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Your data matches 303 different statistics following compositions of up to 3 maps.
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Matching statistic: St001192
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 0
[1,0,1,0]
=> 1
[1,1,0,0]
=> 0
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 2
[1,1,1,0,0,0]
=> 0
[1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> 2
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> 3
[1,1,1,1,0,0,0,0]
=> 0
[1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 2
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 2
[1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,1,1,0,0,0,0]
=> 2
Description
The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.
Matching statistic: St001239
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
St001239: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1 = 0 + 1
[1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,1,1,0,0,0,0]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,0,1,0]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,0,1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,0,1,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,0,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> 4 = 3 + 1
[1,1,0,1,1,1,0,0,0,0]
=> 3 = 2 + 1
Description
The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
Matching statistic: St000308
(load all 22 compositions to match this statistic)
(load all 22 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000308: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 = 0 + 1
[1,0,1,0]
=> [2,1] => 1 = 0 + 1
[1,1,0,0]
=> [1,2] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [3,2,1] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [2,3,1] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [3,1,2] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,1,3] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [1,2,3] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => 3 = 2 + 1
Description
The height of the tree associated to a permutation.
A permutation can be mapped to a rooted tree with vertices $\{0,1,2,\ldots,n\}$ and root $0$ in the following way. Entries of the permutations are inserted one after the other, each child is larger than its parent and the children are in strict order from left to right. Details of the construction are found in [1].
The statistic is given by the height of this tree.
See also [[St000325]] for the width of this tree.
Matching statistic: St000328
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000328: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000328: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[]]
=> 1 = 0 + 1
[1,0,1,0]
=> [[],[]]
=> 2 = 1 + 1
[1,1,0,0]
=> [[[]]]
=> 1 = 0 + 1
[1,0,1,0,1,0]
=> [[],[],[]]
=> 3 = 2 + 1
[1,0,1,1,0,0]
=> [[],[[]]]
=> 2 = 1 + 1
[1,1,0,0,1,0]
=> [[[]],[]]
=> 2 = 1 + 1
[1,1,0,1,0,0]
=> [[[],[]]]
=> 2 = 1 + 1
[1,1,1,0,0,0]
=> [[[[]]]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [[],[],[],[]]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [[],[],[[]]]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [[],[[]],[]]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0]
=> [[],[[],[]]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [[],[[[]]]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [[[]],[],[]]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [[[]],[[]]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [[[],[]],[]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [[[[]]],[]]
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [[],[],[],[],[]]
=> 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [[],[],[],[[]]]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [[],[],[[]],[]]
=> 4 = 3 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [[],[],[[],[]]]
=> 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [[],[],[[[]]]]
=> 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [[],[[]],[],[]]
=> 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [[],[[]],[[]]]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [[],[[],[]],[]]
=> 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [[],[[],[],[]]]
=> 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [[],[[],[[]]]]
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [[],[[[]]],[]]
=> 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [[],[[[]],[]]]
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [[],[[[],[]]]]
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [[[]],[],[],[]]
=> 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [[[]],[],[[]]]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [[[]],[[]],[]]
=> 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [[[]],[[],[]]]
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [[[]],[[[]]]]
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [[[],[]],[],[]]
=> 3 = 2 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [[[],[]],[[]]]
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [[[],[],[]],[]]
=> 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [[[],[[]]],[]]
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> 3 = 2 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> 2 = 1 + 1
Description
The maximum number of child nodes in a tree.
Matching statistic: St000381
(load all 44 compositions to match this statistic)
(load all 44 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000381: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 = 0 + 1
[1,0,1,0]
=> [1,1] => 1 = 0 + 1
[1,1,0,0]
=> [2] => 2 = 1 + 1
[1,0,1,0,1,0]
=> [1,1,1] => 1 = 0 + 1
[1,0,1,1,0,0]
=> [1,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,1] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [3] => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [2,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [3,1] => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [3,1] => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [4] => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 3 = 2 + 1
Description
The largest part of an integer composition.
