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Your data matches 49 different statistics following compositions of up to 3 maps.
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Matching statistic: St000657
(load all 92 compositions to match this statistic)
(load all 92 compositions to match this statistic)
St000657: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1
[1,1] => 1
[2] => 2
[1,1,1] => 1
[1,2] => 1
[2,1] => 1
[3] => 3
[1,1,1,1] => 1
[1,1,2] => 1
[1,2,1] => 1
[1,3] => 1
[2,1,1] => 1
[2,2] => 2
[3,1] => 1
[4] => 4
[1,1,1,1,1] => 1
[1,1,1,2] => 1
[1,1,2,1] => 1
[1,1,3] => 1
[1,2,1,1] => 1
[1,2,2] => 1
[1,3,1] => 1
[1,4] => 1
[2,1,1,1] => 1
[2,1,2] => 1
[2,2,1] => 1
[2,3] => 2
[3,1,1] => 1
[3,2] => 2
[4,1] => 1
[5] => 5
[1,1,1,1,1,1] => 1
[1,1,1,1,2] => 1
[1,1,1,2,1] => 1
[1,1,1,3] => 1
[1,1,2,1,1] => 1
[1,1,2,2] => 1
[1,1,3,1] => 1
[1,1,4] => 1
[1,2,1,1,1] => 1
[1,2,1,2] => 1
[1,2,2,1] => 1
[1,2,3] => 1
[1,3,1,1] => 1
[1,3,2] => 1
[1,4,1] => 1
[1,5] => 1
[2,1,1,1,1] => 1
[2,1,1,2] => 1
[2,1,2,1] => 1
Description
The smallest part of an integer composition.
Matching statistic: St000655
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000655: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1
[1,1] => [1,0,1,0]
=> 1
[2] => [1,1,0,0]
=> 2
[1,1,1] => [1,0,1,0,1,0]
=> 1
[1,2] => [1,0,1,1,0,0]
=> 1
[2,1] => [1,1,0,0,1,0]
=> 1
[3] => [1,1,1,0,0,0]
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 5
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1
Description
The length of the minimal rise of a Dyck path.
For the length of a maximal rise, see [[St000444]].
Matching statistic: St000685
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
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%
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] => [1,0]
=> 1
[1,1] => [2] => [1,1,0,0]
=> 1
[2] => [1,1] => [1,0,1,0]
=> 2
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 1
[1,2] => [2,1] => [1,1,0,0,1,0]
=> 1
[2,1] => [1,2] => [1,0,1,1,0,0]
=> 1
[3] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,2] => [3,1] => [1,1,1,0,0,0,1,0]
=> 1
[1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[1,3] => [2,1,1] => [1,1,0,0,1,0,1,0]
=> 1
[2,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 1
[2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[3,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4
[1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,2] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,2,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,3] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,2,1,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[1,4] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,1,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[2,2,1] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 1
[2,3] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1
[3,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[4,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 1
[5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5
[1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,1,1,2] => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,1,1,2,1] => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
[1,1,1,3] => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 1
[1,1,2,1,1] => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> 1
[1,1,2,2] => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 1
[1,1,3,1] => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> 1
[1,1,4] => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 1
[1,2,1,1,1] => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> 1
[1,2,1,2] => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1
[1,2,2,1] => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> 1
[1,2,3] => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,1,1] => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2] => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1
[1,4,1] => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 1
[1,5] => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1
[2,1,1,1,1] => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[2,1,1,2] => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
[2,1,2,1] => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 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
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000700: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> [[]]
=> 1
[1,1] => [1,0,1,0]
=> [[],[]]
=> 1
[2] => [1,1,0,0]
=> [[[]]]
=> 2
[1,1,1] => [1,0,1,0,1,0]
=> [[],[],[]]
=> 1
[1,2] => [1,0,1,1,0,0]
=> [[],[[]]]
=> 1
[2,1] => [1,1,0,0,1,0]
=> [[[]],[]]
=> 1
[3] => [1,1,1,0,0,0]
=> [[[[]]]]
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 4
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [[[[]]],[],[]]
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [[[[]]],[[]]]
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> 5
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[],[]]
=> 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[],[[]]]
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[],[],[],[[]],[]]
=> 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[],[],[],[[[]]]]
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[],[],[[]],[],[]]
=> 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[],[],[[]],[[]]]
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[],[],[[[]]],[]]
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[],[],[[[[]]]]]
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[],[[]],[],[],[]]
=> 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[],[[]],[],[[]]]
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[],[[]],[[]],[]]
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [[],[[]],[[[]]]]
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [[],[[[]]],[],[]]
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [[],[[[]]],[[]]]
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [[],[[[[]]]],[]]
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [[[]],[],[],[],[]]
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [[[]],[],[],[[]]]
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [[[]],[],[[]],[]]
=> 1
Description
The protection number of an ordered tree.
