> [[],[]]
=> 1
[4,1] => [1,1] => [1,0,1,0]
=> [[],[]]
=> 1
[5] => [1] => [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] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 1
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 1
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 1
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 1
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 1
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 1
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1
[1,5] => [1,1] => [1,0,1,0]
=> [[],[]]
=> 1
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 1
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 1
[1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [[[[[[[[[]]]]]]]]]
=> ? = 8
[1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [[[[[[[[[[]]]]]]]]]]
=> ? = 9
[1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [[[[[[[]]]]]],[],[]]
=> ? = 1
[1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [[[[[[]]]]],[],[[]]]
=> ? = 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [[[[[]]]],[],[[[]]]]
=> ? = 1
[1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [[[[]]],[],[[[[]]]]]
=> ? = 1
Description
The protection number of an ordered tree.
This is the minimal distance from the root to a leaf.
Matching statistic: St000993
Mp00133: Integer compositions —delta morphism⟶ 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
[1,1,1] => [3] => [3]
=> [1,1,1]
=> 3
[1,2] => [1,1] => [1,1]
=> [2]
=> 1
[2,1] => [1,1] => [1,1]
=> [2]
=> 1
[3] => [1] => [1]
=> [1]
=> ? = 1
[1,1,1,1] => [4] => [4]
=> [1,1,1,1]
=> 4
[1,1,2] => [2,1] => [2,1]
=> [2,1]
=> 1
[1,2,1] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[1,3] => [1,1] => [1,1]
=> [2]
=> 1
[2,1,1] => [1,2] => [2,1]
=> [2,1]
=> 1
[2,2] => [2] => [2]
=> [1,1]
=> 2
[3,1] => [1,1] => [1,1]
=> [2]
=> 1
[4] => [1] => [1]
=> [1]
=> ? = 1
[1,1,1,1,1] => [5] => [5]
=> [1,1,1,1,1]
=> 5
[1,1,1,2] => [3,1] => [3,1]
=> [2,1,1]
=> 1
[1,1,2,1] => [2,1,1] => [2,1,1]
=> [3,1]
=> 1
[1,1,3] => [2,1] => [2,1]
=> [2,1]
=> 1
[1,2,1,1] => [1,1,2] => [2,1,1]
=> [3,1]
=> 1
[1,2,2] => [1,2] => [2,1]
=> [2,1]
=> 1
[1,3,1] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[1,4] => [1,1] => [1,1]
=> [2]
=> 1
[2,1,1,1] => [1,3] => [3,1]
=> [2,1,1]
=> 1
[2,1,2] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[2,2,1] => [2,1] => [2,1]
=> [2,1]
=> 1
[2,3] => [1,1] => [1,1]
=> [2]
=> 1
[3,1,1] => [1,2] => [2,1]
=> [2,1]
=> 1
[3,2] => [1,1] => [1,1]
=> [2]
=> 1
[4,1] => [1,1] => [1,1]
=> [2]
=> 1
[5] => [1] => [1]
=> [1]
=> ? = 1
[1,1,1,1,1,1] => [6] => [6]
=> [1,1,1,1,1,1]
=> 6
[1,1,1,1,2] => [4,1] => [4,1]
=> [2,1,1,1]
=> 1
[1,1,1,2,1] => [3,1,1] => [3,1,1]
=> [3,1,1]
=> 1
[1,1,1,3] => [3,1] => [3,1]
=> [2,1,1]
=> 1
[1,1,2,1,1] => [2,1,2] => [2,2,1]
=> [3,2]
=> 1
[1,1,2,2] => [2,2] => [2,2]
=> [2,2]
=> 2
[1,1,3,1] => [2,1,1] => [2,1,1]
=> [3,1]
=> 1
[1,1,4] => [2,1] => [2,1]
=> [2,1]
=> 1
[1,2,1,1,1] => [1,1,3] => [3,1,1]
=> [3,1,1]
=> 1
[1,2,1,2] => [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1
[1,2,2,1] => [1,2,1] => [2,1,1]
=> [3,1]
=> 1
[1,2,3] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[1,3,1,1] => [1,1,2] => [2,1,1]
=> [3,1]
=> 1
[1,3,2] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[1,4,1] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[1,5] => [1,1] => [1,1]
=> [2]
=> 1
[2,1,1,1,1] => [1,4] => [4,1]
=> [2,1,1,1]
=> 1
[2,1,1,2] => [1,2,1] => [2,1,1]
=> [3,1]
=> 1
[2,1,2,1] => [1,1,1,1] => [1,1,1,1]
=> [4]
=> 1
[2,1,3] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[2,2,1,1] => [2,2] => [2,2]
=> [2,2]
=> 2
[2,2,2] => [3] => [3]
=> [1,1,1]
=> 3
[2,3,1] => [1,1,1] => [1,1,1]
=> [3]
=> 1
[2,4] => [1,1] => [1,1]
=> [2]
=> 1
[6] => [1] => [1]
=> [1]
=> ? = 1
[7] => [1] => [1]
=> [1]
=> ? = 1
[8] => [1] => [1]
=> [1]
=> ? = 1
[9] => [1] => [1]
=> [1]
=> ? = 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000685
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000685: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 98%●distinct values known / distinct values provided: 78%
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000685: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 98%●distinct values known / distinct values provided: 78%
Values
[1] => [1] => [1] => [1,0]
=> 1
[1,1] => [2] => [1,1] => [1,0,1,0]
=> 2
[2] => [1] => [1] => [1,0]
=> 1
[1,1,1] => [3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,2] => [1,1] => [2] => [1,1,0,0]
=> 1
[2,1] => [1,1] => [2] => [1,1,0,0]
=> 1
[3] => [1] => [1] => [1,0]
=> 1
[1,1,1,1] => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,2,1] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,3] => [1,1] => [2] => [1,1,0,0]
