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Your data matches 68 different statistics following compositions of up to 3 maps.
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Matching statistic: St000900
(load all 9 compositions to match this statistic)
(load all 9 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 20 compositions to match this statistic)
(load all 20 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 6 compositions to match this statistic)
(load all 6 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]
=> [[[[[[[[[[]]]]]]]]]]
=> ? ∊ {1,1,1,1,9}
[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]
=> [[[[[[[]]]]]],[],[]]
=> ? ∊ {1,1,1,1,9}
[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]
=> [[[[[[]]]]],[],[[]]]
=> ? ∊ {1,1,1,1,9}
[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]
=> [[[[[]]]],[],[[[]]]]
=> ? ∊ {1,1,1,1,9}
[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]
=> [[[[]]],[],[[[[]]]]]
=> ? ∊ {1,1,1,1,9}
Description
The protection number of an ordered tree.
This is the minimal distance from the root to a leaf.
Matching statistic: St000993
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
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
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]
=> ? ∊ {1,1,1,1,1,1,1,1,9}
[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]
=> ? ∊ {1,1,1,1,1,1,1,1,9}
[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]
=> ? ∊ {1,1,1,1,1,1,1,1,9}
[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]
=> ? ∊ {1,1,1,1,1,1,1,1,9}
[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]
=> ? ∊ {1,1,1,1,1,1,1,1,9}
[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]
=> ? ∊ {1,1,1,1,1,1,1,1,9}
[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]
=> ? ∊ {1,1,1,1,1,1,1,1,9}
[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]
=> ? ∊ {1,1,1,1,1,1,1,1,9}
[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]
=> ? ∊ {1,1,1,1,1,1,1,1,9}
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]
=> ? ∊ {1,9}
[9] => [1] => [1]
=> [1,0]
=> ? ∊ {1,9}
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}}
=> ? ∊ {1,1,1,1,1,1,9}
[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}}
=> ? ∊ {1,1,1,1,1,1,9}
[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}}
=> ? ∊ {1,1,1,1,1,1,9}
[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}}
=> ? ∊ {1,1,1,1,1,1,9}
[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}}
=> ? ∊ {1,1,1,1,1,1,9}
[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}}
=> ? ∊ {1,1,1,1,1,1,9}
[9] => [1] => [1,0]
=> {{1}}
=> ? ∊ {1,1,1,1,1,1,9}
Description
The minimal size of a block of a set partition.
Matching statistic: St000667
(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
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]
=> []
=> ? ∊ {1,2}
[2] => [1] => [1]
=> []
=> ? ∊ {1,2}
[1,1,1] => [3] => [3]
=> []
=> ? ∊ {1,3}
[1,2] => [1,1] => [1,1]
=> [1]
=> 1
[2,1] => [1,1] => [1,1]
=> [1]
=> 1
[3] => [1] => [1]
=> []
=> ? ∊ {1,3}
[1,1,1,1] => [4] => [4]
=> []
=> ? ∊ {1,2,4}
[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]
=> []
=> ? ∊ {1,2,4}
[3,1] => [1,1] => [1,1]
=> [1]
=> 1
[4] => [1] => [1]
=> []
=> ? ∊ {1,2,4}
[1,1,1,1,1] => [5] => [5]
=> []
=> ? ∊ {1,5}
[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]
=> []
=> ? ∊ {1,5}
[1,1,1,1,1,1] => [6] => [6]
=> []
=> ? ∊ {1,2,3,6}
[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]
=> []
=> ? ∊ {1,2,3,6}
[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]
=> []
=> ? ∊ {1,2,3,6}
[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]
=> []
=> ? ∊ {1,2,3,6}
[1,1,1,1,1,1,1] => [7] => [7]
=> []
=> ? ∊ {1,7}
[1,1,1,1,1,2] => [5,1] => [5,1]
=> [1]
=> 1
[7] => [1] => [1]
=> []
=> ? ∊ {1,7}
[1,1,1,1,1,1,1,1] => [8] => [8]
=> []
=> ? ∊ {1,2,4,8}
[2,2,2,2] => [4] => [4]
=> []
=> ? ∊ {1,2,4,8}
[4,4] => [2] => [2]
=> []
=> ? ∊ {1,2,4,8}
[8] => [1] => [1]
=> []
=> ? ∊ {1,2,4,8}
[1,1,1,1,1,1,1,1,1] => [9] => [9]
=> []
=> ? ∊ {1,3,9}
[3,3,3] => [3] => [3]
=> []
=> ? ∊ {1,3,9}
[9] => [1] => [1]
=> []
=> ? ∊ {1,3,9}
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] => ? ∊ {1,1,1,1,1,1,8} - 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] => ? ∊ {1,1,1,1,1,1,8} - 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] => ? ∊ {1,1,1,1,1,1,8} - 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] => ? ∊ {1,1,1,1,1,1,8} - 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] => ? ∊ {1,1,1,1,1,1,8} - 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] => ? ∊ {1,1,1,1,1,1,8} - 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] => ? ∊ {1,1,1,1,1,1,8} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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] => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,9} - 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 58 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. St000781The number of proper colouring schemes of a Ferrers diagram. St000326The position of the first one in a binary word after appending a 1 at the end. St000847The number of standard Young tableaux whose descent set is the binary word. St001722The number of minimal chains with small intervals between a binary word and the top element. St001011Number of simple modules of projective dimension 2 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. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000706The product of the factorials of the multiplicities of an integer partition. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. 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. St001568The smallest positive integer that does not appear twice in the partition. St000914The sum of the values of the Möbius function of a poset. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew 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. St000618The number of self-evacuating tableaux of given shape. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001571The Cartan determinant of the integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001890The maximum magnitude of the Möbius function of a poset. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. 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. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000782The indicator function of whether a given perfect matching is an L & P matching. St000627The exponent of a binary word. St001877Number of indecomposable injective modules with projective dimension 2. St000617The number of global maxima of a Dyck path. 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. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph.
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