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Your data matches 15 different statistics following compositions of up to 3 maps.
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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
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 1
[1,1] => [2] => [1,1,0,0]
=> 2
[2] => [1] => [1,0]
=> 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
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 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00039: Integer compositions —complement⟶ Integer compositions
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%
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
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: 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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> [[]]
=> 1
[1,1] => [2] => [1,1,0,0]
=> [[[]]]
=> 2
[2] => [1] => [1,0]
=> [[]]
=> 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
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: 99% ●values known / values provided: 99%●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: 99% ●values known / values provided: 99%●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
[10] => [1] => [1]
=> [1]
=> ? = 1
[12] => [1] => [1]
=> [1]
=> ? = 1
[11] => [1] => [1]
=> [1]
=> ? = 1
Description
The multiplicity of the largest part of an integer partition.
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: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
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
[9] => [1] => [1]
=> [1,0]
=> ? = 1
[10] => [1] => [1]
=> [1,0]
=> ? = 1
[12] => [1] => [1]
=> [1,0]
=> ? = 1
[11] => [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: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001075: Set partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
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
[9] => [1] => [1,0]
=> {{1}}
=> ? = 1
[10] => [1] => [1,0]
=> {{1}}
=> ? = 1
[12] => [1] => [1,0]
=> {{1}}
=> ? = 1
[11] => [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: 43% ●values known / values provided: 98%●distinct values known / distinct values provided: 43%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 43% ●values known / values provided: 98%●distinct values known / distinct values provided: 43%
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
[2,2,2,2] => [4] => [4]
=> []
=> ? = 4
[4,4] => [2] => [2]
=> []
=> ? = 2
[8] => [1] => [1]
=> []
=> ? = 1
[3,3,3] => [3] => [3]
=> []
=> ? = 3
[9] => [1] => [1]
=> []
=> ? = 1
[2,2,2,2,2] => [5] => [5]
=> []
=> ? = 5
[5,5] => [2] => [2]
=> []
=> ? = 2
[10] => [1] => [1]
=> []
=> ? = 1
[2,2,2,2,2,2] => [6] => [6]
=> []
=> ? = 6
[3,3,3,3] => [4] => [4]
=> []
=> ? = 4
[6,6] => [2] => [2]
=> []
=> ? = 2
[4,4,4] => [3] => [3]
=> []
=> ? = 3
[12] => [1] => [1]
=> []
=> ? = 1
[11] => [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: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 86%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000210: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 86%
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
[1,1,1,1,1,1,2] => [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
[1,1,1,1,1,1,3] => [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
[1,1,1,1,1,2,2] => [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
[1,1,1,1,2,1,2] => [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
[1,1,1,2,1,1,2] => [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
[1,1,2,1,1,1,2] => [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
[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
[1,1,1,1,1,1,4] => [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
[1,1,1,1,1,2,3] => [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
[1,1,1,1,1,3,2] => [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
[1,1,1,1,1,4,1] => [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
[1,1,1,1,2,1,3] => [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
[1,1,1,1,2,2,2] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 3 - 1
[1,1,1,1,2,3,1] => [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
[1,1,1,1,3,1,2] => [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
[1,1,1,1,3,2,1] => [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
[1,1,1,1,4,1,1] => [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
[1,1,1,2,1,1,3] => [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
[1,1,1,2,1,2,2] => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,1,2,4,5,7,6] => ? = 1 - 1
[1,1,1,2,1,3,1] => [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
[1,1,1,2,2,1,2] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,1,2,5,4,6,7] => ? = 1 - 1
[1,1,1,2,2,2,1] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 1 - 1
[1,1,1,2,3,1,1] => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,1,2,4,5,7,6] => ? = 1 - 1
[1,1,1,3,1,1,2] => [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
[1,1,1,3,1,2,1] => [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
[1,1,1,3,2,1,1] => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,1,2,4,5,7,6] => ? = 1 - 1
[1,1,1,4,1,1,1] => [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
[1,1,2,1,1,1,3] => [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
[1,1,2,1,1,2,2] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [2,1,3,5,4,7,6] => ? = 1 - 1
[1,1,2,1,1,3,1] => [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
[1,1,2,1,2,1,2] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6,7] => ? = 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.