=> 1
[2,1,1] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,2] => [2] => [1,1] => [1,0,1,0]
=> 2
[3,1] => [1,1] => [2] => [1,1,0,0]
=> 1
[4] => [1] => [1] => [1,0]
=> 1
[1,1,1,1,1] => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1] => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,3] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,2,1,1] => [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,2,2] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[1,3,1] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,4] => [1,1] => [2] => [1,1,0,0]
=> 1
[2,1,1,1] => [1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,2] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[2,2,1] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[2,3] => [1,1] => [2] => [1,1,0,0]
=> 1
[3,1,1] => [1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[3,2] => [1,1] => [2] => [1,1,0,0]
=> 1
[4,1] => [1,1] => [2] => [1,1,0,0]
=> 1
[5] => [1] => [1] => [1,0]
=> 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,2,1] => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,1,3] => [3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1,1] => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,2,2] => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,3,1] => [2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,4] => [2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[1,2,1,1,1] => [1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,2,1,2] => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,2,2,1] => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,2,3] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,3,1,1] => [1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,3,2] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,4,1] => [1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,5] => [1,1] => [2] => [1,1,0,0]
=> 1
[2,1,1,1,1] => [1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,1,2] => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,1,2,1] => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 8
[1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
[1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 1
[1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,3,1] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> ? = 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,3,1,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,3,1,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
[1,1,2,1,1,1,1,1] => [2,1,5] => [1,3,1,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,2,1,1,1,1,1,1] => [1,1,6] => [3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[2,1,1,1,1,1,1,1] => [1,7] => [2,1,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 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: St001038
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 98%●distinct values known / distinct values provided: 89%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 98%●distinct values known / distinct values provided: 89%
Values
[1] => [1] => [1]
=> [1,0]
=> ? = 1
[1,1] => [2] => [2]
=> [1,0,1,0]
=> 2
[2] => [1] => [1]
=> [1,0]
=> ? = 1
[1,1,1] => [3] => [3]
=> [1,0,1,0,1,0]
=> 3
[1,2] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[2,1] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[3] => [1] => [1]
=> [1,0]
=> ? = 1
[1,1,1,1] => [4] => [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[1,1,2] => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,1] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[2,1,1] => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,2] => [2] => [2]
=> [1,0,1,0]
=> 2
[3,1] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[4] => [1] => [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1] => [5] => [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,2] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1] => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,3] => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,1,1] => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,2] => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,3,1] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,4] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[2,1,1,1] => [1,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,2] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,2,1] => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[3,1,1] => [1,2] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[3,2] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[4,1] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[5] => [1] => [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,1] => [6] => [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,1,2] => [4,1] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,2,1] => [3,1,1] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,1,3] => [3,1] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1,1] => [2,1,2] => [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,2,2] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,3,1] => [2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,4] => [2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,1,1,1] => [1,1,3] => [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,1,2] => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[1,2,2,1] => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,2,3] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,3,1,1] => [1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,3,2] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,4,1] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,5] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[2,1,1,1,1] => [1,4] => [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[2,1,1,2] => [1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,2,1] => [1,1,1,1] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[2,1,3] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,2,1,1] => [2,2] => [2,2]
=> [1,1,1,0,0,0]
=> 2
[2,2,2] => [3] => [3]
=> [1,0,1,0,1,0]
=> 3
[2,3,1] => [1,1,1] => [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[2,4] => [1,1] => [1,1]
=> [1,1,0,0]
=> 1
[6] => [1] => [1]
=> [1,0]
=> ? = 1
[7] => [1] => [1]
=> [1,0]
=> ? = 1
[8] => [1] => [1]
=> [1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1] => [9] => [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[9] => [1] => [1]
=> [1,0]
=> ? = 1
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001075
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001075: Set partitions ⟶ ℤResult quality: 89% ●values known / values provided: 97%●distinct values known / distinct values provided: 89%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001075: Set partitions ⟶ ℤResult quality: 89% ●values known / values provided: 97%●distinct values known / distinct values provided: 89%
Values
[1] => [1] => [1,0]
=> {{1}}
=> ? = 1
[1,1] => [2] => [1,1,0,0]
=> {{1,2}}
=> 2
[2] => [1] => [1,0]
=> {{1}}
=> ? = 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[1,2] => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1
[2,1] => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1
[3] => [1] => [1,0]
=> {{1}}
=> ? = 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[1,3] => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[2,2] => [2] => [1,1,0,0]
=> {{1,2}}
=> 2
[3,1] => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1
[4] => [1] => [1,0]
=> {{1}}
=> ? = 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 5
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[1,4] => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[2,3] => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[3,2] => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1
[4,1] => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1
[5] => [1] => [1,0]
=> {{1}}
=> ? = 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,6}}
=> 6
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 1
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 1
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 1
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[1,5] => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1
[2,1,3] => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[2,2,1,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[2,2,2] => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[2,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[2,4] => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1
[6] => [1] => [1,0]
=> {{1}}
=> ? = 1
[7] => [1] => [1,0]
=> {{1}}
=> ? = 1
[8] => [1] => [1,0]
=> {{1}}
=> ? = 1
[1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> {{1,2,3,4,5,6,7,8,9}}
=> ? = 9
[1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> {{1,2,3,4,5,6},{7},{8}}
=> ? = 1
[1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> {{1,2,3,4,5},{6},{7,8}}
=> ? = 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> {{1,2,3,4},{5},{6,7,8}}
=> ? = 1
[1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> {{1,2,3},{4},{5,6,7,8}}
=> ? = 1
[1,1,2,1,1,1,1,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1,2},{3},{4,5,6,7,8}}
=> ? = 1
[9] => [1] => [1,0]
=> {{1}}
=> ? = 1
Description
The minimal size of a block of a set partition.