This is the minimal distance from the root to a leaf.
Matching statistic: St000908
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000908: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> ([],1)
=> 1
[1,1] => [1,0,1,0]
=> ([(0,1)],2)
=> 1
[2] => [1,1,0,0]
=> ([],2)
=> 2
[1,1,1] => [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 1
[1,2] => [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 1
[2,1] => [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 1
[3] => [1,1,1,0,0,0]
=> ([],3)
=> 3
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
[1,3] => [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 1
[2,2] => [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[3,1] => [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 1
[4] => [1,1,1,1,0,0,0,0]
=> ([],4)
=> 4
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> ([(0,4),(1,4),(2,4),(4,3)],5)
=> 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(4,1),(4,2),(4,3)],5)
=> 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> 2
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4)],5)
=> 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> ([],5)
=> 5
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> ([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
=> 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,5),(3,5),(4,5)],6)
=> 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> ([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> ([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> 1
Description
The length of the shortest maximal antichain in a poset.
Matching statistic: St000297
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> 10 => 1
[1,1] => [1,1]
=> [2]
=> 100 => 1
[2] => [2]
=> [1,1]
=> 110 => 2
[1,1,1] => [1,1,1]
=> [3]
=> 1000 => 1
[1,2] => [2,1]
=> [2,1]
=> 1010 => 1
[2,1] => [2,1]
=> [2,1]
=> 1010 => 1
[3] => [3]
=> [1,1,1]
=> 1110 => 3
[1,1,1,1] => [1,1,1,1]
=> [4]
=> 10000 => 1
[1,1,2] => [2,1,1]
=> [3,1]
=> 10010 => 1
[1,2,1] => [2,1,1]
=> [3,1]
=> 10010 => 1
[1,3] => [3,1]
=> [2,1,1]
=> 10110 => 1
[2,1,1] => [2,1,1]
=> [3,1]
=> 10010 => 1
[2,2] => [2,2]
=> [2,2]
=> 1100 => 2
[3,1] => [3,1]
=> [2,1,1]
=> 10110 => 1
[4] => [4]
=> [1,1,1,1]
=> 11110 => 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [5]
=> 100000 => 1
[1,1,1,2] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[1,1,2,1] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[1,1,3] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[1,2,1,1] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[1,2,2] => [2,2,1]
=> [3,2]
=> 10100 => 1
[1,3,1] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[1,4] => [4,1]
=> [2,1,1,1]
=> 101110 => 1
[2,1,1,1] => [2,1,1,1]
=> [4,1]
=> 100010 => 1
[2,1,2] => [2,2,1]
=> [3,2]
=> 10100 => 1
[2,2,1] => [2,2,1]
=> [3,2]
=> 10100 => 1
[2,3] => [3,2]
=> [2,2,1]
=> 11010 => 2
[3,1,1] => [3,1,1]
=> [3,1,1]
=> 100110 => 1
[3,2] => [3,2]
=> [2,2,1]
=> 11010 => 2
[4,1] => [4,1]
=> [2,1,1,1]
=> 101110 => 1
[5] => [5]
=> [1,1,1,1,1]
=> 111110 => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [6]
=> 1000000 => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [5,1]
=> 1000010 => 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [5,1]
=> 1000010 => 1
[1,1,1,3] => [3,1,1,1]
=> [4,1,1]
=> 1000110 => 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1000010 => 1
[1,1,2,2] => [2,2,1,1]
=> [4,2]
=> 100100 => 1
[1,1,3,1] => [3,1,1,1]
=> [4,1,1]
=> 1000110 => 1
[1,1,4] => [4,1,1]
=> [3,1,1,1]
=> 1001110 => 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1000010 => 1
[1,2,1,2] => [2,2,1,1]
=> [4,2]
=> 100100 => 1
[1,2,2,1] => [2,2,1,1]
=> [4,2]
=> 100100 => 1
[1,2,3] => [3,2,1]
=> [3,2,1]
=> 101010 => 1
[1,3,1,1] => [3,1,1,1]
=> [4,1,1]
=> 1000110 => 1
[1,3,2] => [3,2,1]
=> [3,2,1]
=> 101010 => 1
[1,4,1] => [4,1,1]
=> [3,1,1,1]
=> 1001110 => 1
[1,5] => [5,1]
=> [2,1,1,1,1]
=> 1011110 => 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [5,1]
=> 1000010 => 1
[2,1,1,2] => [2,2,1,1]
=> [4,2]
=> 100100 => 1
[2,1,2,1] => [2,2,1,1]
=> [4,2]
=> 100100 => 1
Description
The number of leading ones in a binary word.