Matching statistic: St000392
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00093: Dyck paths —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 10 => 1 = 0 + 1
[1,0,1,0]
=> 1010 => 1 = 0 + 1
[1,1,0,0]
=> 1100 => 2 = 1 + 1
[1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[1,0,1,1,0,0]
=> 101100 => 2 = 1 + 1
[1,1,0,0,1,0]
=> 110010 => 2 = 1 + 1
[1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[1,1,1,0,0,0]
=> 111000 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 3 = 2 + 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001235
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001235: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => 1 = 0 + 1
[1,0,1,0]
=> [1,1] => 2 = 1 + 1
[1,1,0,0]
=> [2] => 1 = 0 + 1
[1,0,1,0,1,0]
=> [1,1,1] => 3 = 2 + 1
[1,0,1,1,0,0]
=> [1,2] => 2 = 1 + 1
[1,1,0,0,1,0]
=> [2,1] => 2 = 1 + 1
[1,1,0,1,0,0]
=> [2,1] => 2 = 1 + 1
[1,1,1,0,0,0]
=> [3] => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 4 = 3 + 1
[1,0,1,0,1,1,0,0]
=> [1,1,2] => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
=> [1,2,1] => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [1,2,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,3] => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [2,1,1] => 3 = 2 + 1
[1,1,0,0,1,1,0,0]
=> [2,2] => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [2,1,1] => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
=> [2,1,1] => 3 = 2 + 1
[1,1,0,1,1,0,0,0]
=> [2,2] => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> [3,1] => 2 = 1 + 1
[1,1,1,0,0,1,0,0]
=> [3,1] => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
=> [3,1] => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
=> [4] => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1] => 5 = 4 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 3 = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => 3 = 2 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => 4 = 3 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 3 = 2 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 4 = 3 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,1,1,1] => 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 3 = 2 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 2 = 1 + 1
Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St001372
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00093: Dyck paths —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 10 => 1 = 0 + 1
[1,0,1,0]
=> 1010 => 1 = 0 + 1
[1,1,0,0]
=> 1100 => 2 = 1 + 1
[1,0,1,0,1,0]
=> 101010 => 1 = 0 + 1
[1,0,1,1,0,0]
=> 101100 => 2 = 1 + 1
[1,1,0,0,1,0]
=> 110010 => 2 = 1 + 1
[1,1,0,1,0,0]
=> 110100 => 2 = 1 + 1
[1,1,1,0,0,0]
=> 111000 => 3 = 2 + 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 1 = 0 + 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 3 = 2 + 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> 11010010 => 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 3 + 1
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 1 = 0 + 1
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 3 = 2 + 1
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 3 = 2 + 1
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 4 = 3 + 1
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 2 = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 2 = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 2 = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 3 = 2 + 1
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 2 = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 3 = 2 + 1
Description
The length of a longest cyclic run of ones of a binary word.
Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St000141
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000141: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1]]
=> [1] => 0
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 0
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 1
[1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => 2
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 2
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 2
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 1
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [2,1,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => 3
Description
The maximum drop size of a permutation.
The maximum drop size of a permutation $\pi$ of $[n]=\{1,2,\ldots, n\}$ is defined to be the maximum value of $i-\pi(i)$.