Matching statistic: St000667
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 95%●distinct values known / distinct values provided: 33%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 95%●distinct values known / distinct values provided: 33%
Values
[1] => [1] => [1]
=> []
=> ? = 1
[1,1] => [2] => [2]
=> []
=> ? = 2
[2] => [1] => [1]
=> []
=> ? = 1
[1,1,1] => [3] => [3]
=> []
=> ? = 3
[1,2] => [1,1] => [1,1]
=> [1]
=> 1
[2,1] => [1,1] => [1,1]
=> [1]
=> 1
[3] => [1] => [1]
=> []
=> ? = 1
[1,1,1,1] => [4] => [4]
=> []
=> ? = 4
[1,1,2] => [2,1] => [2,1]
=> [1]
=> 1
[1,2,1] => [1,1,1] => [1,1,1]
=> [1,1]
=> 1
[1,3] => [1,1] => [1,1]
=> [1]
=> 1
[2,1,1] => [1,2] => [2,1]
=> [1]
=> 1
[2,2] => [2] => [2]
=> []
=> ? = 2
[3,1] => [1,1] => [1,1]
=> [1]
=> 1
[4] => [1] => [1]
=> []
=> ? = 1
[1,1,1,1,1] => [5] => [5]
=> []
=> ? = 5
[1,1,1,2] => [3,1] => [3,1]
=> [1]
=> 1
[1,1,2,1] => [2,1,1] => [2,1,1]
=> [1,1]
=> 1
[1,1,3] => [2,1] => [2,1]
=> [1]
=> 1
[1,2,1,1] => [1,1,2] => [2,1,1]
=> [1,1]
=> 1
[1,2,2] => [1,2] => [2,1]
=> [1]
=> 1
[1,3,1] => [1,1,1] => [1,1,1]
=> [1,1]
=> 1
[1,4] => [1,1] => [1,1]
=> [1]
=> 1
[2,1,1,1] => [1,3] => [3,1]
=> [1]
=> 1
[2,1,2] => [1,1,1] => [1,1,1]
=> [1,1]
=> 1
[2,2,1] => [2,1] => [2,1]
=> [1]
=> 1
[2,3] => [1,1] => [1,1]
=> [1]
=> 1
[3,1,1] => [1,2] => [2,1]
=> [1]
=> 1
[3,2] => [1,1] => [1,1]
=> [1]
=> 1
[4,1] => [1,1] => [1,1]
=> [1]
=> 1
[5] => [1] => [1]
=> []
=> ? = 1
[1,1,1,1,1,1] => [6] => [6]
=> []
=> ? = 6
[1,1,1,1,2] => [4,1] => [4,1]
=> [1]
=> 1
[1,1,1,2,1] => [3,1,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,3] => [3,1] => [3,1]
=> [1]
=> 1
[1,1,2,1,1] => [2,1,2] => [2,2,1]
=> [2,1]
=> 1
[1,1,2,2] => [2,2] => [2,2]
=> [2]
=> 2
[1,1,3,1] => [2,1,1] => [2,1,1]
=> [1,1]
=> 1
[1,1,4] => [2,1] => [2,1]
=> [1]
=> 1
[1,2,1,1,1] => [1,1,3] => [3,1,1]
=> [1,1]
=> 1
[1,2,1,2] => [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,2,1] => [1,2,1] => [2,1,1]
=> [1,1]
=> 1
[1,2,3] => [1,1,1] => [1,1,1]
=> [1,1]
=> 1
[1,3,1,1] => [1,1,2] => [2,1,1]
=> [1,1]
=> 1
[1,3,2] => [1,1,1] => [1,1,1]
=> [1,1]
=> 1
[1,4,1] => [1,1,1] => [1,1,1]
=> [1,1]
=> 1
[1,5] => [1,1] => [1,1]
=> [1]
=> 1
[2,1,1,1,1] => [1,4] => [4,1]
=> [1]
=> 1
[2,1,1,2] => [1,2,1] => [2,1,1]
=> [1,1]
=> 1
[2,1,2,1] => [1,1,1,1] => [1,1,1,1]
=> [1,1,1]
=> 1
[2,1,3] => [1,1,1] => [1,1,1]
=> [1,1]
=> 1
[2,2,1,1] => [2,2] => [2,2]
=> [2]
=> 2
[2,2,2] => [3] => [3]
=> []
=> ? = 3
[2,3,1] => [1,1,1] => [1,1,1]
=> [1,1]
=> 1
[2,4] => [1,1] => [1,1]
=> [1]
=> 1
[3,1,1,1] => [1,3] => [3,1]
=> [1]
=> 1
[3,1,2] => [1,1,1] => [1,1,1]
=> [1,1]
=> 1
[3,2,1] => [1,1,1] => [1,1,1]
=> [1,1]
=> 1
[3,3] => [2] => [2]
=> []
=> ? = 2
[4,1,1] => [1,2] => [2,1]
=> [1]
=> 1
[4,2] => [1,1] => [1,1]
=> [1]
=> 1
[5,1] => [1,1] => [1,1]
=> [1]
=> 1
[6] => [1] => [1]
=> []
=> ? = 1
[1,1,1,1,1,1,1] => [7] => [7]
=> []
=> ? = 7
[1,1,1,1,1,2] => [5,1] => [5,1]
=> [1]
=> 1
[7] => [1] => [1]
=> []
=> ? = 1
[1,1,1,1,1,1,1,1] => [8] => [8]
=> []
=> ? = 8
[2,2,2,2] => [4] => [4]
=> []
=> ? = 4
[4,4] => [2] => [2]
=> []
=> ? = 2
[8] => [1] => [1]
=> []
=> ? = 1
[1,1,1,1,1,1,1,1,1] => [9] => [9]
=> []
=> ? = 9
[3,3,3] => [3] => [3]
=> []
=> ? = 3
[9] => [1] => [1]
=> []
=> ? = 1
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000210