Matching statistic: St000326
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 10 => 10 => 1
[1,1] => [1,1]
=> 110 => 110 => 1
[2] => [2]
=> 100 => 010 => 2
[1,1,1] => [1,1,1]
=> 1110 => 1110 => 1
[1,2] => [2,1]
=> 1010 => 1100 => 1
[2,1] => [2,1]
=> 1010 => 1100 => 1
[3] => [3]
=> 1000 => 0010 => 3
[1,1,1,1] => [1,1,1,1]
=> 11110 => 11110 => 1
[1,1,2] => [2,1,1]
=> 10110 => 11010 => 1
[1,2,1] => [2,1,1]
=> 10110 => 11010 => 1
[1,3] => [3,1]
=> 10010 => 10100 => 1
[2,1,1] => [2,1,1]
=> 10110 => 11010 => 1
[2,2] => [2,2]
=> 1100 => 0110 => 2
[3,1] => [3,1]
=> 10010 => 10100 => 1
[4] => [4]
=> 10000 => 00010 => 4
[1,1,1,1,1] => [1,1,1,1,1]
=> 111110 => 111110 => 1
[1,1,1,2] => [2,1,1,1]
=> 101110 => 110110 => 1
[1,1,2,1] => [2,1,1,1]
=> 101110 => 110110 => 1
[1,1,3] => [3,1,1]
=> 100110 => 101010 => 1
[1,2,1,1] => [2,1,1,1]
=> 101110 => 110110 => 1
[1,2,2] => [2,2,1]
=> 11010 => 11100 => 1
[1,3,1] => [3,1,1]
=> 100110 => 101010 => 1
[1,4] => [4,1]
=> 100010 => 100100 => 1
[2,1,1,1] => [2,1,1,1]
=> 101110 => 110110 => 1
[2,1,2] => [2,2,1]
=> 11010 => 11100 => 1
[2,2,1] => [2,2,1]
=> 11010 => 11100 => 1
[2,3] => [3,2]
=> 10100 => 01100 => 2
[3,1,1] => [3,1,1]
=> 100110 => 101010 => 1
[3,2] => [3,2]
=> 10100 => 01100 => 2
[4,1] => [4,1]
=> 100010 => 100100 => 1
[5] => [5]
=> 100000 => 000010 => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 1111110 => 1111110 => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> 1011110 => 1101110 => 1
[1,1,1,2,1] => [2,1,1,1,1]
=> 1011110 => 1101110 => 1
[1,1,1,3] => [3,1,1,1]
=> 1001110 => 1010110 => 1
[1,1,2,1,1] => [2,1,1,1,1]
=> 1011110 => 1101110 => 1
[1,1,2,2] => [2,2,1,1]
=> 110110 => 111010 => 1
[1,1,3,1] => [3,1,1,1]
=> 1001110 => 1010110 => 1
[1,1,4] => [4,1,1]
=> 1000110 => 1001010 => 1
[1,2,1,1,1] => [2,1,1,1,1]
=> 1011110 => 1101110 => 1
[1,2,1,2] => [2,2,1,1]
=> 110110 => 111010 => 1
[1,2,2,1] => [2,2,1,1]
=> 110110 => 111010 => 1
[1,2,3] => [3,2,1]
=> 101010 => 111000 => 1
[1,3,1,1] => [3,1,1,1]
=> 1001110 => 1010110 => 1
[1,3,2] => [3,2,1]
=> 101010 => 111000 => 1
[1,4,1] => [4,1,1]
=> 1000110 => 1001010 => 1
[1,5] => [5,1]
=> 1000010 => 1000100 => 1
[2,1,1,1,1] => [2,1,1,1,1]
=> 1011110 => 1101110 => 1
[2,1,1,2] => [2,2,1,1]
=> 110110 => 111010 => 1
[2,1,2,1] => [2,2,1,1]
=> 110110 => 111010 => 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000382
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [1] => 1
[1,1] => [1,1]
=> [[1],[2]]
=> [1,1] => 1
[2] => [2]
=> [[1,2]]
=> [2] => 2
[1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[1,2] => [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
[2,1] => [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
[3] => [3]
=> [[1,2,3]]
=> [3] => 3
[1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[1,1,2] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[1,2,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[1,3] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[2,1,1] => [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
[2,2] => [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[3,1] => [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
[4] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[1,1,1,2] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,1,2,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,1,3] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,2,1,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[1,2,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[1,3,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[1,4] => [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
[2,1,1,1] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
[2,1,2] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[2,2,1] => [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
[2,3] => [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
[3,1,1] => [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
[3,2] => [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
[4,1] => [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
[5] => [5]
=> [[1,2,3,4,5]]
=> [5] => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[1,1,1,3] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[1,1,2,2] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 1
[1,1,3,1] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 1
[1,1,4] => [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[1,2,1,2] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 1
[1,2,2,1] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 1
[1,2,3] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => 1
[1,3,1,1] => [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 1
[1,3,2] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => 1
[1,4,1] => [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 1
[1,5] => [5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
[2,1,1,2] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 1
[2,1,2,1] => [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [1] => 1
[1,1] => [1,1]
=> [[1],[2]]
=> [1,1] => 1
[2] => [2]
=> [[1,2]]
=> [2] => 2
[1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
[1,2] => [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[2,1] => [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[3] => [3]
=> [[1,2,3]]
=> [3] => 3
[1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
[1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[1,3] => [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
[2,2] => [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
[3,1] => [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
[4] => [4]
=> [[1,2,3,4]]
=> [4] => 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
[1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
[1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[1,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
[2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
[2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
[2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
[2,3] => [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 2
[3,1,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
[3,2] => [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 2