Matching statistic: St000209
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00137: Dyck paths —to symmetric ASM⟶ Alternating sign matrices
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00002: Alternating sign matrices —to left key permutation⟶ Permutations
St000209: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [[1]]
=> [1] => 0
[1,0,1,0]
=> [[1,0],[0,1]]
=> [1,2] => 0
[1,1,0,0]
=> [[0,1],[1,0]]
=> [2,1] => 1
[1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> [1,2,3] => 0
[1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> [1,3,2] => 1
[1,1,0,0,1,0]
=> [[0,1,0],[1,0,0],[0,0,1]]
=> [2,1,3] => 1
[1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> [1,3,2] => 1
[1,1,1,0,0,0]
=> [[0,0,1],[0,1,0],[1,0,0]]
=> [3,2,1] => 2
[1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> [1,2,3,4] => 0
[1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[1,0,1,1,0,0,1,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1
[1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 2
[1,1,0,0,1,0,1,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
=> [2,1,3,4] => 1
[1,1,0,0,1,1,0,0]
=> [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
=> [2,1,4,3] => 1
[1,1,0,1,0,0,1,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,0,0],[0,0,0,1]]
=> [1,3,2,4] => 1
[1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> [1,2,4,3] => 1
[1,1,0,1,1,0,0,0]
=> [[0,1,0,0],[1,-1,0,1],[0,0,1,0],[0,1,0,0]]
=> [1,4,3,2] => 2
[1,1,1,0,0,0,1,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
=> [3,2,1,4] => 2
[1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[0,1,0,0],[1,0,-1,1],[0,0,1,0]]
=> [2,1,4,3] => 1
[1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[0,1,-1,1],[1,-1,1,0],[0,1,0,0]]
=> [1,4,3,2] => 2
[1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [4,3,2,1] => 3
[1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,2,3,4,5] => 0
[1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,0,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 1
[1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 1
[1,0,1,1,0,0,1,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 1
[1,0,1,1,0,1,0,0,1,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 1
[1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,0,1,1,0,1,1,0,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 2
[1,0,1,1,1,0,0,0,1,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 1
[1,0,1,1,1,0,1,0,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 2
[1,0,1,1,1,1,0,0,0,0]
=> [[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => 3
[1,1,0,0,1,0,1,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [2,1,3,4,5] => 1
[1,1,0,0,1,0,1,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 1
[1,1,0,0,1,1,0,0,1,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [2,1,4,3,5] => 1
[1,1,0,0,1,1,0,1,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [2,1,3,5,4] => 1
[1,1,0,0,1,1,1,0,0,0]
=> [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [2,1,5,4,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> [1,3,2,4,5] => 1
[1,1,0,1,0,0,1,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 1
[1,1,0,1,0,1,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,0,0],[0,0,0,0,1]]
=> [1,2,4,3,5] => 1
[1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> [1,2,3,5,4] => 1
[1,1,0,1,0,1,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 2
[1,1,0,1,1,0,0,0,1,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [1,4,3,2,5] => 2
[1,1,0,1,1,0,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,0,0],[0,1,0,-1,1],[0,0,0,1,0]]
=> [1,3,2,5,4] => 1
[1,1,0,1,1,0,1,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,1,0],[0,0,1,-1,1],[0,1,-1,1,0],[0,0,1,0,0]]
=> [1,2,5,4,3] => 2
[1,1,0,1,1,1,0,0,0,0]
=> [[0,1,0,0,0],[1,-1,0,0,1],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [1,5,4,3,2] => 3
Description
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 maximum of this value over all cycles in the permutation.