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: St000487
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000487: Permutations ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 86%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000487: Permutations ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 86%
Values
[1] => [1] => [1,0]
=> [1] => ? = 1
[1,1] => [2] => [1,1,0,0]
=> [2,1] => 2
[2] => [1] => [1,0]
=> [1] => ? = 1
[1,1,1] => [3] => [1,1,1,0,0,0]
=> [3,1,2] => 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]
=> [4,1,2,3] => 4
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 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]
=> [1,3,2] => 1
[2,2] => [2] => [1,1,0,0]
=> [2,1] => 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]
=> [5,1,2,3,4] => 5
[1,1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
[1,1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,3] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 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]
=> [1,4,2,3] => 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]
=> [2,1,3] => 1
[2,3] => [1,1] => [1,0,1,0]
=> [1,2] => 1
[3,1,1] => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 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]
=> [6,1,2,3,4,5] => 6
[1,1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => 1
[1,1,1,2,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => 1
[1,1,1,3] => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,1,2,4] => 1
[1,1,2,1,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
[1,1,2,2] => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
[1,1,4] => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
[1,2,1,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 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]
=> [1,3,2,4] => 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]
=> [1,2,4,3] => 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]
=> [1,5,2,3,4] => 1
[2,1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 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]
=> [2,1,4,3] => 2
[2,2,2] => [3] => [1,1,1,0,0,0]
=> [3,1,2] => 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
[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
[7] => [1] => [1,0]
=> [1] => ? = 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
[1,1,1,1,1,2,1] => [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
[1,1,1,1,2,1,1] => [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
[1,1,1,2,1,1,1] => [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
[1,1,2,1,1,1,1] => [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
[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
[8] => [1] => [1,0]
=> [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
[1,1,1,1,1,2,2] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,1,2,3,4,7,6] => ? = 2
[1,1,1,1,1,3,1] => [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
[1,1,1,1,2,1,2] => [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
[1,1,1,1,2,2,1] => [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
[1,1,1,1,3,1,1] => [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
[1,1,1,2,1,1,2] => [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
[1,1,1,2,1,2,1] => [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
[1,1,1,2,2,1,1] => [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
[1,1,1,3,1,1,1] => [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
[1,1,2,1,1,1,2] => [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
[1,1,2,1,1,2,1] => [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
[1,1,2,1,2,1,1] => [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
[1,1,2,2,1,1,1] => [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
[1,1,3,1,1,1,1] => [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
[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
[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
[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
[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
[9] => [1] => [1,0]
=> [1] => ? = 1
[1,1,1,1,1,1,4] => [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1,2,3,4,5,7] => ? = 1
[1,1,1,1,1,2,3] => [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
[1,1,1,1,1,3,2] => [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
[1,1,1,1,1,4,1] => [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
[1,1,1,1,2,1,3] => [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
[1,1,1,1,2,2,2] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> [4,1,2,3,7,5,6] => ? = 3
[1,1,1,1,2,3,1] => [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
[1,1,1,1,3,1,2] => [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
[1,1,1,1,3,2,1] => [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
[1,1,1,1,4,1,1] => [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
[1,1,1,2,1,1,3] => [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
[1,1,1,2,1,2,2] => [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [3,1,2,4,5,7,6] => ? = 1
[1,1,1,2,1,3,1] => [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
[1,1,1,2,2,1,2] => [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [3,1,2,5,4,6,7] => ? = 1
[1,1,1,2,2,2,1] => [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> [3,1,2,6,4,5,7] => ? = 1
Description
The length of the shortest cycle of a permutation.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000900The minimal number of repetitions of a part in an integer composition. St000455The second largest eigenvalue of a graph if it is integral. St000456The monochromatic index of a connected graph.
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