[4,1] => [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
[5] => [5]
=> [[1,2,3,4,5]]
=> [5] => 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 1
[1,1,1,3] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 1
[1,1,2,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
[1,1,3,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 1
[1,1,4] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 1
[1,2,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
[1,2,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
[1,2,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[1,3,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 1
[1,3,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
[1,4,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
[1,5] => [5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 1
[2,1,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
[2,1,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
Description
The last part of an integer composition.
Matching statistic: St000733
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [[1]]
=> 1
[1,1] => [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1
[2] => [2]
=> [[1,2]]
=> [[1],[2]]
=> 2
[1,1,1] => [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
[1,2] => [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
[2,1] => [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
[3] => [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 3
[1,1,1,1] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[1,1,2] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[1,2,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[1,3] => [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 1
[2,1,1] => [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[2,2] => [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[3,1] => [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 1
[4] => [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4
[1,1,1,1,1] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
[1,1,1,2] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
[1,1,2,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
[1,1,3] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
[1,2,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
[1,2,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
[1,3,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
[1,4] => [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
[2,1,1,1] => [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
[2,1,2] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
[2,2,1] => [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
[2,3] => [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 2
[3,1,1] => [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
[3,2] => [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 2
[4,1] => [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
[5] => [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 5
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 1
[1,1,1,1,2] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 1
[1,1,1,2,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 1
[1,1,1,3] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 1
[1,1,2,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 1
[1,1,2,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 1
[1,1,3,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 1
[1,1,4] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 1
[1,2,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 1
[1,2,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 1
[1,2,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 1
[1,2,3] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 1
[1,3,1,1] => [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 1
[1,3,2] => [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 1
[1,4,1] => [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 1
[1,5] => [5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 1
[2,1,1,1,1] => [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 1
[2,1,1,2] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 1
[2,1,2,1] => [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 1
Description
The row containing the largest entry of a standard tableau.
The following 39 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001829The common independence number of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St000993The multiplicity of the largest part of an integer partition. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001075The minimal size of a block of a set partition. St001119The length of a shortest maximal path in a graph. St001316The domatic number of a graph. St000667The greatest common divisor of the parts of the partition. St000990The first ascent of a permutation. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001322The size of a minimal independent dominating set in a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000617The number of global maxima of a Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000210Minimum over maximum difference of elements in cycles. St000487The length of the shortest cycle of a permutation. St001571The Cartan determinant of the integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000654The first descent of a permutation. St000906The length of the shortest maximal chain in a poset. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000314The number of left-to-right-maxima of a permutation. St000310The minimal degree of a vertex of a graph. St000090The variation of a composition. St000478Another weight of a partition according to Alladi. St000699The toughness times the least common multiple of 1,. St000934The 2-degree of an integer partition. St000264The girth of a graph, which is not a tree. St001570The minimal number of edges to add to make a graph Hamiltonian. St001060The distinguishing index of a graph. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001884The number of borders of a binary word. St001330The hat guessing number of a graph. St000907The number of maximal antichains of minimal length in a poset.
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