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000210: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 93%●distinct values known / distinct values provided: 67%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000210: Permutations ⟶ ℤResult quality: 67% ●values known / values provided: 93%●distinct values known / distinct values provided: 67%
Values
[1] => [1] => [1,0]
=> [1] => 0 = 1 - 1
[1,1] => [2] => [1,1,0,0]
=> [2,1] => 1 = 2 - 1
[2] => [1] => [1,0]
=> [1] => 0 = 1 - 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [3,1,2] => 2 = 3 - 1
[1,2] => [1,1] => [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[2,1] => [1,1] => [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[3] => [1] => [1,0]
=> [1] => 0 = 1 - 1
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 3 = 4 - 1
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,3] => [1,1] => [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[2,2] => [2] => [1,1,0,0]
=> [2,1] => 1 = 2 - 1
[3,1] => [1,1] => [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[4] => [1] => [1,0]
=> [1] => 0 = 1 - 1
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => 4 = 5 - 1
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0 = 1 - 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,4] => [1,1] => [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0 = 1 - 1
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[2,3] => [1,1] => [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
[3,2] => [1,1] => [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[4,1] => [1,1] => [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[5] => [1] => [1,0]
=> [1] => 0 = 1 - 1
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => 5 = 6 - 1
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 0 = 1 - 1
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 0 = 1 - 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 0 = 1 - 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 0 = 1 - 1
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1 = 2 - 1
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0 = 1 - 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0 = 1 - 1
[1,2,1,2] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,2,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[1,2,3] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,3,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0 = 1 - 1
[1,3,2] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,4,1] => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
[1,5] => [1,1] => [1,0,1,0]
=> [1,2] => 0 = 1 - 1
[2,1,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => 0 = 1 - 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0 = 1 - 1
[2,1,2,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0 = 1 - 1
[1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7,1,2,3,4,5,6] => ? = 7 - 1
[1,1,1,1,1,1,1,1] => [8] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [8,1,2,3,4,5,6,7] => ? = 8 - 1
[1,1,1,1,1,1,2] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 1 - 1
[1,1,1,1,1,2,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => ? = 1 - 1
[1,1,1,1,2,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => ? = 1 - 1
[1,1,1,2,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 1 - 1