The following 293 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000662The staircase size of the code of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St000013The height of a Dyck path. St000147The largest part of an integer partition. St000444The length of the maximal rise of a Dyck path. St000451The length of the longest pattern of the form k 1 2. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000846The maximal number of elements covering an element of a poset. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St001530The depth of a Dyck path. St000021The number of descents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000171The degree of the graph. St000272The treewidth of a graph. St000306The bounce count of a Dyck path. St000374The number of exclusive right-to-left minima of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000651The maximal size of a rise in a permutation. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001117The game chromatic index of a graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001644The dimension of a graph. St001826The maximal number of leaves on a vertex of a graph. St001962The proper pathwidth of a graph. St000010The length of the partition. St000025The number of initial rises of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000166The depth minus 1 of an ordered tree. St000172The Grundy number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000325The width of the tree associated to a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000470The number of runs in a permutation. St000527The width of the poset. St000542The number of left-to-right-minima of a permutation. St000628The balance of a binary word. St000676The number of odd rises of a Dyck path. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000734The last entry in the first row of a standard tableau. St000822The Hadwiger number of the graph. St000845The maximal number of elements covered by an element in a poset. St000982The length of the longest constant subword. St000983The length of the longest alternating subword. St001029The size of the core of a graph. St001058The breadth of the ordered tree. St001062The maximal size of a block of a set partition. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001268The size of the largest ordinal summand in the poset. St001330The hat guessing number of a graph. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001399The distinguishing number of a poset. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001652The length of a longest interval of consecutive numbers. St001674The number of vertices of the largest induced star graph in the graph. St001809The index of the step at the first peak of maximal height in a Dyck path. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St000094The depth of an ordered tree. St000326The position of the first one in a binary word after appending a 1 at the end. St000439The position of the first down step of a Dyck path. St000521The number of distinct subtrees of an ordered tree. St000485The length of the longest cycle of a permutation. St000956The maximal displacement of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St000503The maximal difference between two elements in a common block. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000730The maximal arc length of a set partition. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001118The acyclic chromatic index of a graph. St001933The largest multiplicity of a part in an integer partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000681The Grundy value of Chomp on Ferrers diagrams. St000937The number of positive values of the symmetric group character corresponding to the partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St001389The number of partitions of the same length below the given integer partition. St000015The number of peaks of a Dyck path. St000288The number of ones in a binary word. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000335The difference of lower and upper interactions. St000443The number of long tunnels of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000733The row containing the largest entry of a standard tableau. St000738The first entry in the last row of a standard tableau. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001471The magnitude of a Dyck path. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001523The degree of symmetry of a Dyck path. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001814The number of partitions interlacing the given partition. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001488The number of corners of a skew partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000271The chromatic index of a graph. St001432The order dimension of the partition. St001638The book thickness of a graph. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000366The number of double descents of a permutation. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000732The number of double deficiencies of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001075The minimal size of a block of a set partition. St001566The length of the longest arithmetic progression in a permutation. St000731The number of double exceedences of a permutation. St000007The number of saliances of the permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000307The number of rowmotion orbits of a poset. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000456The monochromatic index of a connected graph. St000155The number of exceedances (also excedences) of a permutation. St000245The number of ascents of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000710The number of big deficiencies of a permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St000035The number of left outer peaks of a permutation. St000153The number of adjacent cycles of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000778The metric dimension of a graph. St000991The number of right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001589The nesting number of a perfect matching. St000533The minimum of the number of parts and the size of the first part of an integer partition. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000654The first descent of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000667The greatest common divisor of the parts of the partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. 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. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000080The rank of the poset. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000528The height of a poset. St000907The number of maximal antichains of minimal length in a poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001782The order of rowmotion on the set of order ideals of a poset. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000781The number of proper colouring schemes of a Ferrers diagram. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000145The Dyson rank of a partition. St000284The Plancherel distribution on integer partitions. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000706The product of the factorials of the multiplicities of an integer partition. 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. St000934The 2-degree of an integer partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001890The maximum magnitude of the Möbius function of a poset. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001864The number of excedances of a signed permutation. St001060The distinguishing index of a graph. St001877Number of indecomposable injective modules with projective dimension 2. 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. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. St001730The number of times the path corresponding to a binary word crosses the base line. St000264The girth of a graph, which is not a tree. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001557The number of inversions of the second entry of a permutation. St001578The minimal number of edges to add or remove to make a graph a line graph. St001960The number of descents of a permutation minus one if its first entry is not one. St000120The number of left tunnels of a Dyck path. St000225Difference between largest and smallest parts in a partition. St000670The reversal length of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St000808The number of up steps of the associated bargraph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001742The difference of the maximal and the minimal degree in a graph. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001875The number of simple modules with projective dimension at most 1. St001624The breadth of a lattice. St000386The number of factors DDU in a Dyck path. St000884The number of isolated descents of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001435The number of missing boxes in the first row. St000891The number of distinct diagonal sums of a permutation matrix. St000022The number of fixed points of a permutation. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000703The number of deficiencies of a permutation. St000390The number of runs of ones in a binary word.
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