[1,1,2,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,4,5,6] => ? = 1 - 1
[2,1,1,1,1,1,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => ? = 1 - 1
[1,1,1,1,1,1,1,1,1] => [9] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [9,1,2,3,4,5,6,7,8] => ? = 9 - 1
[1,1,1,1,1,1,1,2] => [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,6,8] => ? = 1 - 1
[1,1,1,1,1,1,2,1] => [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [6,1,2,3,4,5,7,8] => ? = 1 - 1
[1,1,1,1,1,1,3] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 1 - 1
[1,1,1,1,1,2,1,1] => [5,1,2] => [1,1,1,1,1,0,0,0,0,0,1,0,1,1,0,0]
=> [5,1,2,3,4,6,8,7] => ? = 1 - 1
[1,1,1,1,1,2,2] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 2 - 1
[1,1,1,1,1,3,1] => [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [5,1,2,3,4,6,7] => ? = 1 - 1
[1,1,1,1,2,1,1,1] => [4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [4,1,2,3,5,8,6,7] => ? = 1 - 1
[1,1,1,1,2,1,2] => [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [4,1,2,3,5,6,7] => ? = 1 - 1
[1,1,1,1,2,2,1] => [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> [4,1,2,3,6,5,7] => ? = 1 - 1
[1,1,1,1,3,1,1] => [4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> [4,1,2,3,5,7,6] => ? = 1 - 1
[1,1,1,2,1,1,1,1] => [3,1,4] => [1,1,1,0,0,0,1,0,1,1,1,1,0,0,0,0]
=> [3,1,2,4,8,5,6,7] => ? = 1 - 1
[1,1,1,2,1,1,2] => [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [3,1,2,4,6,5,7] => ? = 1 - 1
[1,1,1,2,1,2,1] => [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [3,1,2,4,5,6,7] => ? = 1 - 1
[1,1,1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [3,1,2,5,4,7,6] => ? = 2 - 1
[1,1,1,3,1,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,2,4,7,5,6] => ? = 1 - 1
[1,1,2,1,1,1,1,1] => [2,1,5] => [1,1,0,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,8,4,5,6,7] => ? = 1 - 1
[1,1,2,1,1,1,2] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,6,4,5,7] => ? = 1 - 1
[1,1,2,1,1,2,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [2,1,3,5,4,6,7] => ? = 1 - 1
[1,1,2,1,2,1,1] => [2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,5,7,6] => ? = 1 - 1
[1,1,2,2,1,1,1] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [2,1,4,3,7,5,6] => ? = 2 - 1
[1,1,3,1,1,1,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,7,4,5,6] => ? = 1 - 1
[1,2,1,1,1,1,1,1] => [1,1,6] => [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,8,3,4,5,6,7] => ? = 1 - 1
[2,1,1,1,1,1,1,1] => [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,2,3,4,5,6,7] => ? = 1 - 1
[2,1,1,1,1,1,2] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,6,2,3,4,5,7] => ? = 1 - 1
[2,1,1,1,1,2,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,5,2,3,4,6,7] => ? = 1 - 1
[2,2,1,1,1,1,1] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? = 2 - 1
[3,1,1,1,1,1,1] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,2,3,4,5,6] => ? = 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 $n-1$.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000487The length of the shortest cycle of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000264The girth of a graph, which is not a tree. St000455The second largest eigenvalue of a graph if it is integral. St000456The monochromatic index of a connected graph.
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